http://ballistipedia.com/api.php?action=feedcontributions&user=Herb&feedformat=atomShotStat - User contributions [en]2020-09-26T14:57:42ZUser contributionsMediaWiki 1.31.6http://ballistipedia.com/index.php?title=Talk:Measuring_Precision&diff=1296Talk:Measuring Precision2015-06-26T20:04:41Z<p>Herb: </p>
<hr />
<div>Ok, I'm happy with title on this and general layout. Still some clean up to do in the various measures. My notion is just to present the conceptual idea for the measures on this page. Each dispersion measure title would be linked to a wiki page describing that measure in more detail.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
----<br />
3.4 Elliptical Error Probable (EEP)<br />
<br />
Need to do so more literature searching on this. I suspect that this includes Hoyt case where rho <> 0. I'm thinking of constraining it to ellipses oriented along horizontal and vertical axis. You could then convert to circle with a simple rescaling.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
: EEP is fully analogous to CEP: For a given ellipse and a fixed bivariate normal shot distribution, you can either ask "what is the probability of a hit whithin my ellipse? Or you can ask "without changing the shape of the ellipse (its aspect ratio), how much do I have to blow it up (scale both axes uniformly) such that it is expected to cover, say, 50% of the hits?" In the latter case, the length of the major and minor axis is calculated by applying the scaling factor to the lengths of the ellipse that was originally specified. A special case results if the ellipse has the same center, shape and orientation as the ellipse that characterizes the bivariate shot distribution.<br />
<br />
: EEP thus has nothing to do with the Hoyt distribution. The Hoyt distribution is the distribution of radial error with respect to the true center of a bivariate normal shot distribution that is correlated and has unequal variances. This is fully analogous to the Rayleigh distribution that is also the distribution of radial error, but in the more restricted case of equal variances and 0 correlation. Just like the Rayleigh distribution, the Hoyt distribution can be used to calculate CEP, but under more liberal assumptions.<br />
<br />
:[[User:armadillo|armadillo]]<br />
<br />
::&nbsp;&nbsp;&nbsp;Sorry that I'm expressed myself so poorly. I'll refer to an orthogonal ellipse as an ellipse oriented along horizontal or vertical axis. By a nonorthogonal ellipse I mean one for which the major axis is at an angle to both the horizontal or vertical axis. <br />
::&nbsp;&nbsp;&nbsp;For an orthogonal ellipse then <math>\sigma_h \neq \sigma_v</math>. For such ellipses you could turn them into a circle with a simple rescaling. Easy enough to do on paper, but a bit laborious. The scaling factor is from ratio of variances.<br />
::&nbsp;&nbsp;&nbsp;For an nonorthogonal ellipse then still <math>\sigma_h \neq \sigma_v</math> but "conversion" to a circle is a lot messier. You really need a computer program to do it. The issue isn't really the need for a computer but the idea that the computer is fitting multiple parameters and making multiple clip level decisions. A lot more mysticism as to what Oz is doing behind the curtain. <br />
::&nbsp;&nbsp;&nbsp;So the issue that I'm wonder about is if it is worthwhile to set up the orthogonal ellipses as a special case since you're just "fitting" the one scaling constant around COI. For instance muzzle velocity variations causing vertical stringing on the target. You'd probably use a computer to do it anyway, it is just that the orthogonal constraint greatly reduces the amount of mysticism of what the computer is doing. <br />
<br />
:::: thanks! I now understand that an ellipse along axes doesn't imply correlation.<br />
<br />
:::: Still wondering if special cases should be broken out of Hoyt for what I have as [http://ballistipedia.com/index.php?title=Dispersion_Assumptions#Case_2.2C_Equal_variances_and_correlated Case 2] and [http://ballistipedia.com/index.php?title=Dispersion_Assumptions#Case_3.2C_Unequal_variances_and_uncorrelated Case 3]. <br />
<br />
:::: [[User:Herb|Herb]] ([[User talk:Herb|talk]]) 01:53, 6 June 2015 (EDT) <br />
<br />
----<br />
3.8 Hoyt Distribution Parameters (Bivariate Normal Distribution Parameters)<br />
<br />
Think changing name for this to "Hoyt Error Probable" to fit in with other such measures.<br /><br />
[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
----<br />
I think the two sections <br />
: 3.10 Radial Standard Deviation of the Rayleigh Distribution<br />
: 3.11 Rayleigh Distribution Shape Parameter<br />
Should probably be merged. My notion is to swizzle Radial Standard Deviation so that it fits Rayleigh Distribution better. ie sqrt(2) factor. <br /><br />
[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
:: ''But why are you interested in keeping RSD alive despite all the confusion surrounding it? [[User:David|David]] ([[User talk:David|talk]]) 22:49, 3 June 2015 (EDT)''<br />
<br />
::: Rayleigh Distribution Shape Parameter just doesn't sound like a measure. I am just seduced by the sexy name. Inclusion of "standard deviation" is nice. "Radial" meaning a circle is good too. Mulling this over for the 43rd time in my mind, how about "Rayleigh Radial Scale Factor"?? "Radial" is really the keyword. Fits in with wikipedia use too. What do you think?<br />
<br />
::: You're right, bad terminology is hard to overcome. Electric circuits still are calculated as if a positive charge is moving instead of the electron. So a somewhat fresh start might be good. <br />
<br />
::: I don't like calling the measure "Rayleigh Distribution" either since we aren't calculating the distribution per sey, but fitting a parameter which it uses. <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 23:50, 3 June 2015 (EDT) <br />
<br />
:::: ''"Rayleigh Parameter" would be better, but that's why I often just use "σ": Its meaning is consistent and its usage pervasive throughout the site. When I began I was calling the Rayleigh parameter the RSD until I realized that it wasn't! [[User:David|David]] ([[User talk:David|talk]]) 12:18, 4 June 2015 (EDT)''<br />
<br />
::::: Just to be clear, at the top of this thread I was thinking of redefining RSD as <math>r = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (r_i - \bar{r})^2}</math>,<br /><br />
:::::where <math>\bar{r}</math> is the mean radius. Thus Mean Radius and Radial Standard deviation would relate back to the Rayleigh Distribution. The point is that if <math>\sigma_h \neq \sigma_v</math> then you shouldn't be assuming that the Rayleigh Distribution models the shots. "Radius" and "Radial" inherently make you think of a circular pattern. Maybe "Mean Radius Variance"? That would really do what I want and make the two tie together. To let you in on a secret, I think the mean radius would be a better predictor of variance than the actual variance measurement (with a big assumption that the pattern is truly round. If pattern isn't round then the MR estimator of RSD wouldn't be robust.) <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 13:27, 4 June 2015 (EDT)<br />
<br />
<br />
----<br />
<br />
3.12 String Method<br />
<br />
Need some literature reference for this. No point in just pulling crap measures out of thin air. Found a couple of forum references. I remember it being something like mean radius and looked it up. Sexy in that it is a nice example of Rice Distribution.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 23:50, 3 June 2015 (EDT) <br />
<br />
: ''This is just ''Mean Radius * n'', so I would note it under the Mean Radius section as evidence that MR is nothing new. [[User:David|David]] ([[User talk:David|talk]]) 12:18, 4 June 2015 (EDT)''<br />
<br />
:: No the "original" string measurement was from POA to shots which gives Rice Distribution. It was a measure much more weighed towards accuracy than precision. The two variations are mean radius variations. Sorry to have confused you. <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 12:38, 4 June 2015 (EDT) <br />
<br />
::: ''Oh yeah, that is good. Worth covering! [[User:David|David]] ([[User talk:David|talk]]) 12:43, 4 June 2015 (EDT)''<br />
----<br />
<br />
----<br />
<br />
4 Dispersion Measures about COI<br />
<br />
... being more repetitive: The figure caption ''"A circular dispersion is the Rayleigh distribution."'' is a bit misleading: The distribution of shots (in the sense of (h,v) coordinates) is assumed to be circular normal. Then the distances to the true COI follows a Rayleigh distribution.<br />
<br />
4.4 Elliptical Error Probable (EEP)<br />
<br />
Also here: ''"Elliptical Error Probable assumes that the shots follow the Hoyt distribution"'' - if the shots (in the sense of (h,v) coordinates) are assumed to follow a bivariate normal distribution, the their distances to the true COI follows a Hoyt distribution.<br />
<br />
''"The EEP is the only measurement considered which is appropriate for a non-circular distribution.''" - I wouldn't say that. CEP for elliptical shot distributions is fine if you're interested in the probability of hitting a disc around the COI. If you want to characterize dispersion of the shots ((h,v) coordinates), then CEP does of course not provide as much information about an elliptical shot group as the EEP.<br />
<br />
''"The EEP(50) measurement were based on the median values then it would be a robust estimator."'' - I find this problematic since the empirical median is a bad estimator for the true median. That's why there are things like the Hodges-Lehmann pseudo-median or the Hogg (1967) estimator. There are robust estimators for the covariance matrix like the MCD estimator.<br />
<br />
5.1 Dispersion Measures From POA<br />
<br />
Figure caption ''"Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA."'' - see above: The distances of the shots to the true COI follow a Rayleigh distribution (not the shots themselves), and the distances of the shots to the origin follow a Rice distribution.<br />
<br />
:[[User:armadillo|armadillo]]<br />
<br />
:: I think I've fixed most of the flaws you've pointed out. I think the point about the median values being robust should stand. I think more details about the point should be on a wiki page specific for the EEP measurement. I did try to reword the point a bit. <br />http://ballistipedia.com/index.php?title=Measuring_Precision&diff=1294&oldid=1245<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 16:01, 26 June 2015 (EDT)</div>Herbhttp://ballistipedia.com/index.php?title=Talk:Measuring_Precision&diff=1295Talk:Measuring Precision2015-06-26T20:01:07Z<p>Herb: </p>
<hr />
<div>Ok, I'm happy with title on this and general layout. Still some clean up to do in the various measures. My notion is just to present the conceptual idea for the measures on this page. Each dispersion measure title would be linked to a wiki page describing that measure in more detail.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
----<br />
3.4 Elliptical Error Probable (EEP)<br />
<br />
Need to do so more literature searching on this. I suspect that this includes Hoyt case where rho <> 0. I'm thinking of constraining it to ellipses oriented along horizontal and vertical axis. You could then convert to circle with a simple rescaling.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
: EEP is fully analogous to CEP: For a given ellipse and a fixed bivariate normal shot distribution, you can either ask "what is the probability of a hit whithin my ellipse? Or you can ask "without changing the shape of the ellipse (its aspect ratio), how much do I have to blow it up (scale both axes uniformly) such that it is expected to cover, say, 50% of the hits?" In the latter case, the length of the major and minor axis is calculated by applying the scaling factor to the lengths of the ellipse that was originally specified. A special case results if the ellipse has the same center, shape and orientation as the ellipse that characterizes the bivariate shot distribution.<br />
<br />
: EEP thus has nothing to do with the Hoyt distribution. The Hoyt distribution is the distribution of radial error with respect to the true center of a bivariate normal shot distribution that is correlated and has unequal variances. This is fully analogous to the Rayleigh distribution that is also the distribution of radial error, but in the more restricted case of equal variances and 0 correlation. Just like the Rayleigh distribution, the Hoyt distribution can be used to calculate CEP, but under more liberal assumptions.<br />
<br />
:[[User:armadillo|armadillo]]<br />
<br />
::&nbsp;&nbsp;&nbsp;Sorry that I'm expressed myself so poorly. I'll refer to an orthogonal ellipse as an ellipse oriented along horizontal or vertical axis. By a nonorthogonal ellipse I mean one for which the major axis is at an angle to both the horizontal or vertical axis. <br />
::&nbsp;&nbsp;&nbsp;For an orthogonal ellipse then <math>\sigma_h \neq \sigma_v</math>. For such ellipses you could turn them into a circle with a simple rescaling. Easy enough to do on paper, but a bit laborious. The scaling factor is from ratio of variances.<br />
::&nbsp;&nbsp;&nbsp;For an nonorthogonal ellipse then still <math>\sigma_h \neq \sigma_v</math> but "conversion" to a circle is a lot messier. You really need a computer program to do it. The issue isn't really the need for a computer but the idea that the computer is fitting multiple parameters and making multiple clip level decisions. A lot more mysticism as to what Oz is doing behind the curtain. <br />
::&nbsp;&nbsp;&nbsp;So the issue that I'm wonder about is if it is worthwhile to set up the orthogonal ellipses as a special case since you're just "fitting" the one scaling constant around COI. For instance muzzle velocity variations causing vertical stringing on the target. You'd probably use a computer to do it anyway, it is just that the orthogonal constraint greatly reduces the amount of mysticism of what the computer is doing. <br />
<br />
:::: thanks! I now understand that an ellipse along axes doesn't imply correlation.<br />
<br />
:::: Still wondering if special cases should be broken out of Hoyt for what I have as [http://ballistipedia.com/index.php?title=Dispersion_Assumptions#Case_2.2C_Equal_variances_and_correlated Case 2] and [http://ballistipedia.com/index.php?title=Dispersion_Assumptions#Case_3.2C_Unequal_variances_and_uncorrelated Case 3]. <br />
<br />
:::: [[User:Herb|Herb]] ([[User talk:Herb|talk]]) 01:53, 6 June 2015 (EDT) <br />
<br />
----<br />
3.8 Hoyt Distribution Parameters (Bivariate Normal Distribution Parameters)<br />
<br />
Think changing name for this to "Hoyt Error Probable" to fit in with other such measures.<br /><br />
[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
----<br />
I think the two sections <br />
: 3.10 Radial Standard Deviation of the Rayleigh Distribution<br />
: 3.11 Rayleigh Distribution Shape Parameter<br />
Should probably be merged. My notion is to swizzle Radial Standard Deviation so that it fits Rayleigh Distribution better. ie sqrt(2) factor. <br /><br />
[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 22:11, 3 June 2015 (EDT)<br />
<br />
:: ''But why are you interested in keeping RSD alive despite all the confusion surrounding it? [[User:David|David]] ([[User talk:David|talk]]) 22:49, 3 June 2015 (EDT)''<br />
<br />
::: Rayleigh Distribution Shape Parameter just doesn't sound like a measure. I am just seduced by the sexy name. Inclusion of "standard deviation" is nice. "Radial" meaning a circle is good too. Mulling this over for the 43rd time in my mind, how about "Rayleigh Radial Scale Factor"?? "Radial" is really the keyword. Fits in with wikipedia use too. What do you think?<br />
<br />
::: You're right, bad terminology is hard to overcome. Electric circuits still are calculated as if a positive charge is moving instead of the electron. So a somewhat fresh start might be good. <br />
<br />
::: I don't like calling the measure "Rayleigh Distribution" either since we aren't calculating the distribution per sey, but fitting a parameter which it uses. <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 23:50, 3 June 2015 (EDT) <br />
<br />
:::: ''"Rayleigh Parameter" would be better, but that's why I often just use "σ": Its meaning is consistent and its usage pervasive throughout the site. When I began I was calling the Rayleigh parameter the RSD until I realized that it wasn't! [[User:David|David]] ([[User talk:David|talk]]) 12:18, 4 June 2015 (EDT)''<br />
<br />
::::: Just to be clear, at the top of this thread I was thinking of redefining RSD as <math>r = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (r_i - \bar{r})^2}</math>,<br /><br />
:::::where <math>\bar{r}</math> is the mean radius. Thus Mean Radius and Radial Standard deviation would relate back to the Rayleigh Distribution. The point is that if <math>\sigma_h \neq \sigma_v</math> then you shouldn't be assuming that the Rayleigh Distribution models the shots. "Radius" and "Radial" inherently make you think of a circular pattern. Maybe "Mean Radius Variance"? That would really do what I want and make the two tie together. To let you in on a secret, I think the mean radius would be a better predictor of variance than the actual variance measurement (with a big assumption that the pattern is truly round. If pattern isn't round then the MR estimator of RSD wouldn't be robust.) <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 13:27, 4 June 2015 (EDT)<br />
<br />
<br />
----<br />
<br />
3.12 String Method<br />
<br />
Need some literature reference for this. No point in just pulling crap measures out of thin air. Found a couple of forum references. I remember it being something like mean radius and looked it up. Sexy in that it is a nice example of Rice Distribution.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 23:50, 3 June 2015 (EDT) <br />
<br />
: ''This is just ''Mean Radius * n'', so I would note it under the Mean Radius section as evidence that MR is nothing new. [[User:David|David]] ([[User talk:David|talk]]) 12:18, 4 June 2015 (EDT)''<br />
<br />
:: No the "original" string measurement was from POA to shots which gives Rice Distribution. It was a measure much more weighed towards accuracy than precision. The two variations are mean radius variations. Sorry to have confused you. <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 12:38, 4 June 2015 (EDT) <br />
<br />
::: ''Oh yeah, that is good. Worth covering! [[User:David|David]] ([[User talk:David|talk]]) 12:43, 4 June 2015 (EDT)''<br />
----<br />
<br />
----<br />
<br />
4 Dispersion Measures about COI<br />
<br />
... being more repetitive: The figure caption ''"A circular dispersion is the Rayleigh distribution."'' is a bit misleading: The distribution of shots (in the sense of (h,v) coordinates) is assumed to be circular normal. Then the distances to the true COI follows a Rayleigh distribution.<br />
<br />
4.4 Elliptical Error Probable (EEP)<br />
<br />
Also here: ''"Elliptical Error Probable assumes that the shots follow the Hoyt distribution"'' - if the shots (in the sense of (h,v) coordinates) are assumed to follow a bivariate normal distribution, the their distances to the true COI follows a Hoyt distribution.<br />
<br />
''"The EEP is the only measurement considered which is appropriate for a non-circular distribution.''" - I wouldn't say that. CEP for elliptical shot distributions is fine if you're interested in the probability of hitting a disc around the COI. If you want to characterize dispersion of the shots ((h,v) coordinates), then CEP does of course not provide as much information about an elliptical shot group as the EEP.<br />
<br />
''"The EEP(50) measurement were based on the median values then it would be a robust estimator."'' - I find this problematic since the empirical median is a bad estimator for the true median. That's why there are things like the Hodges-Lehmann pseudo-median or the Hogg (1967) estimator. There are robust estimators for the covariance matrix like the MCD estimator.<br />
<br />
5.1 Dispersion Measures From POA<br />
<br />
Figure caption ''"Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA."'' - see above: The distances of the shots to the true COI follow a Rayleigh distribution (not the shots themselves), and the distances of the shots to the origin follow a Rice distribution.<br />
<br />
:[[User:armadillo|armadillo]]<br />
<br />
:: I think I've fixed most of the flaws you've pointed out. I think the point about the median values being robust should stand. I think more details about the point should be on a wiki page specific for the EEP measurement. I did try to reword the point a bit. <br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 16:01, 26 June 2015 (EDT)</div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1294Measuring Precision2015-06-26T20:00:27Z<p>Herb: /* Elliptical Error Probable (EEP) */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| When shots are dispersed circularly about the true COI, then the distance from each shot to the true COI follows a Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that is expected to cover proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the true ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set.<br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - If the shots dispersed about the true COI in an elliptical pattern which has its major axis at an angle to the coordinate axes, then the distances from the shots to the true COI follow the Hoyt distribution.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the distances from the shots to the true COI follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the "best" measurement considered which is appropriate for a non-circular distribution. Best in the sense that it would provide the most precise statistical estimates. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
If the EEP(50) measurement is based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would use the medians so that it would be robust since that would require an extraordinary amount of experimental data to get a good value. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - The distances of the shots to the true COI follow a Rayleigh distribution, but the distances of the shots to the offset center of the target follow a Rice distribution. ]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough precise experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] "The difference between theory and practice is larger in practice than in theory."<br />
|}<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1293Measuring Precision2015-06-26T18:05:35Z<p>Herb: /* Elliptical Error Probable (EEP) */ fixed some wording problems</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| When shots are dispersed circularly about the true COI, then the distance from each shot to the true COI follows a Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that is expected to cover proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the true ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set.<br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - If the shots dispersed about the true COI in an elliptical pattern which has its major axis at an angle to the coordinate axes, then the distances from the shots to the true COI follow the Hoyt distribution.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the distances from the shots to the true COI follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the "best" measurement considered which is appropriate for a non-circular distribution. Best in the sense that it would provide the most precise statistical estimates. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - The distances of the shots to the true COI follow a Rayleigh distribution, but the distances of the shots to the offset center of the target follow a Rice distribution. ]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough precise experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] "The difference between theory and practice is larger in practice than in theory."<br />
|}<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1292Measuring Precision2015-06-26T17:42:21Z<p>Herb: /* String Length (SL) Method */ fixed problem with Rice distribution caption.</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| When shots are dispersed circularly about the true COI, then the distance from each shot to the true COI follows a Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that is expected to cover proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the true ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set.<br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - The distances of the shots to the true COI follow a Rayleigh distribution, but the distances of the shots to the offset center of the target follow a Rice distribution. ]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough precise experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] "The difference between theory and practice is larger in practice than in theory."<br />
|}<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1291Measuring Precision2015-06-26T17:39:22Z<p>Herb: /* Dispersion Measures about COI */ fixed problem with first figure caption.</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| When shots are dispersed circularly about the true COI, then the distance from each shot to the true COI follows a Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that is expected to cover proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the true ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set.<br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA.]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough precise experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] "The difference between theory and practice is larger in practice than in theory."<br />
|}<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Talk:Projectile_Dispersion_Classifications&diff=1290Talk:Projectile Dispersion Classifications2015-06-25T17:27:26Z<p>Herb: </p>
<hr />
<div>''"then the general bivariate normal equation becomes the Hoyt distribution"'' is a bit misleading. The distribution of (h,v) '''coordinates''' is assumed to be bivariate normal. Then the Hoyt distribution is the (univariate) distribution of the '''distances''' (radial error) of each point to the true COI.<br />
<br />
Likewise, ''"special case is the Rayleigh Distribution"'' should make more explicit that the Rayleigh distribution concerns the '''distances''' of each point to the true COI.<br />
<br />
The same applies to the sentence ''"this distribution will be called the Orthogonal Elliptical Distribution. It is obviously a special case of the Hoyt distribution which in turn is a special case of the bivariate normal distribution."''' - the Hoyt distribution is not a special case of the bivariate normal distribution.<br />
<br />
:[[User:armadillo|armadillo]]<br />
<br />
: You raise good points. The wording is a bit sloppy. I did glossed over the "true COI" which is something that ought to be discussed.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 20:26, 24 June 2015 (EDT)<br />
<br />
: Actually I now realize that the wording isn't just sloppy, but seriously flawed.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 13:27, 25 June 2015 (EDT)</div>Herbhttp://ballistipedia.com/index.php?title=Talk:Projectile_Dispersion_Classifications&diff=1289Talk:Projectile Dispersion Classifications2015-06-25T00:26:36Z<p>Herb: </p>
<hr />
<div>''"then the general bivariate normal equation becomes the Hoyt distribution"'' is a bit misleading. The distribution of (h,v) '''coordinates''' is assumed to be bivariate normal. Then the Hoyt distribution is the (univariate) distribution of the '''distances''' (radial error) of each point to the true COI.<br />
<br />
Likewise, ''"special case is the Rayleigh Distribution"'' should make more explicit that the Rayleigh distribution concerns the '''distances''' of each point to the true COI.<br />
<br />
The same applies to the sentence ''"this distribution will be called the Orthogonal Elliptical Distribution. It is obviously a special case of the Hoyt distribution which in turn is a special case of the bivariate normal distribution."''' - the Hoyt distribution is not a special case of the bivariate normal distribution.<br />
<br />
:[[User:armadillo|armadillo]]<br />
<br />
: You raise good points. The wording is a bit sloppy. I did glossed over the "true COI" which is something that ought to be discussed.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 20:26, 24 June 2015 (EDT)</div>Herbhttp://ballistipedia.com/index.php?title=Talk:Derivation_of_the_Rayleigh_Distribution_Equation&diff=1288Talk:Derivation of the Rayleigh Distribution Equation2015-06-25T00:13:11Z<p>Herb: </p>
<hr />
<div>''' rev 13:28, 3 June 2015 Herb '''<br />
<br />
Ok, name of page needs a bit of fixing. <br />
<br />
more verbosity in text<br />
<br />
I goofed in accuracy section. "PDF" functions are set up wrong. Should be something like "PDF of X as x"<br />
<br />
Been about 45 years since I took calculus in college. I'll have to look up conversion from Cartesian to Polar coordinates, but I know the conversion will swizzle to the right answer...<br />
<br />
----<br />
<br />
This is topic that I think bears further investigation...<br />
<br />
is it <math>\sigma_h = \sigma_v</math> or <math>\sigma_h^2 = \sigma_v^2</math> ??<br />
<br />
Of course if they are equal then both equations are true. The problem is in pooling the values if:<br /><br />
<br />
<math>\sigma_h \approx \sigma_v</math><br />
<br />
which equation do we use to pool the values?<br />
<br />
* <math>(\sigma_h + \sigma_v)/2</math><br />
* <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math><br />
<br />
I think <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> should be corrected by factor <math>\frac{1}{\sqrt{2}}</math> since <math>(\sigma + \sigma)/2 = \sigma</math> but <math>\sqrt{\sigma^2 + \sigma^2} = \sigma \sqrt{2}</math><br />
<br />
In general it would seem that the ratio <math>(\sigma_h / \sigma_v)</math> could be useful as a guide to stay out of trouble. Obviously the ratio should depend on sample size, ''n''. Think this is sort of the idea, but it doesn't right (two limits should converge) ...<br />
<br />
&nbsp;&nbsp;&nbsp; <math> .33\frac{(\sigma_h + \sigma_v)}{\sqrt{n}} \leq (\sigma_h / \sigma_v) \leq 0.75\frac{(\sigma_h + \sigma_v)}{\sqrt{n}} </math> <br />
<br />
----<br />
<br />
I also really don't like using a <math>\sigma</math> as a factor in the equation. If you think about radial values then there is a <math>\sigma</math> which can be calculated from the <math>r_i</math> values. The two <math>\sigma</math>'s aren't equal. <br />
<br />
----<br />
<br />
The !@#$%^&* literature really messes up calculation of radial standard deviation. needs <math>\frac{1}{\sqrt{2}}</math> correction. <br />
<br />
-----<br />
<br />
4.1 Derivation From the Bivariate Normal distribution<br />
<br />
... being more repetitive: ''"a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution"'' is misleading: The bivariate normal distribution does not become the (univariate) Hoyt distribution. After converting (h,v) coordinates to polar (r,azimuth) coordinates, the r-coordinate (distance to true COI) follows a Hoyt distribution.<br />
<br />
5.1 Derivation OF Single Shot PDF From the Bivariate Normal distribution<br />
<br />
''"The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:"'' is a bit weird because a distribution does not make assumptions. Making the assumption that shots (in the sense of (h,v) coordinates) follow a restricted (circular) bivariate normal distribution implies that the distance to the true COI follows a Rayleigh distribution.<br />
<br />
:: You're right, the wording is sloppy.<br />[[User:Herb|Herb]] ([[User talk:Herb|talk]]) 20:13, 24 June 2015 (EDT)</div>Herbhttp://ballistipedia.com/index.php?title=Mean_Radius&diff=1271Mean Radius2015-06-22T14:53:59Z<p>Herb: /* Theoretical r_{\Re} Distribution */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Mean Radius<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
= Experimental Summary =<br />
<br />
yada yada <br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* The dispersion of shot <math>i</math> follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius.<br />
** <math>h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to fit <math>\sigma</math> to calculate the Mean Radius.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Measure positions <math>(h_i, v_i)</math> for each shot, <math>i</math>.<br />
|-<br />
| Experimental Measure<br />
| <math>MR = \overline{r_n}</math> method<br />
Preliminary Cartesian Calculations<br />
* <math>\bar{h} = \frac{1}{n} \sum_{i=1}^n h_i </math><br />
* <math>\bar{v} = \frac{1}{n} \sum_{i=1}^n v_i </math><br />
Shot impact positions converted from Cartesian Coordinates<br />
* <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br /><br />
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).<br />
<br />
<math>\overline{r_n}</math> - the average radius of ''n'' shots<br />
<br />
<math>\overline{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
<hr /><br />
<math>MR = f(\Re)</math> method<br />
|-<br />
| Outlier Tests<br />
|<br />
|}<br />
<br />
== Given ==<br />
<br />
== Assumptions ==<br />
<br />
== Data transformation ==<br />
<br />
== Experimental Measure ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>r_i</math> Distribution =<br />
<br />
Distribution for a single shot as a function of r. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\Re</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
|-<br />
| <math>PDF_{r(1)}(r; \Re)</math><br />
| <math>\frac {r}{\Re^2} \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \Re)</math><br />
| <math> 1 - \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\Re</math><br />
|-<br />
| Median of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)}</math><br />
| <math>\frac{(4-\pi)}{2}\Re^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>\overline{r(n)}</math> Distribution =<br />
Given:<br />
* <math>n</math> shots were taken on a target<br />
* The average mean radius, <math>\overline{r(n)}</math>, was calculated<br />
* The Rayleigh shape parameter <math>\Re</math> for an individual shot is known.<br />
then using <math>r</math> as a variable, the properties of the distribution of the average mean radius for <math>n</math> shots is investigated in this section. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>\bar{r_n}</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>n</math> - n of shots in sample<br />
<math>\Re</math> - Rayleigh shape parameter from individual shot distribution <br />
|-<br />
| <math>PDF(\bar{r_n}; n, \Re)</math><br />
| <math>\frac{\Gamma(n,2\Re^2)}{n}</math><br><br />
where <math>\Gamma(n,2\Re^2)</math> is the Gamma Distribution<br />
|-<br />
| <math>CDF(r; n, \Re)</math><br />
| <br />
|-<br />
| Mode of PDF)<br />
| <math>\bar{r_n}</math><br />
|-<br />
| Median of PDF<br />
| no closed form, but <math>\approx 1.177\bar{r_n}</math><br />
|-<br />
| Mean of PDF<br />
| <math>\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}</math><br />
|-<br />
| Variance<br />
| <br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|- <br />
| NSPG Invariant<br />
| Yes<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
<br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
== CDF ==<br />
yada yada <br />
<br />
== Mode, Median, Mean ==<br />
yada yada <br />
<br />
== Outlier Tests ==<br />
yada yada<br />
<br />
<br />
= Theoretical <math>r_{\Re}</math> Distribution =<br />
<br />
Distribution for MR where MR calculated from <math>\Re</math><br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\Re</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
<math>n</math> - the number of shots in the group<br />
|-<br />
| <math>PDF_{r(1)}(r; \Re)</math><br />
| <math>\frac {nr}{\Re^2} \exp\Big \{-\frac {nr^2}{2\Re^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \Re)</math><br />
| <math> 1 - \exp\Big \{-\frac {nr^2}{2\Re^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\Re</math><br />
|-<br />
| Median of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)}</math><br />
| <math>\frac{(4-\pi)}{2}\Re^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= ''Studentized'' Mean Radius =<br />
<br />
'''need table for this...''' <br />
<br />
== Outlier Tests ==<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - Discussion of other models for shot dispersion<br />
<br />
<!--<br />
[[Data Transformations to Rayleigh Distribution]] - Methods to transform non-conforming data to Rayleigh Distribution<br />
--></div>Herbhttp://ballistipedia.com/index.php?title=Projectile_Dispersion_Classifications&diff=1270Projectile Dispersion Classifications2015-06-20T16:05:02Z<p>Herb: /* Simplification of the Hoyt distribution into Special Cases */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Before considering the measurements that will be used for the actual statistical analysis, let's consider the assumptions about projectile dispersion about the Center of Impact (COI) and how sets of those assumptions might be grouped into different classifications. The various classifications will offer insight as to the fundamental patterns expected for shots and insights to the interactions of various measures. Thus an understanding of the basic assumptions about projectile dispersion is key in being able to effectively use the measures. <br />
<br />
The COI is the only true point of reference which can be calculated from the pattern of shots on a target. Thus the COI is the reference point for precision measurements. The overall error that we are interested in measuring is the sum of all the various interactions that make multiple projectiles shot to the same point of aim (POA) disperse about the COI. <br />
<br />
Since we are primarily interested in the dispersion relative to the COI, the overall assumption is that the weapon could be properly sighted so that the COI would be the same as the POA. In practice this is achieved by [[FAQ#How_many_shots_do_I_need_to_sight_in.3F| adjusting the weapon's sights]]. Thus in order to isolate projectile dispersion, all of the factors of internal and external ballistics that cause a bias to the COI on a target will be ignored. For example, for the purposes of classifying projectile dispersion, accuracy errors due to POA errors will be ignored. <br />
<br />
The [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution] is the broadly assumed probability model used for a single random variable and it is characterized by its mean <math>(\bar{x})</math> and standard deviation <math>(\sigma)</math>. The [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem] shows that for measures for the "average" shot, or averages of multiple targets are used, then for "large" samples the averages will conform to Normal distribution even if the fundamental distribution is not a normal distribution. <br />
<br />
[[File:Bivariate.png|400px|thumb|right|Distribution of samples from a symmetric bivariate normal distribution. Axis units are multiples of σ.]]<br />
<br />
Since we are interested in shot dispersion on a two-dimensional target we will assume that the horizontal and vertical dispersions of the population of shots are each Normal distributions. Thus the horizontal dispersion will have mean <math>\mu_H</math> and standard deviation <math>\sigma_H</math>. The vertical dispersion will have mean <math>\mu_V</math> and standard deviation <math>\sigma_V</math>. Then a further assumption is made by assuming that the two dimensional expansion of the Normal distribution the [http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Non-degenerate_case Bivariate Normal distribution], applies. This adds an additional term the [http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient correlation parameter ''ρ'']. (See also: [[What is ρ in the Bivariate Normal distribution?]]) Thus the expectation is that distribution should then describe, the dispersion of a gunshots about the COI, (<math>\mu_H</math> and <math>\mu_V</math>). The full bivariate normal distribution is thus:<br ><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(H,V; \mu_H, \mu_V, \sigma_H, \sigma_V, \rho) =<br />
\frac{1}{2 \pi \sigma_H \sigma_V \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(H-\mu_H)^2}{\sigma_H^2} +<br />
\frac{(V-\mu_V)^2}{\sigma_V^2} -<br />
\frac{2\rho(H-\mu_H)(V-\mu_V)}{\sigma_H \sigma_V}<br />
\right]<br />
\right)<br />
</math><br />
<br />
where:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>-1 &le; \rho &le; 1</math><br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> \sigma_H>0 </math> and <math> \sigma_V>0 </math><br />
<br />
Note that the above restrictions are not additional restrictions on the model, but rather simply pointing out how the mathematics works. Thus they are more analogous to the mathematical notion that a person can't have a negative age. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] An ancillary point worth mentioning is that the assuming the Normal distribution in three dimensions leads to the [http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell–Boltzmann distribution] which is the foundation of the ideal gas laws. <br />
|}<br />
<br />
= Simplification of the Hoyt distribution into Special Cases =<br />
<br />
To eliminate the COI (<math>\mu_H</math>, <math>\mu_V</math>) which makes the equations "messier", a translation of the coordinate system to the COI is desired. Thus:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\mu_h = 0</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>\mu_v = 0</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>h = H - \mu_H</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>v = V - \mu_V</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;<math>\sigma_h = \sigma_H </math>&nbsp;&nbsp; and &nbsp;&nbsp;<math>\sigma_v = \sigma_V </math><br /><br />
<br />
This is a very pragmatic and justifiable consideration since the COI can be measured on the target, and the dispersion about the COI is the aspect of interest when measuring precision. As noted before, by adjusting the weapon's sights the POA can be made to coincide with the COI. Thus this simplification of the dispersion equations is strictly for ease of understanding as is not a limitation on the nature of the dispersion classifications. With the translation of the coordinate system to the COI, then the general bivariate normal equation becomes the Hoyt distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v; \sigma_h, \sigma_v, \rho) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho hv}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Looking at this equation two other different mutually exclusive simplifications can be readily seen:<br />
<br />
* '''Either''' <math>\sigma_h = \sigma_v</math> (equal standard deviations) '''or''' <math>\sigma_h \neq \sigma_v</math> (unequal standard deviations).<br />
: Obviously if we could measure both <math>\sigma_h</math> and <math>\sigma_v</math> with a very high precision (e.g 6 significant figures), then the two quantities would never really be equal. But in many cases the assumption is reasonable. In reality since shooters typically collect only a small amount of data, statistical tests will fail to detect a difference unless the difference is great. In such cases the shot pattern would be noticeably elliptical, not round. <br />
<br />
* '''Either''' <math>\rho = 0</math> (uncorrelated) '''or''' <math>\rho \neq 0</math> (correlated). <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: !! CAREFUL !! '''[http://en.wikipedia.org/wiki/Correlation_does_not_imply_causation Correlation does not imply causation]''' <br />
<br />
: There is somewhat famous example. A researcher gathered statistics for stork sightings and births in a particular county over a twenty year period. Analysis of the data showed that over the twenty year period both stork sightings and births had increased with a very significant linear correlation. From the data you might erroneously infer that storks do bring babies! <br />
|}<br />
<br />
The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation as shown in the Table below. There is one case that results in circular groups, and three that result in elliptical groups. As the different in variances gets greater, or the further <math>\rho</math> is from 0, then the ellipse will be more pronounced. <br />
<br />
{| class="wikitable" <br />
|+ Group Shape vs. Assumptions (COI at Origin)<br />
|-<br />
|<br />
| <math>\sigma_h \approx \sigma_v</math><br />
| <math>\sigma_h \neq \sigma_v</math><br />
|-<br />
| <math>\rho \approx 0</math><br />
| Case 1 - Circular Groups<br />
* special case is the Rayleigh Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
| Case 3 - Elliptical Groups<br />
* special case is the Orthogonal Elliptical Distribution<br />
* Major axis of ellipse along<br /> horizontal or vertical axis<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
|-<br />
| <math>\rho \neq 0</math><br />
* Major axis of ellipse at an angle to<br />both the horizontal and vertical axes<br />
| Case 2- Elliptical Groups<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
: - <math>\rho</math><br />
| Case 4 - Elliptical Groups<br />
* general case of the Hoyt distribution required<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
: - <math>\rho</math><br />
|}<br />
<br />
== Experimental reality of Comparing <math>s_h</math> and <math>s_v</math>==<br />
<br />
The table above uses ''approximately equal to'' <math>(\approx)</math> rather than ''strictly equal to'' <math>( = )</math>. This is an acknowledgement that we are dividing the cases into ones that are close enough to be useful, even though they most certainly are not exact. To be overly persnickety there are two considerations. <br />
<br />
First we can only get experimental estimates from calculations based on sample data for the factors <math>\sigma_h</math>, <math>\sigma_v</math>, <math>\rho</math> and these estimates are at best only good to a scant few significant figures. Thus even though the difference between ''approximately equal to'' and ''strictly equal to'' is under some experimental control there are practical limits. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but it is quickly impractical to do so. (Assume that to double the precision that we have to quadruple the sample size. This exponential increase quickly becomes unmanageable. It is easy to pontificate about averaging over a million targets, but no one is going to shoot that many.) Thus even if <math>\sigma_h \equiv \sigma_v</math> we'd never expect that we'd experimentally get <math>s_h = s_v</math> due to experimental error. <br />
<br />
Second there is the good enough. Shooting by definition is going to have fairly small sample sizes. So if <math>0.66s_h < s_v < 1.5 s_h</math> then, as a rule of thumb, that is probably good enough. Of course for large sample we would want to tighten the window. The harsh reality is that if <math>s_h</math> and <math>s_v</math> could be measured with great precision (e.g. to ten significant figures), then two values would always be statistically significantly different. <br />
<br />
Thus the approximation that <math>\sigma_h \approx \sigma_v</math> will be used unless the variances are known to be statistically significantly different. On the experimental data it is possible to test for a statistically significant difference by using a ratio of <math>s_h^2</math> and <math>s_v^2</math> via the F-Test. The "catch" in using the F test is that the variance has poor precision for small samples. Thus the difference must be great for the F-Test to detect that the two variances are not equal.<br />
<br />
== Simplifications Reduce Number of Coefficients to Fit ==<br />
<br />
The Hoyt distribution is general enough to be able to fit all four of the special cases in the table above. The point in making special cases of the Hoyt distribution is to reduce the number of coefficients to fit to the data. In general the more coefficients to be fit, the more data is required. Also when fitting multiple coefficients some of the coefficients are determined with greater precision than others. Thus to get a "good" fit for multiple coefficients a lot more data is required not just the minimum. <br />
<br />
Thus to fit the COI at least two shots are required. To fit the constant for the Rayleigh equation another shot would be required for a total of three shots. To fit the Hoyt distribution an additional five shots would be required for a total of seven shots. Probably 10 shots would be required to get a "decent" fit for the Rayleigh distribution, and at least 25 for the Hoyt distribution.<br />
<br />
== Notation in Simplified Cases ==<br />
<br />
The formulas for the distributions in the cases detailed in subsequent parts of this page are given in terms of the population parameters (i.e. <math>\mu_h, \mu_v, \sigma_h, \mbox{and } \sigma_v</math>) rather than the experimentally determined factors (i.e. <math>\bar{h}, \bar{v}, s_h, \mbox{and } s_v</math>) on purpose to emphasize the theoretical nature of the assumptions. Of course the "true" population parameters are unknown, and they could only be estimated with the corresponding experimentally fitted values about which there is some error.<br />
<br />
= Conformance Testing =<br />
<br />
== <math>\rho \approx 0</math> ==<br />
<br />
only way linear least squares<br />
<br />
== <math>\sigma_h \approx \sigma_v </math>==<br />
<br />
# F-Test <math>\frac{s_h^2}{s_v^2}</math>&nbsp;&nbsp;&nbsp;&nbsp; if &nbsp;&nbsp;&nbsp;<math>s_h < s_v</math>&nbsp;&nbsp;&nbsp; else &nbsp;&nbsp;&nbsp;<math>\frac{s_v^2}{s_h^2}</math><br />
# Studentized Ranges<br />
# Chi-Squared <math>(n-1) \frac{s^2}{\hat{\sigma}^2}</math><br />
<br />
= Circular Shot Distribution about COI =<br />
<br />
== Case 1, Rayleigh Distribution == <br />
[[File:raleigh.jpg|250px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
Given: <br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \approx 0</math><br /><br />
then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>f(r) = \frac{r}{\Re^2} e^{-r^2/(2\Re^2)}, \quad r \geq 0,</math> and <math>\Re</math> is the shape factor of the Rayleigh distribution.<br /><br />
<br />
This is really the best case for shot dispersion. Shot groups would be round. <br />
<br />
Strictly, for the Rayleigh distribution to apply, then <math>\sigma_h = \sigma_v</math>, in which case <math>\Re = \sigma_h = \sigma_v</math>. For the "loose" application of the Rayleigh distribution to apply, then <math>\Re \approx (\sigma_h + \sigma_v)/2 \approx \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math>.<br />
<br />
The following statistical measurements are appropriate:<br />
* Circular Error Probable (CEP)<br />
* Covering Circle Radius (CCR)<br />
* Diagonal<br />
* Extreme Spread (ES) <br />
* Figure of Merit (FOM)<br />
* Mean Radius (MR)<br />
* Radial Standard deviation<br />
<br />
'''Notes:'''<br />
# The Diagonal, the Extreme Spread and the FOM are different measurements, even though they conceivable could be based on the same two shots! The Extreme Spread would only depend on the two shots most distant in separation. The the Diagonal and the FOM would depend on two to four shots. For a large number of shots we'd typically expect four different shots to define the extremes for horizontal and vertical deflection.<br />
# For the measures for the CCR, the Diagonal, the GS and the FOM measurements a target would a ragged hole would be acceptable, but for the rest of the measures the {''h,v''} positions of each shot must be known.<br />
# Experimentally the radial distance for each shot, ''i'', is <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
# The conversion to polar coordinates results in each shot having coordinates <math>(r, \theta)</math>. (a) The conversion implicitly assumes that the polar coordinates have been translated so that the center is at the Cartesian Coordinate of the true center of the population <math>(\bar{h}, \bar{v})</math>. (b) The distribution of <math>\theta</math> is assumed to be entirely random and hence irrelevant. This assumption is testable. (c) The distribution is thus converted from a two-variable distribution to a one-variable distribution.<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] Note that there is a conundrum in how we are "averaging" the horizontal and vertical standard deviations to get <math>\sigma_{\Re}</math>. Look at the two expressions. They lead to two choices, either of which may casually seem valid.<br />
* <math>\sigma_h = \sigma_v</math><br />
* <math>\sigma_h^2 = \sigma_v^2</math><br />
<br />
In general if we look at the first formula "averaging" it leads to using:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\Re = \frac{\sigma_h + \sigma_v}{2} = \sigma_h</math> &nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
However in statistics standard deviations are "averaged" (pooled) by taking the square root of the average of their variances:<br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re^2 = {\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}} = \sigma_h </math>&nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
but:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
&nbsp;&nbsp;if and only if <math>\sigma_h \equiv \sigma_v</math><br />
<br />
Thus we should take the extent that:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} \neq \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
<br />
as a severe warning that we can not push the assumption that <math>\sigma_h \approx \sigma_v</math> too far if we expect the simplification of the general Hoyt distribution to the Rayleigh distribution to give meaningful results. <br />
<br />
The situation is even more tenuous given the small samples that shooters typically use. In general the relative precision of the variance value about a mean is much less precise than the relative precision of the mean value. The statistical test to compare two experimental variance values (i.e. <math>\sigma_h^2, \text{and} \sigma_v^2</math> in our case) is the F-Test which uses the ratios of the variances. For small samples a large difference would need to be observed before the ratio would be statistically significantly because of the imprecision in the individual experimental variance values. <br />
|}<br />
<br />
= Elliptical Shot Distribution about COI =<br />
<br />
== Case 2, Equal variances and correlated == <br />
Given:<br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
<br />
The following statistical measurement is appropriate:<br />
* Elliptic Error Probable<br />
<br />
==Case 3, Unequal variances and uncorrelated (Orthogonal Elliptical Distribution)== <br />
Given:<br /> <br />
# <math>\sigma_h \neq \sigma_v</math><br /><br />
# <math>\rho \approx 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v}<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
For the purposes of this wiki, this distribution will be called the '''Orthogonal Elliptical Distribution'''. It is obviously a special case of the Hoyt distribution which in turn is a special case of the bivariate normal distribution. <br />
<br />
In order of the model complexity, the following statistical measurements are appropriate:<br />
* Individual Horizontal and Vertical variances<br />
* Elliptic Error Probable<br />
<br />
In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively.<br />
<br />
==Case 4, Unequal variances and correlated (Hoyt Distribution)== <br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
<br />
Given:<br /> <br />
#<math>\sigma_h \neq \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be the Hoyt distribution with no simplifications:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Shot groups would be elliptical or egg-shaped if either the horizontal range or vertical range were large. The following statistical measurements are appropriate:<br />
* Elliptic Error Probable<br />
<br />
= Related topics =<br />
<br />
See also the following topics which are closely related:<br />
* [[Error Propagation]] - A basic discussion of how errors propagate when making measurements. <br />
* [[Stringing]] - Definition of stringing and how it can be handled<br />
<br />
= References =<br />
<references /><br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1269Herb References2015-06-19T01:52:58Z<p>Herb: </p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| (Abstract @ http://www.jstor.org/stable/2282775)]<br />
<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 (Abstract @ http://www.jstor.org/stable/1266181)]<br />
<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 (PDF @ http://handle.dtic.mil/100.2/ADA059117)]<br />
<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333. [http://www.jstor.org/stable/2346911?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2346911?seq=1#page_scan_tab_contents)] [http://www.robertnz.net/pdf/lc_chisq.pdf (PDF @ http://www.robertnz.net/pdf/lc_chisq.pdf)]<br />
<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf (PDF @ http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf)]<br />
<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA476368 (PDF @ http://handle.dtic.mil/100.2/ADA476368)]<br />
<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA476368 (PDF @ http://handle.dtic.mil/100.2/ADA476368)] <br />
<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. [http://www.jstor.org/stable/2003026 (READ @ http://www.jstor.org/stable/2003026)] [http://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0129116-8/ (PDF @ http://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0129116-8/)]<br />
<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. [http://www.jstor.org/stable/2004054 (READ @ http://www.jstor.org/stable/2004054)]<br />
<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory. [http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0270739 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0270739)]<br />
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<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
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* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. [http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257)]<br />
<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339. [http://www.jstor.org/stable/2347721?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2347721?seq=1#page_scan_tab_contents)] [http://www.robertnz.net/pdf/lc_chisq.pdf (PDF @ http://www.robertnz.net/pdf/lc_chisq.pdf)]<br />
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* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309. [http://www.jstor.org/stable/2347778?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2347778?seq=1#page_scan_tab_contents)] <br />
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* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| (Cached private monograph)]]. <br />
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* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. [http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf (PDF @ http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf)]<br />
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* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. [http://www.rand.org/pubs/reports/2008/R234.pdf (PDF @ http://www.rand.org/pubs/reports/2008/R234.pdf)]<br />
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* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
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* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. [http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf (PDF @ http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf)]<br />
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* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98. [http://www.jstor.org/stable/2346884 (Read @ http://www.jstor.org/stable/2346884)]<br />
<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
<br />
* <div id="Siddiqui1961"></div>Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(Cached PDF)]]<br />
<br />
* <div id="Siddiqui1964"></div>Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(Cached PDF)]]<br />
<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. [http://www.jstor.org/stable/25052751 (Abstract @ http://www.jstor.org/stable/25052751)]<br />
<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. [http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html (HTML Fulltext @ http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html)]<br />
<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its estimation. Metrika, 15 (1), 149–163. [http://link.springer.com/article/10.1007%2FBF02613568 (Abstract @ http://link.springer.com/article/10.1007%2FBF02613568)]<br />
<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. [http://www.jstor.org/stable/2290205 (Abstract @ http://www.jstor.org/stable/2290205)]<br />
<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA153828 (PDF @ http://handle.dtic.mil/100.2/ADA153828)]<br />
<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA153828 (PDF @ http://handle.dtic.mil/100.2/ADA153828)]<br />
<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Memorandum Rept. ADA006586, Army Ballistic Research Lab, Aberdeen Proving Ground [http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf (PDF @ http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf)]<br />
<br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. [http://dx.doi.org/10.1002/nav.3800220407 Abstract @ http://dx.doi.org/10.1002/nav.3800220407] [https://archive.org/details/navalresearchlog2241975offi (PDF of Naval Logistics Quarterly issue @ https://archive.org/details/navalresearchlog2241975offi0]<br />
<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. [http://handle.dtic.mil/100.2/AD0759989 (PDF @ http://handle.dtic.mil/100.2/AD0759989)]<br />
<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. [http://handle.dtic.mil/100.2/ADA266528 (PDF @ http://handle.dtic.mil/100.2/ADA266528)]<br />
<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960. [http://www.scientific.net/AMM.341-342.955 (Abstract @ http://www.scientific.net/AMM.341-342.955)]<br />
<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228. [http://www.scientific.net/AMR.765-767.2224 (Abstract @ http://www.scientific.net/AMR.765-767.2224)]<br />
<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. [http://handle.dtic.mil/100.2/ADA324337 (PDF @ http://handle.dtic.mil/100.2/ADA324337)]<br />
<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. [http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797 (Abstract @ http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797)]<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
Siddiqui had a two part series on the Rayleigh distribution ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).<br />
<br />
=== Derivation ===<br />
<br />
== Sample Range ==<br />
[[#Harter1960b | Harter (1964) ]] gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentized range. <br />
<br />
= Reference Data =<br />
<br />
* [[File:Confidence Interval Convergence.xlsx]]: Shows how precision confidence intervals shrink as sample size increases.<br />
<br />
* [[File:Sigma1RangeStatistics.xls]]: Simulated median, 50%, 80%, and 95% quantiles, plus first four sample moments, for shot groups containing 2 to 100 shots, of: Extreme Spread, Diagonal, Figure of Merit.<br />
<br />
* [[File:SymmetricBivariateSigma1.xls]]: Monte Carlo simulation results validating the [[Closed Form Precision]] math.<br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].</div>Herbhttp://ballistipedia.com/index.php?title=Projectile_Dispersion_Classifications&diff=1268Projectile Dispersion Classifications2015-06-17T16:24:38Z<p>Herb: /* Simplifications Reduce Number of Coefficients to Fit */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Before considering the measurements that will be used for the actual statistical analysis, let's consider the assumptions about projectile dispersion about the Center of Impact (COI) and how sets of those assumptions might be grouped into different classifications. The various classifications will offer insight as to the fundamental patterns expected for shots and insights to the interactions of various measures. Thus an understanding of the basic assumptions about projectile dispersion is key in being able to effectively use the measures. <br />
<br />
The COI is the only true point of reference which can be calculated from the pattern of shots on a target. Thus the COI is the reference point for precision measurements. The overall error that we are interested in measuring is the sum of all the various interactions that make multiple projectiles shot to the same point of aim (POA) disperse about the COI. <br />
<br />
Since we are primarily interested in the dispersion relative to the COI, the overall assumption is that the weapon could be properly sighted so that the COI would be the same as the POA. In practice this is achieved by [[FAQ#How_many_shots_do_I_need_to_sight_in.3F| adjusting the weapon's sights]]. Thus in order to isolate projectile dispersion, all of the factors of internal and external ballistics that cause a bias to the COI on a target will be ignored. For example, for the purposes of classifying projectile dispersion, accuracy errors due to POA errors will be ignored. <br />
<br />
The [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution] is the broadly assumed probability model used for a single random variable and it is characterized by its mean <math>(\bar{x})</math> and standard deviation <math>(\sigma)</math>. The [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem] shows that for measures for the "average" shot, or averages of multiple targets are used, then for "large" samples the averages will conform to Normal distribution even if the fundamental distribution is not a normal distribution. <br />
<br />
[[File:Bivariate.png|400px|thumb|right|Distribution of samples from a symmetric bivariate normal distribution. Axis units are multiples of σ.]]<br />
<br />
Since we are interested in shot dispersion on a two-dimensional target we will assume that the horizontal and vertical dispersions of the population of shots are each Normal distributions. Thus the horizontal dispersion will have mean <math>\mu_H</math> and standard deviation <math>\sigma_H</math>. The vertical dispersion will have mean <math>\mu_V</math> and standard deviation <math>\sigma_V</math>. Then a further assumption is made by assuming that the two dimensional expansion of the Normal distribution the [http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Non-degenerate_case Bivariate Normal distribution], applies. This adds an additional term the [http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient correlation parameter ''ρ'']. (See also: [[What is ρ in the Bivariate Normal distribution?]]) Thus the expectation is that distribution should then describe, the dispersion of a gunshots about the COI, (<math>\mu_H</math> and <math>\mu_V</math>). The full bivariate normal distribution is thus:<br ><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(H,V; \mu_H, \mu_V, \sigma_H, \sigma_V, \rho) =<br />
\frac{1}{2 \pi \sigma_H \sigma_V \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(H-\mu_H)^2}{\sigma_H^2} +<br />
\frac{(V-\mu_V)^2}{\sigma_V^2} -<br />
\frac{2\rho(H-\mu_H)(V-\mu_V)}{\sigma_H \sigma_V}<br />
\right]<br />
\right)<br />
</math><br />
<br />
where:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>-1 &le; \rho &le; 1</math><br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> \sigma_H>0 </math> and <math> \sigma_V>0 </math><br />
<br />
Note that the above restrictions are not additional restrictions on the model, but rather simply pointing out how the mathematics works. Thus they are more analogous to the mathematical notion that a person can't have a negative age. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] An ancillary point worth mentioning is that the assuming the Normal distribution in three dimensions leads to the [http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell–Boltzmann distribution] which is the foundation of the ideal gas laws. <br />
|}<br />
<br />
= Simplification of the Hoyt distribution into Special Cases =<br />
<br />
To eliminate the COI (<math>\mu_H</math>, <math>\mu_V</math>) which makes the equations "messier", a translation of the coordinate system to the COI is desired. Thus:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\mu_h = 0</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>\mu_v = 0</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>h = H - \mu_H</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>v = V - \mu_V</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;<math>\sigma_h = \sigma_H </math>&nbsp;&nbsp; and &nbsp;&nbsp;<math>\sigma_v = \sigma_V </math><br /><br />
<br />
This is a very pragmatic and justifiable consideration since the COI can be measured on the target, and the dispersion about the COI is the aspect of interest when measuring precision. As noted before, by adjusting the weapon's sights the POA can be made to coincide with the COI. Thus this simplification of the dispersion equations is strictly for ease of understanding as is not a limitation on the nature of the dispersion classifications. With the translation of the coordinate system to the COI, then the general bivariate normal equation becomes the Hoyt distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v; \sigma_h, \sigma_v, \rho) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho hv}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Looking at this equation two other different mutually exclusive simplifications can be readily seen:<br />
<br />
* '''Either''' <math>\sigma_h = \sigma_v</math> (equal standard deviations) '''or''' <math>\sigma_h \neq \sigma_v</math> (unequal standard deviations).<br />
: Obviously if we could measure both <math>\sigma_h</math> and <math>\sigma_v</math> with a very high precision (e.g 6 significant figures), then the two quantities would never really be equal. But in many cases the assumption is reasonable. In reality since shooters typically collect only a small amount of data, statistical tests will fail to detect a difference unless the difference is great. In such cases the shot pattern would be noticeably elliptical, not round. <br />
<br />
* '''Either''' <math>\rho = 0</math> (uncorrelated) '''or''' <math>\rho \neq 0</math> (correlated). <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: !! CAREFUL !! '''[http://en.wikipedia.org/wiki/Correlation_does_not_imply_causation Correlation does not imply causation]''' <br />
<br />
: There is somewhat famous example. A researcher gathered statistics for stork sightings and births in a particular county over a twenty year period. Analysis of the data showed that over the twenty year period both stork sightings and births had increased with a very significant linear correlation. From the data you might erroneously infer that storks do bring babies! <br />
|}<br />
<br />
The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation as shown in the Table below. There is one case that results in circular groups, and three that result in elliptical groups. As the different in variances gets greater, or the further <math>\rho</math> is from 0, then the ellipse will be more pronounced. <br />
<br />
{| class="wikitable" <br />
|+ Group Shape vs. Assumptions (COI at Origin)<br />
|-<br />
|<br />
| <math>\sigma_h \approx \sigma_v</math><br />
| <math>\sigma_h \neq \sigma_v</math><br />
|-<br />
| <math>\rho \approx 0</math><br />
| Case 1 - Circular Groups<br />
* special case is the Rayleigh Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
| Case 3 - Elliptical Groups<br />
* Major axis of ellipse along<br /> horizontal or vertical axis<br />
* special case is the Orthogonal Elliptical Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
|-<br />
| <math>\rho \neq 0</math><br />
* Major axis of ellipse at an angle to<br />both the horizontal and vertical axes<br />
| Case 2- Elliptical Groups<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
: - <math>\rho</math><br />
| Case 4 - Elliptical Groups<br />
* general case of the Hoyt distribution required<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
: - <math>\rho</math><br />
|}<br />
<br />
== Experimental reality of Comparing <math>s_h</math> and <math>s_v</math>==<br />
<br />
The table above uses ''approximately equal to'' <math>(\approx)</math> rather than ''strictly equal to'' <math>( = )</math>. This is an acknowledgement that we are dividing the cases into ones that are close enough to be useful, even though they most certainly are not exact. To be overly persnickety there are two considerations. <br />
<br />
First we can only get experimental estimates from calculations based on sample data for the factors <math>\sigma_h</math>, <math>\sigma_v</math>, <math>\rho</math> and these estimates are at best only good to a scant few significant figures. Thus even though the difference between ''approximately equal to'' and ''strictly equal to'' is under some experimental control there are practical limits. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but it is quickly impractical to do so. (Assume that to double the precision that we have to quadruple the sample size. This exponential increase quickly becomes unmanageable. It is easy to pontificate about averaging over a million targets, but no one is going to shoot that many.) Thus even if <math>\sigma_h \equiv \sigma_v</math> we'd never expect that we'd experimentally get <math>s_h = s_v</math> due to experimental error. <br />
<br />
Second there is the good enough. Shooting by definition is going to have fairly small sample sizes. So if <math>0.66s_h < s_v < 1.5 s_h</math> then, as a rule of thumb, that is probably good enough. Of course for large sample we would want to tighten the window. The harsh reality is that if <math>s_h</math> and <math>s_v</math> could be measured with great precision (e.g. to ten significant figures), then two values would always be statistically significantly different. <br />
<br />
Thus the approximation that <math>\sigma_h \approx \sigma_v</math> will be used unless the variances are known to be statistically significantly different. On the experimental data it is possible to test for a statistically significant difference by using a ratio of <math>s_h^2</math> and <math>s_v^2</math> via the F-Test. The "catch" in using the F test is that the variance has poor precision for small samples. Thus the difference must be great for the F-Test to detect that the two variances are not equal.<br />
<br />
== Simplifications Reduce Number of Coefficients to Fit ==<br />
<br />
The Hoyt distribution is general enough to be able to fit all four of the special cases in the table above. The point in making special cases of the Hoyt distribution is to reduce the number of coefficients to fit to the data. In general the more coefficients to be fit, the more data is required. Also when fitting multiple coefficients some of the coefficients are determined with greater precision than others. Thus to get a "good" fit for multiple coefficients a lot more data is required not just the minimum. <br />
<br />
Thus to fit the COI at least two shots are required. To fit the constant for the Rayleigh equation another shot would be required for a total of three shots. To fit the Hoyt distribution an additional five shots would be required for a total of seven shots. Probably 10 shots would be required to get a "decent" fit for the Rayleigh distribution, and at least 25 for the Hoyt distribution.<br />
<br />
== Notation in Simplified Cases ==<br />
<br />
The formulas for the distributions in the cases detailed in subsequent parts of this page are given in terms of the population parameters (i.e. <math>\mu_h, \mu_v, \sigma_h, \mbox{and } \sigma_v</math>) rather than the experimentally determined factors (i.e. <math>\bar{h}, \bar{v}, s_h, \mbox{and } s_v</math>) on purpose to emphasize the theoretical nature of the assumptions. Of course the "true" population parameters are unknown, and they could only be estimated with the corresponding experimentally fitted values about which there is some error.<br />
<br />
= Conformance Testing =<br />
<br />
== <math>\rho \approx 0</math> ==<br />
<br />
only way linear least squares<br />
<br />
== <math>\sigma_h \approx \sigma_v </math>==<br />
<br />
# F-Test <math>\frac{s_h^2}{s_v^2}</math>&nbsp;&nbsp;&nbsp;&nbsp; if &nbsp;&nbsp;&nbsp;<math>s_h < s_v</math>&nbsp;&nbsp;&nbsp; else &nbsp;&nbsp;&nbsp;<math>\frac{s_v^2}{s_h^2}</math><br />
# Studentized Ranges<br />
# Chi-Squared <math>(n-1) \frac{s^2}{\hat{\sigma}^2}</math><br />
<br />
= Circular Shot Distribution about COI =<br />
<br />
== Case 1, Rayleigh Distribution == <br />
[[File:raleigh.jpg|250px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
Given: <br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \approx 0</math><br /><br />
then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>f(r) = \frac{r}{\Re^2} e^{-r^2/(2\Re^2)}, \quad r \geq 0,</math> and <math>\Re</math> is the shape factor of the Rayleigh distribution.<br /><br />
<br />
This is really the best case for shot dispersion. Shot groups would be round. <br />
<br />
Strictly, for the Rayleigh distribution to apply, then <math>\sigma_h = \sigma_v</math>, in which case <math>\Re = \sigma_h = \sigma_v</math>. For the "loose" application of the Rayleigh distribution to apply, then <math>\Re \approx (\sigma_h + \sigma_v)/2 \approx \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math>.<br />
<br />
The following statistical measurements are appropriate:<br />
* Circular Error Probable (CEP)<br />
* Covering Circle Radius (CCR)<br />
* Diagonal<br />
* Extreme Spread (ES) <br />
* Figure of Merit (FOM)<br />
* Mean Radius (MR)<br />
* Radial Standard deviation<br />
<br />
'''Notes:'''<br />
# The Diagonal, the Extreme Spread and the FOM are different measurements, even though they conceivable could be based on the same two shots! The Extreme Spread would only depend on the two shots most distant in separation. The the Diagonal and the FOM would depend on two to four shots. For a large number of shots we'd typically expect four different shots to define the extremes for horizontal and vertical deflection.<br />
# For the measures for the CCR, the Diagonal, the GS and the FOM measurements a target would a ragged hole would be acceptable, but for the rest of the measures the {''h,v''} positions of each shot must be known.<br />
# Experimentally the radial distance for each shot, ''i'', is <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
# The conversion to polar coordinates results in each shot having coordinates <math>(r, \theta)</math>. (a) The conversion implicitly assumes that the polar coordinates have been translated so that the center is at the Cartesian Coordinate of the true center of the population <math>(\bar{h}, \bar{v})</math>. (b) The distribution of <math>\theta</math> is assumed to be entirely random and hence irrelevant. This assumption is testable. (c) The distribution is thus converted from a two-variable distribution to a one-variable distribution.<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] Note that there is a conundrum in how we are "averaging" the horizontal and vertical standard deviations to get <math>\sigma_{\Re}</math>. Look at the two expressions. They lead to two choices, either of which may casually seem valid.<br />
* <math>\sigma_h = \sigma_v</math><br />
* <math>\sigma_h^2 = \sigma_v^2</math><br />
<br />
In general if we look at the first formula "averaging" it leads to using:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\Re = \frac{\sigma_h + \sigma_v}{2} = \sigma_h</math> &nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
However in statistics standard deviations are "averaged" (pooled) by taking the square root of the average of their variances:<br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re^2 = {\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}} = \sigma_h </math>&nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
but:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
&nbsp;&nbsp;if and only if <math>\sigma_h \equiv \sigma_v</math><br />
<br />
Thus we should take the extent that:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} \neq \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
<br />
as a severe warning that we can not push the assumption that <math>\sigma_h \approx \sigma_v</math> too far if we expect the simplification of the general Hoyt distribution to the Rayleigh distribution to give meaningful results. <br />
<br />
The situation is even more tenuous given the small samples that shooters typically use. In general the relative precision of the variance value about a mean is much less precise than the relative precision of the mean value. The statistical test to compare two experimental variance values (i.e. <math>\sigma_h^2, \text{and} \sigma_v^2</math> in our case) is the F-Test which uses the ratios of the variances. For small samples a large difference would need to be observed before the ratio would be statistically significantly because of the imprecision in the individual experimental variance values. <br />
|}<br />
<br />
= Elliptical Shot Distribution about COI =<br />
<br />
== Case 2, Equal variances and correlated == <br />
Given:<br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
<br />
The following statistical measurement is appropriate:<br />
* Elliptic Error Probable<br />
<br />
==Case 3, Unequal variances and uncorrelated (Orthogonal Elliptical Distribution)== <br />
Given:<br /> <br />
# <math>\sigma_h \neq \sigma_v</math><br /><br />
# <math>\rho \approx 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v}<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
For the purposes of this wiki, this distribution will be called the '''Orthogonal Elliptical Distribution'''. It is obviously a special case of the Hoyt distribution which in turn is a special case of the bivariate normal distribution. <br />
<br />
In order of the model complexity, the following statistical measurements are appropriate:<br />
* Individual Horizontal and Vertical variances<br />
* Elliptic Error Probable<br />
<br />
In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively.<br />
<br />
==Case 4, Unequal variances and correlated (Hoyt Distribution)== <br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
<br />
Given:<br /> <br />
#<math>\sigma_h \neq \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be the Hoyt distribution with no simplifications:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Shot groups would be elliptical or egg-shaped if either the horizontal range or vertical range were large. The following statistical measurements are appropriate:<br />
* Elliptic Error Probable<br />
<br />
= Related topics =<br />
<br />
See also the following topics which are closely related:<br />
* [[Error Propagation]] - A basic discussion of how errors propagate when making measurements. <br />
* [[Stringing]] - Definition of stringing and how it can be handled<br />
<br />
= References =<br />
<references /><br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Projectile_Dispersion_Classifications&diff=1267Projectile Dispersion Classifications2015-06-17T14:55:49Z<p>Herb: /* Experimental reality of Comparing s_h and s_v */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Before considering the measurements that will be used for the actual statistical analysis, let's consider the assumptions about projectile dispersion about the Center of Impact (COI) and how sets of those assumptions might be grouped into different classifications. The various classifications will offer insight as to the fundamental patterns expected for shots and insights to the interactions of various measures. Thus an understanding of the basic assumptions about projectile dispersion is key in being able to effectively use the measures. <br />
<br />
The COI is the only true point of reference which can be calculated from the pattern of shots on a target. Thus the COI is the reference point for precision measurements. The overall error that we are interested in measuring is the sum of all the various interactions that make multiple projectiles shot to the same point of aim (POA) disperse about the COI. <br />
<br />
Since we are primarily interested in the dispersion relative to the COI, the overall assumption is that the weapon could be properly sighted so that the COI would be the same as the POA. In practice this is achieved by [[FAQ#How_many_shots_do_I_need_to_sight_in.3F| adjusting the weapon's sights]]. Thus in order to isolate projectile dispersion, all of the factors of internal and external ballistics that cause a bias to the COI on a target will be ignored. For example, for the purposes of classifying projectile dispersion, accuracy errors due to POA errors will be ignored. <br />
<br />
The [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution] is the broadly assumed probability model used for a single random variable and it is characterized by its mean <math>(\bar{x})</math> and standard deviation <math>(\sigma)</math>. The [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem] shows that for measures for the "average" shot, or averages of multiple targets are used, then for "large" samples the averages will conform to Normal distribution even if the fundamental distribution is not a normal distribution. <br />
<br />
[[File:Bivariate.png|400px|thumb|right|Distribution of samples from a symmetric bivariate normal distribution. Axis units are multiples of σ.]]<br />
<br />
Since we are interested in shot dispersion on a two-dimensional target we will assume that the horizontal and vertical dispersions of the population of shots are each Normal distributions. Thus the horizontal dispersion will have mean <math>\mu_H</math> and standard deviation <math>\sigma_H</math>. The vertical dispersion will have mean <math>\mu_V</math> and standard deviation <math>\sigma_V</math>. Then a further assumption is made by assuming that the two dimensional expansion of the Normal distribution the [http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Non-degenerate_case Bivariate Normal distribution], applies. This adds an additional term the [http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient correlation parameter ''ρ'']. (See also: [[What is ρ in the Bivariate Normal distribution?]]) Thus the expectation is that distribution should then describe, the dispersion of a gunshots about the COI, (<math>\mu_H</math> and <math>\mu_V</math>). The full bivariate normal distribution is thus:<br ><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(H,V; \mu_H, \mu_V, \sigma_H, \sigma_V, \rho) =<br />
\frac{1}{2 \pi \sigma_H \sigma_V \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(H-\mu_H)^2}{\sigma_H^2} +<br />
\frac{(V-\mu_V)^2}{\sigma_V^2} -<br />
\frac{2\rho(H-\mu_H)(V-\mu_V)}{\sigma_H \sigma_V}<br />
\right]<br />
\right)<br />
</math><br />
<br />
where:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>-1 &le; \rho &le; 1</math><br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> \sigma_H>0 </math> and <math> \sigma_V>0 </math><br />
<br />
Note that the above restrictions are not additional restrictions on the model, but rather simply pointing out how the mathematics works. Thus they are more analogous to the mathematical notion that a person can't have a negative age. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] An ancillary point worth mentioning is that the assuming the Normal distribution in three dimensions leads to the [http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell–Boltzmann distribution] which is the foundation of the ideal gas laws. <br />
|}<br />
<br />
= Simplification of the Hoyt distribution into Special Cases =<br />
<br />
To eliminate the COI (<math>\mu_H</math>, <math>\mu_V</math>) which makes the equations "messier", a translation of the coordinate system to the COI is desired. Thus:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\mu_h = 0</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>\mu_v = 0</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>h = H - \mu_H</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>v = V - \mu_V</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;<math>\sigma_h = \sigma_H </math>&nbsp;&nbsp; and &nbsp;&nbsp;<math>\sigma_v = \sigma_V </math><br /><br />
<br />
This is a very pragmatic and justifiable consideration since the COI can be measured on the target, and the dispersion about the COI is the aspect of interest when measuring precision. As noted before, by adjusting the weapon's sights the POA can be made to coincide with the COI. Thus this simplification of the dispersion equations is strictly for ease of understanding as is not a limitation on the nature of the dispersion classifications. With the translation of the coordinate system to the COI, then the general bivariate normal equation becomes the Hoyt distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v; \sigma_h, \sigma_v, \rho) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho hv}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Looking at this equation two other different mutually exclusive simplifications can be readily seen:<br />
<br />
* '''Either''' <math>\sigma_h = \sigma_v</math> (equal standard deviations) '''or''' <math>\sigma_h \neq \sigma_v</math> (unequal standard deviations).<br />
: Obviously if we could measure both <math>\sigma_h</math> and <math>\sigma_v</math> with a very high precision (e.g 6 significant figures), then the two quantities would never really be equal. But in many cases the assumption is reasonable. In reality since shooters typically collect only a small amount of data, statistical tests will fail to detect a difference unless the difference is great. In such cases the shot pattern would be noticeably elliptical, not round. <br />
<br />
* '''Either''' <math>\rho = 0</math> (uncorrelated) '''or''' <math>\rho \neq 0</math> (correlated). <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: !! CAREFUL !! '''[http://en.wikipedia.org/wiki/Correlation_does_not_imply_causation Correlation does not imply causation]''' <br />
<br />
: There is somewhat famous example. A researcher gathered statistics for stork sightings and births in a particular county over a twenty year period. Analysis of the data showed that over the twenty year period both stork sightings and births had increased with a very significant linear correlation. From the data you might erroneously infer that storks do bring babies! <br />
|}<br />
<br />
The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation as shown in the Table below. There is one case that results in circular groups, and three that result in elliptical groups. As the different in variances gets greater, or the further <math>\rho</math> is from 0, then the ellipse will be more pronounced. <br />
<br />
{| class="wikitable" <br />
|+ Group Shape vs. Assumptions (COI at Origin)<br />
|-<br />
|<br />
| <math>\sigma_h \approx \sigma_v</math><br />
| <math>\sigma_h \neq \sigma_v</math><br />
|-<br />
| <math>\rho \approx 0</math><br />
| Case 1 - Circular Groups<br />
* special case is the Rayleigh Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
| Case 3 - Elliptical Groups<br />
* Major axis of ellipse along<br /> horizontal or vertical axis<br />
* special case is the Orthogonal Elliptical Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
|-<br />
| <math>\rho \neq 0</math><br />
* Major axis of ellipse at an angle to<br />both the horizontal and vertical axes<br />
| Case 2- Elliptical Groups<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
: - <math>\rho</math><br />
| Case 4 - Elliptical Groups<br />
* general case of the Hoyt distribution required<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
: - <math>\rho</math><br />
|}<br />
<br />
== Experimental reality of Comparing <math>s_h</math> and <math>s_v</math>==<br />
<br />
The table above uses ''approximately equal to'' <math>(\approx)</math> rather than ''strictly equal to'' <math>( = )</math>. This is an acknowledgement that we are dividing the cases into ones that are close enough to be useful, even though they most certainly are not exact. To be overly persnickety there are two considerations. <br />
<br />
First we can only get experimental estimates from calculations based on sample data for the factors <math>\sigma_h</math>, <math>\sigma_v</math>, <math>\rho</math> and these estimates are at best only good to a scant few significant figures. Thus even though the difference between ''approximately equal to'' and ''strictly equal to'' is under some experimental control there are practical limits. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but it is quickly impractical to do so. (Assume that to double the precision that we have to quadruple the sample size. This exponential increase quickly becomes unmanageable. It is easy to pontificate about averaging over a million targets, but no one is going to shoot that many.) Thus even if <math>\sigma_h \equiv \sigma_v</math> we'd never expect that we'd experimentally get <math>s_h = s_v</math> due to experimental error. <br />
<br />
Second there is the good enough. Shooting by definition is going to have fairly small sample sizes. So if <math>0.66s_h < s_v < 1.5 s_h</math> then, as a rule of thumb, that is probably good enough. Of course for large sample we would want to tighten the window. The harsh reality is that if <math>s_h</math> and <math>s_v</math> could be measured with great precision (e.g. to ten significant figures), then two values would always be statistically significantly different. <br />
<br />
Thus the approximation that <math>\sigma_h \approx \sigma_v</math> will be used unless the variances are known to be statistically significantly different. On the experimental data it is possible to test for a statistically significant difference by using a ratio of <math>s_h^2</math> and <math>s_v^2</math> via the F-Test. The "catch" in using the F test is that the variance has poor precision for small samples. Thus the difference must be great for the F-Test to detect that the two variances are not equal.<br />
<br />
== Simplifications Reduce Number of Coefficients to Fit ==<br />
<br />
The Hoyt distribution is general enough to be able to fit all four of the special cases in the table above. The point in making special cases of the Hoyt distribution is to reduce the number of coefficients to fit to the data. In general the more coefficients to be fit, the more data is required. Also when fitting multiple coefficients some of the coefficients are determined with greater precision than others. Thus to get a "good" fit for multiple coefficients a lot more data is required not just the minimum. <br />
<br />
Thus to fit the COI at least two shots are required. To fit the constant for the Rayleigh equation another shot would be required for a total of three shots. To fit the Hoyt distribution an additional five shots would be required for a total of seven shots. In reality 10 shots would be required to get a "decent" fit for the Rayleigh distribution, and at least 25 for the Hoyt distribution.<br />
<br />
== Notation in Simplified Cases ==<br />
<br />
The formulas for the distributions in the cases detailed in subsequent parts of this page are given in terms of the population parameters (i.e. <math>\mu_h, \mu_v, \sigma_h, \mbox{and } \sigma_v</math>) rather than the experimentally determined factors (i.e. <math>\bar{h}, \bar{v}, s_h, \mbox{and } s_v</math>) on purpose to emphasize the theoretical nature of the assumptions. Of course the "true" population parameters are unknown, and they could only be estimated with the corresponding experimentally fitted values about which there is some error.<br />
<br />
= Conformance Testing =<br />
<br />
== <math>\rho \approx 0</math> ==<br />
<br />
only way linear least squares<br />
<br />
== <math>\sigma_h \approx \sigma_v </math>==<br />
<br />
# F-Test <math>\frac{s_h^2}{s_v^2}</math>&nbsp;&nbsp;&nbsp;&nbsp; if &nbsp;&nbsp;&nbsp;<math>s_h < s_v</math>&nbsp;&nbsp;&nbsp; else &nbsp;&nbsp;&nbsp;<math>\frac{s_v^2}{s_h^2}</math><br />
# Studentized Ranges<br />
# Chi-Squared <math>(n-1) \frac{s^2}{\hat{\sigma}^2}</math><br />
<br />
= Circular Shot Distribution about COI =<br />
<br />
== Case 1, Rayleigh Distribution == <br />
[[File:raleigh.jpg|250px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
Given: <br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \approx 0</math><br /><br />
then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>f(r) = \frac{r}{\Re^2} e^{-r^2/(2\Re^2)}, \quad r \geq 0,</math> and <math>\Re</math> is the shape factor of the Rayleigh distribution.<br /><br />
<br />
This is really the best case for shot dispersion. Shot groups would be round. <br />
<br />
Strictly, for the Rayleigh distribution to apply, then <math>\sigma_h = \sigma_v</math>, in which case <math>\Re = \sigma_h = \sigma_v</math>. For the "loose" application of the Rayleigh distribution to apply, then <math>\Re \approx (\sigma_h + \sigma_v)/2 \approx \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math>.<br />
<br />
The following statistical measurements are appropriate:<br />
* Circular Error Probable (CEP)<br />
* Covering Circle Radius (CCR)<br />
* Diagonal<br />
* Extreme Spread (ES) <br />
* Figure of Merit (FOM)<br />
* Mean Radius (MR)<br />
* Radial Standard deviation<br />
<br />
'''Notes:'''<br />
# The Diagonal, the Extreme Spread and the FOM are different measurements, even though they conceivable could be based on the same two shots! The Extreme Spread would only depend on the two shots most distant in separation. The the Diagonal and the FOM would depend on two to four shots. For a large number of shots we'd typically expect four different shots to define the extremes for horizontal and vertical deflection.<br />
# For the measures for the CCR, the Diagonal, the GS and the FOM measurements a target would a ragged hole would be acceptable, but for the rest of the measures the {''h,v''} positions of each shot must be known.<br />
# Experimentally the radial distance for each shot, ''i'', is <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
# The conversion to polar coordinates results in each shot having coordinates <math>(r, \theta)</math>. (a) The conversion implicitly assumes that the polar coordinates have been translated so that the center is at the Cartesian Coordinate of the true center of the population <math>(\bar{h}, \bar{v})</math>. (b) The distribution of <math>\theta</math> is assumed to be entirely random and hence irrelevant. This assumption is testable. (c) The distribution is thus converted from a two-variable distribution to a one-variable distribution.<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] Note that there is a conundrum in how we are "averaging" the horizontal and vertical standard deviations to get <math>\sigma_{\Re}</math>. Look at the two expressions. They lead to two choices, either of which may casually seem valid.<br />
* <math>\sigma_h = \sigma_v</math><br />
* <math>\sigma_h^2 = \sigma_v^2</math><br />
<br />
In general if we look at the first formula "averaging" it leads to using:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\Re = \frac{\sigma_h + \sigma_v}{2} = \sigma_h</math> &nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
However in statistics standard deviations are "averaged" (pooled) by taking the square root of the average of their variances:<br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re^2 = {\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}} = \sigma_h </math>&nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
but:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
&nbsp;&nbsp;if and only if <math>\sigma_h \equiv \sigma_v</math><br />
<br />
Thus we should take the extent that:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} \neq \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
<br />
as a severe warning that we can not push the assumption that <math>\sigma_h \approx \sigma_v</math> too far if we expect the simplification of the general Hoyt distribution to the Rayleigh distribution to give meaningful results. <br />
<br />
The situation is even more tenuous given the small samples that shooters typically use. In general the relative precision of the variance value about a mean is much less precise than the relative precision of the mean value. The statistical test to compare two experimental variance values (i.e. <math>\sigma_h^2, \text{and} \sigma_v^2</math> in our case) is the F-Test which uses the ratios of the variances. For small samples a large difference would need to be observed before the ratio would be statistically significantly because of the imprecision in the individual experimental variance values. <br />
|}<br />
<br />
= Elliptical Shot Distribution about COI =<br />
<br />
== Case 2, Equal variances and correlated == <br />
Given:<br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
<br />
The following statistical measurement is appropriate:<br />
* Elliptic Error Probable<br />
<br />
==Case 3, Unequal variances and uncorrelated (Orthogonal Elliptical Distribution)== <br />
Given:<br /> <br />
# <math>\sigma_h \neq \sigma_v</math><br /><br />
# <math>\rho \approx 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v}<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
For the purposes of this wiki, this distribution will be called the '''Orthogonal Elliptical Distribution'''. It is obviously a special case of the Hoyt distribution which in turn is a special case of the bivariate normal distribution. <br />
<br />
In order of the model complexity, the following statistical measurements are appropriate:<br />
* Individual Horizontal and Vertical variances<br />
* Elliptic Error Probable<br />
<br />
In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively.<br />
<br />
==Case 4, Unequal variances and correlated (Hoyt Distribution)== <br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
<br />
Given:<br /> <br />
#<math>\sigma_h \neq \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be the Hoyt distribution with no simplifications:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Shot groups would be elliptical or egg-shaped if either the horizontal range or vertical range were large. The following statistical measurements are appropriate:<br />
* Elliptic Error Probable<br />
<br />
= Related topics =<br />
<br />
See also the following topics which are closely related:<br />
* [[Error Propagation]] - A basic discussion of how errors propagate when making measurements. <br />
* [[Stringing]] - Definition of stringing and how it can be handled<br />
<br />
= References =<br />
<references /><br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Projectile_Dispersion_Classifications&diff=1266Projectile Dispersion Classifications2015-06-17T14:51:19Z<p>Herb: /* Simplifications Reduce Number of Coefficients to Fit */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Before considering the measurements that will be used for the actual statistical analysis, let's consider the assumptions about projectile dispersion about the Center of Impact (COI) and how sets of those assumptions might be grouped into different classifications. The various classifications will offer insight as to the fundamental patterns expected for shots and insights to the interactions of various measures. Thus an understanding of the basic assumptions about projectile dispersion is key in being able to effectively use the measures. <br />
<br />
The COI is the only true point of reference which can be calculated from the pattern of shots on a target. Thus the COI is the reference point for precision measurements. The overall error that we are interested in measuring is the sum of all the various interactions that make multiple projectiles shot to the same point of aim (POA) disperse about the COI. <br />
<br />
Since we are primarily interested in the dispersion relative to the COI, the overall assumption is that the weapon could be properly sighted so that the COI would be the same as the POA. In practice this is achieved by [[FAQ#How_many_shots_do_I_need_to_sight_in.3F| adjusting the weapon's sights]]. Thus in order to isolate projectile dispersion, all of the factors of internal and external ballistics that cause a bias to the COI on a target will be ignored. For example, for the purposes of classifying projectile dispersion, accuracy errors due to POA errors will be ignored. <br />
<br />
The [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution] is the broadly assumed probability model used for a single random variable and it is characterized by its mean <math>(\bar{x})</math> and standard deviation <math>(\sigma)</math>. The [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem] shows that for measures for the "average" shot, or averages of multiple targets are used, then for "large" samples the averages will conform to Normal distribution even if the fundamental distribution is not a normal distribution. <br />
<br />
[[File:Bivariate.png|400px|thumb|right|Distribution of samples from a symmetric bivariate normal distribution. Axis units are multiples of σ.]]<br />
<br />
Since we are interested in shot dispersion on a two-dimensional target we will assume that the horizontal and vertical dispersions of the population of shots are each Normal distributions. Thus the horizontal dispersion will have mean <math>\mu_H</math> and standard deviation <math>\sigma_H</math>. The vertical dispersion will have mean <math>\mu_V</math> and standard deviation <math>\sigma_V</math>. Then a further assumption is made by assuming that the two dimensional expansion of the Normal distribution the [http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Non-degenerate_case Bivariate Normal distribution], applies. This adds an additional term the [http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient correlation parameter ''ρ'']. (See also: [[What is ρ in the Bivariate Normal distribution?]]) Thus the expectation is that distribution should then describe, the dispersion of a gunshots about the COI, (<math>\mu_H</math> and <math>\mu_V</math>). The full bivariate normal distribution is thus:<br ><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(H,V; \mu_H, \mu_V, \sigma_H, \sigma_V, \rho) =<br />
\frac{1}{2 \pi \sigma_H \sigma_V \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(H-\mu_H)^2}{\sigma_H^2} +<br />
\frac{(V-\mu_V)^2}{\sigma_V^2} -<br />
\frac{2\rho(H-\mu_H)(V-\mu_V)}{\sigma_H \sigma_V}<br />
\right]<br />
\right)<br />
</math><br />
<br />
where:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>-1 &le; \rho &le; 1</math><br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> \sigma_H>0 </math> and <math> \sigma_V>0 </math><br />
<br />
Note that the above restrictions are not additional restrictions on the model, but rather simply pointing out how the mathematics works. Thus they are more analogous to the mathematical notion that a person can't have a negative age. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] An ancillary point worth mentioning is that the assuming the Normal distribution in three dimensions leads to the [http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell–Boltzmann distribution] which is the foundation of the ideal gas laws. <br />
|}<br />
<br />
= Simplification of the Hoyt distribution into Special Cases =<br />
<br />
To eliminate the COI (<math>\mu_H</math>, <math>\mu_V</math>) which makes the equations "messier", a translation of the coordinate system to the COI is desired. Thus:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\mu_h = 0</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>\mu_v = 0</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>h = H - \mu_H</math>&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;<math>v = V - \mu_V</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;<math>\sigma_h = \sigma_H </math>&nbsp;&nbsp; and &nbsp;&nbsp;<math>\sigma_v = \sigma_V </math><br /><br />
<br />
This is a very pragmatic and justifiable consideration since the COI can be measured on the target, and the dispersion about the COI is the aspect of interest when measuring precision. As noted before, by adjusting the weapon's sights the POA can be made to coincide with the COI. Thus this simplification of the dispersion equations is strictly for ease of understanding as is not a limitation on the nature of the dispersion classifications. With the translation of the coordinate system to the COI, then the general bivariate normal equation becomes the Hoyt distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v; \sigma_h, \sigma_v, \rho) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho hv}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Looking at this equation two other different mutually exclusive simplifications can be readily seen:<br />
<br />
* '''Either''' <math>\sigma_h = \sigma_v</math> (equal standard deviations) '''or''' <math>\sigma_h \neq \sigma_v</math> (unequal standard deviations).<br />
: Obviously if we could measure both <math>\sigma_h</math> and <math>\sigma_v</math> with a very high precision (e.g 6 significant figures), then the two quantities would never really be equal. But in many cases the assumption is reasonable. In reality since shooters typically collect only a small amount of data, statistical tests will fail to detect a difference unless the difference is great. In such cases the shot pattern would be noticeably elliptical, not round. <br />
<br />
* '''Either''' <math>\rho = 0</math> (uncorrelated) '''or''' <math>\rho \neq 0</math> (correlated). <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: !! CAREFUL !! '''[http://en.wikipedia.org/wiki/Correlation_does_not_imply_causation Correlation does not imply causation]''' <br />
<br />
: There is somewhat famous example. A researcher gathered statistics for stork sightings and births in a particular county over a twenty year period. Analysis of the data showed that over the twenty year period both stork sightings and births had increased with a very significant linear correlation. From the data you might erroneously infer that storks do bring babies! <br />
|}<br />
<br />
The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation as shown in the Table below. There is one case that results in circular groups, and three that result in elliptical groups. As the different in variances gets greater, or the further <math>\rho</math> is from 0, then the ellipse will be more pronounced. <br />
<br />
{| class="wikitable" <br />
|+ Group Shape vs. Assumptions (COI at Origin)<br />
|-<br />
|<br />
| <math>\sigma_h \approx \sigma_v</math><br />
| <math>\sigma_h \neq \sigma_v</math><br />
|-<br />
| <math>\rho \approx 0</math><br />
| Case 1 - Circular Groups<br />
* special case is the Rayleigh Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
| Case 3 - Elliptical Groups<br />
* Major axis of ellipse along<br /> horizontal or vertical axis<br />
* special case is the Orthogonal Elliptical Distribution<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
|-<br />
| <math>\rho \neq 0</math><br />
* Major axis of ellipse at an angle to<br />both the horizontal and vertical axes<br />
| Case 2- Elliptical Groups<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_{\Re}</math> (pooled value of <math>\sigma_h</math> and <math>\sigma_v</math>)<br />
: - <math>\rho</math><br />
| Case 4 - Elliptical Groups<br />
* general case of the Hoyt distribution required<br />
* Parameter(s) to fit (other than COI):<br />
: - <math>\sigma_h</math><br />
: - <math>\sigma_v</math><br />
: - <math>\rho</math><br />
|}<br />
<br />
== Experimental reality of Comparing <math>s_h</math> and <math>s_v</math>==<br />
<br />
The table above uses ''approximately equal to'' <math>(\approx)</math> rather than ''strictly equal to'' <math>( = )</math>. This is an acknowledgement that we are dividing the cases into ones that are close enough to be useful, even though they most certainly are not exact. To be overly persnickety there are two considerations. <br />
<br />
First we can only get experimental estimates from calculations based on sample data for the factors <math>\sigma_h</math>, <math>\sigma_v</math>, <math>\rho</math> and these estimates are at best only good to a scant few significant figures. Thus even though the difference between ''approximately equal to'' and ''strictly equal to'' is under some experimental control there are practical limits. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but it is quickly impractical to do so. (Assume that to double the precision that we have to quadruple the sample size. This exponential increase quickly becomes unmanageable. It is easy to pontificate about averaging over a million targets, but no one is going to shoot that many.) Thus even if <math>\sigma_h \equiv \sigma_v</math> we'd never expect that we'd experimentally get <math>s_h = s_v</math> due to experimental error. <br />
<br />
Second there is the good enough. Shooting by definition is going to have fairly small sample sizes. So if <math>0.66s_h < s_v < 1.5 s_h</math> then, as a rule of thumb, that is probably good enough. Of course for large sample we would want to tighten the window. The harsh reality is that if <math>s_h</math> and <math>s_v</math> could be measured with great precision (e.g. to ten significant figures), then two values would always be statistically significantly different. <br />
<br />
Thus the approximation that <math>\sigma_h \approx \sigma_v</math> will be used unless the variances are known to be statistically significantly different. On the experimental data it is possible to test for a statistically significant difference by using a ratio of <math>s_h^2</math> and <math>s_v^2</math> via the F-Test.<br />
<br />
== Simplifications Reduce Number of Coefficients to Fit ==<br />
<br />
The Hoyt distribution is general enough to be able to fit all four of the special cases in the table above. The point in making special cases of the Hoyt distribution is to reduce the number of coefficients to fit to the data. In general the more coefficients to be fit, the more data is required. Also when fitting multiple coefficients some of the coefficients are determined with greater precision than others. Thus to get a "good" fit for multiple coefficients a lot more data is required not just the minimum. <br />
<br />
Thus to fit the COI at least two shots are required. To fit the constant for the Rayleigh equation another shot would be required for a total of three shots. To fit the Hoyt distribution an additional five shots would be required for a total of seven shots. In reality 10 shots would be required to get a "decent" fit for the Rayleigh distribution, and at least 25 for the Hoyt distribution.<br />
<br />
== Notation in Simplified Cases ==<br />
<br />
The formulas for the distributions in the cases detailed in subsequent parts of this page are given in terms of the population parameters (i.e. <math>\mu_h, \mu_v, \sigma_h, \mbox{and } \sigma_v</math>) rather than the experimentally determined factors (i.e. <math>\bar{h}, \bar{v}, s_h, \mbox{and } s_v</math>) on purpose to emphasize the theoretical nature of the assumptions. Of course the "true" population parameters are unknown, and they could only be estimated with the corresponding experimentally fitted values about which there is some error.<br />
<br />
= Conformance Testing =<br />
<br />
== <math>\rho \approx 0</math> ==<br />
<br />
only way linear least squares<br />
<br />
== <math>\sigma_h \approx \sigma_v </math>==<br />
<br />
# F-Test <math>\frac{s_h^2}{s_v^2}</math>&nbsp;&nbsp;&nbsp;&nbsp; if &nbsp;&nbsp;&nbsp;<math>s_h < s_v</math>&nbsp;&nbsp;&nbsp; else &nbsp;&nbsp;&nbsp;<math>\frac{s_v^2}{s_h^2}</math><br />
# Studentized Ranges<br />
# Chi-Squared <math>(n-1) \frac{s^2}{\hat{\sigma}^2}</math><br />
<br />
= Circular Shot Distribution about COI =<br />
<br />
== Case 1, Rayleigh Distribution == <br />
[[File:raleigh.jpg|250px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
Given: <br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \approx 0</math><br /><br />
then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>f(r) = \frac{r}{\Re^2} e^{-r^2/(2\Re^2)}, \quad r \geq 0,</math> and <math>\Re</math> is the shape factor of the Rayleigh distribution.<br /><br />
<br />
This is really the best case for shot dispersion. Shot groups would be round. <br />
<br />
Strictly, for the Rayleigh distribution to apply, then <math>\sigma_h = \sigma_v</math>, in which case <math>\Re = \sigma_h = \sigma_v</math>. For the "loose" application of the Rayleigh distribution to apply, then <math>\Re \approx (\sigma_h + \sigma_v)/2 \approx \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math>.<br />
<br />
The following statistical measurements are appropriate:<br />
* Circular Error Probable (CEP)<br />
* Covering Circle Radius (CCR)<br />
* Diagonal<br />
* Extreme Spread (ES) <br />
* Figure of Merit (FOM)<br />
* Mean Radius (MR)<br />
* Radial Standard deviation<br />
<br />
'''Notes:'''<br />
# The Diagonal, the Extreme Spread and the FOM are different measurements, even though they conceivable could be based on the same two shots! The Extreme Spread would only depend on the two shots most distant in separation. The the Diagonal and the FOM would depend on two to four shots. For a large number of shots we'd typically expect four different shots to define the extremes for horizontal and vertical deflection.<br />
# For the measures for the CCR, the Diagonal, the GS and the FOM measurements a target would a ragged hole would be acceptable, but for the rest of the measures the {''h,v''} positions of each shot must be known.<br />
# Experimentally the radial distance for each shot, ''i'', is <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
# The conversion to polar coordinates results in each shot having coordinates <math>(r, \theta)</math>. (a) The conversion implicitly assumes that the polar coordinates have been translated so that the center is at the Cartesian Coordinate of the true center of the population <math>(\bar{h}, \bar{v})</math>. (b) The distribution of <math>\theta</math> is assumed to be entirely random and hence irrelevant. This assumption is testable. (c) The distribution is thus converted from a two-variable distribution to a one-variable distribution.<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] Note that there is a conundrum in how we are "averaging" the horizontal and vertical standard deviations to get <math>\sigma_{\Re}</math>. Look at the two expressions. They lead to two choices, either of which may casually seem valid.<br />
* <math>\sigma_h = \sigma_v</math><br />
* <math>\sigma_h^2 = \sigma_v^2</math><br />
<br />
In general if we look at the first formula "averaging" it leads to using:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\Re = \frac{\sigma_h + \sigma_v}{2} = \sigma_h</math> &nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
However in statistics standard deviations are "averaged" (pooled) by taking the square root of the average of their variances:<br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re^2 = {\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br /><br />
&nbsp;&nbsp;&nbsp;<math>\Re = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}} = \sigma_h </math>&nbsp;&nbsp;(with substituting <math>\sigma_h</math> for <math>\sigma_v</math>)<br />
<br />
but:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} = \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
&nbsp;&nbsp;if and only if <math>\sigma_h \equiv \sigma_v</math><br />
<br />
Thus we should take the extent that:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\frac{\sigma_h + \sigma_v}{2} \neq \sqrt{\frac{\sigma_h^2 + \sigma_v^2}{2}}</math><br />
<br />
as a severe warning that we can not push the assumption that <math>\sigma_h \approx \sigma_v</math> too far if we expect the simplification of the general Hoyt distribution to the Rayleigh distribution to give meaningful results. <br />
<br />
The situation is even more tenuous given the small samples that shooters typically use. In general the relative precision of the variance value about a mean is much less precise than the relative precision of the mean value. The statistical test to compare two experimental variance values (i.e. <math>\sigma_h^2, \text{and} \sigma_v^2</math> in our case) is the F-Test which uses the ratios of the variances. For small samples a large difference would need to be observed before the ratio would be statistically significantly because of the imprecision in the individual experimental variance values. <br />
|}<br />
<br />
= Elliptical Shot Distribution about COI =<br />
<br />
== Case 2, Equal variances and correlated == <br />
Given:<br /><br />
#<math>\sigma_h \approx \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
<br />
The following statistical measurement is appropriate:<br />
* Elliptic Error Probable<br />
<br />
==Case 3, Unequal variances and uncorrelated (Orthogonal Elliptical Distribution)== <br />
Given:<br /> <br />
# <math>\sigma_h \neq \sigma_v</math><br /><br />
# <math>\rho \approx 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v}<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
For the purposes of this wiki, this distribution will be called the '''Orthogonal Elliptical Distribution'''. It is obviously a special case of the Hoyt distribution which in turn is a special case of the bivariate normal distribution. <br />
<br />
In order of the model complexity, the following statistical measurements are appropriate:<br />
* Individual Horizontal and Vertical variances<br />
* Elliptic Error Probable<br />
<br />
In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively.<br />
<br />
==Case 4, Unequal variances and correlated (Hoyt Distribution)== <br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
<br />
Given:<br /> <br />
#<math>\sigma_h \neq \sigma_v</math><br /><br />
#<math>\rho \neq 0</math><br /><br />
# The {''h,v''} position of each shot must be known. <br />
then the mathematical formula for the dispersion distribution would be the Hoyt distribution with no simplifications:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Shot groups would be elliptical or egg-shaped if either the horizontal range or vertical range were large. The following statistical measurements are appropriate:<br />
* Elliptic Error Probable<br />
<br />
= Related topics =<br />
<br />
See also the following topics which are closely related:<br />
* [[Error Propagation]] - A basic discussion of how errors propagate when making measurements. <br />
* [[Stringing]] - Definition of stringing and how it can be handled<br />
<br />
= References =<br />
<references /><br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Extreme_Spread&diff=1265Extreme Spread2015-06-17T04:34:35Z<p>Herb: /* Experimental Summary */</p>
<hr />
<div> {|align=right<br />
|__TOC__ <br />
|}<br />
= Experimental Summary =<br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /><br />
All of the (''h'',''v'') positions do not need to be known so a ragged hole will suffice. <br />
|-<br />
| Assumptions<br />
|<br />
* Ideally the shots would have a circular pattern (i.e. the shots would have the Rayleigh Distribution).<br />
** <math>h \sim \mathcal{N}(\bar{h},\sigma_h^2), v \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\Re^2}e^{-r^2/2\Re^2}</math><br />
:: '''Note:''' It is not necessary to calculate the COI, nor the constant <math>\Re</math>, to calculate the Extreme Spread.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Identify two holes, <math>i, j</math> which are the farthest apart and measure <math>ES</math>.<br />
&nbsp;<math>ES = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}</math><br />
|-<br />
| Experimental Measure<br />
| <math>ES</math><br />
|}<br />
<br />
== Given ==<br />
<br />
The requirements for this test are very basic. Just a target with <math>n</math> shots, and some measuring device. Assuming an Extreme spread of under 6 inches then a vernier caliper is used. A measurement is possible to a few thousandths of an inch which is vastly more precision than is usually required. From longer distance a ruler, or perhaps a tape measure.<br />
<br />
== Assumptions ==<br />
<br />
None are needed to make measurement. However some points are worth considering.<br />
<br />
* The same ES measurement could result from a vertical group to a round group. If the shooting process can vary that much then the ES measurement won't give any indication of the change. <br />
<br />
:: If the shot patterns aren't "fairly" round, then using the measurement makes little sense. For instance if muzzle velocity variations are severe, then the vertical range will dominate the ES measurement. Muzzle velocity variations would correlate better with vertical range than with ES. <br />
<br />
* Making assumptions about the dispersion will enable theoretical predictions about the ES measurement. It must be realized that the theoretical solution, assuming the Rayleigh distribution and using Monte Carlo simulation, isn't some arbitrary goal, it is the best case scenario.<br />
<br />
== Data transformation ==<br />
<br />
The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart and measure the distance between them. If the target has a ragged hole it can be a bit tricky, but the edges of the hole should have enough curvature to make shot location possible.<br />
<br />
If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper (cheap is fine!) would be used to measure the distance from the outside edges of the holes, then the bullet caliber subtracted to get a c-t-c measurement. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] A cheap ($10-$20) vernier caliper works fine. There is no need for a $2,000 one that measures to 1/10,000<sup>th</sup> of an inch and has National Bureau of Standards calibration. The vernier caliper is nice for the c-t-c measurement because the knife edges will be parallel and won't obscure the edges of the bullet hole. Thus it is easy to accurately place both of the knife edges on a tangent to the curved bullet holes. <br />
|}<br />
<br />
If using a computer then the center location would be a matter programming. For example a mouse might be used simply to point out the holes, or to drop a dot at the center of the hole, or to drag a circle over the hole. The computer would then make the c-t-c measurement.<br />
<br />
== Experimental Measure ==<br />
<br />
No calculation needs to be done to get the measurement. The single physical measurement is the data sought for the target.<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical Evaluations =<br />
<br />
== Dispersion Follows Rayleigh Distribution ==<br />
<br />
Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>ES</math> Distribution of <math>n</math> shots<br />
|-<br />
| Parameters Needed<br />
| <br />
|-<br />
| <math>PDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| <math>CDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Mode of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Mode increases as number of shots increases. <br />
|-<br />
| Median of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases.<br />
|-<br />
| Mean of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases<br />
|-<br />
| Variance<br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Efficiency<br />
| depends on <math>n</math>, best about 5-7 shots<br />
|-<br />
| (h,v) for all points?<br />
| yes for simulation. <br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric about mean as the number of shots increases. <br />
|}<br />
<br />
<br />
=== Parameters Needed ===<br />
<br />
=== PDF ===<br />
<br />
=== CDF ===<br />
<br />
=== Mode, Median, Mean, Standard Deviation, %Rel Std Dev ===<br />
<br />
Since the distribution is positively skewed: <br />
<blockquote>Mean > Median > Mode</blockquote><br />
<br />
"Normality Error"<br />
As sort of a crude indication of normality let's use the value:<br />
<br />
"Normality Error" = <math> \frac{\frac{CDF(5) - CDF(95)}{2} - Mean}{Mean} {\cdot 100}</math><br />
<br />
So we measure half the distance between the 5<sup>th</sup> percentile and the 95<sup>th</sup> percentile to determine where the Mean should be if the distribution was symmetrical, and determine the % error based on the actual value of the mean. <br />
* + value means positively skewed, <br />
* - value means negatively skewed. <br />
<br />
The point of the "Normality Error" is to give the reader a quinsy feeling about using Student's T-Test for groups with few shots, or the average of a small number of targets. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical ES Values from Monte Carlo Simulation Distribution<br />
|-<br />
! number of shots<br />
! Mode<br />
! Median<br />
! Mean<br />
! "Normality Error"<br />
! Standard<br />
Deviation<br />
! %Rel Std Dev<br />
|-<br />
| 2<br />
| <br />
|<br />
| 1.772<br />
| <br />
| 0.932<br />
| 52.6%<br />
|-<br />
| 3<br />
|<br />
|<br />
| 2.406<br />
| 4.95%<br />
| 0.887<br />
| 36.9%<br />
|-<br />
| 4<br />
|<br />
|<br />
| 2.787<br />
| <br />
| 0.856<br />
| 30.7%<br />
|-<br />
| 5<br />
| <br />
|<br />
| 3.066<br />
| 3.06%<br />
| 0.828<br />
| 27.0%<br />
|-<br />
| 6<br />
| <br />
| <br />
| 3.277<br />
| <br />
| 0.806<br />
|<br />
|-<br />
| 7<br />
| <br />
| <br />
| 3.443<br />
| <br />
| 0.783<br />
|<br />
|-<br />
| 9<br />
|<br />
|<br />
| 3.710<br />
| <br />
| 0.754<br />
|<br />
|-<br />
| 10<br />
| <br />
|<br />
| 3.813<br />
|<br />
| 0.745<br />
|<br />
|-<br />
| 20<br />
|<br />
| <br />
|<br />
| <br />
|<br />
|<br />
|-<br />
| 30<br />
| <br />
|<br />
| 4.788<br />
| 1.63%<br />
| 0.745<br />
| 15.6%<br />
|}<br />
<br />
The tabular values can be used in a number of ways:<br />
<br />
'''Estimate a 95% confidence Interval for Given 2-shot groups based on one ES measuremnt'''<br />
<br />
So a 2-shot group has been measured. If the measured value is accepted as the true value, what would the standard deviation of multiple 2-shot groups be? <br />
<br />
This is another example to warn the reader. Just because you can calculate a standard deviation doesn't mean that a Student's T Test will work. A typical 95% confidence Interval for an individual ES measurement is <math>\pm 1.96 \sigma</math> and for a 2-shot group that is:<br />
:&nbsp;<math>\pm 1.96 \cdot 52.6\% = \pm 103.1\%</math> of the measurement<br />
so the lower confidence interval would be at '''-3.1% !!!''' The nonsensical result is because the distribution is skewed. A negative extreme spread measurement is impossible. It isn't the standard deviation that is wrong, it is the assumption that the confidence interval would be <math>\pm 1.96 \sigma</math> that is the problem. Since the distribution is skewed, the low side of the confidence interval at the 2.5 percentile is at ?? and the high side of the confidence interval at the 97.5 percentile is at +??. <br />
<br />
At 5 shots the T-test is reasonable, and at 10 shots pretty good. <br />
<br />
'''Given ES of one 5-shot group is 1.53 inches'''<br />
<br />
* Estimate ES values for different group sizes.<br />
:: A 3-shot group would be given by measured size times ratios of the Means from the table<br />
:::&nbsp;<math>1.53 \frac{2.406}{3.066} = 1.20</math> inches<br />
:: A 10-shot group would be given by measured size times ratios of Means from the table<br />
:::&nbsp;<math>1.53 \frac{3.813}{3.066} = 1.90</math> inches<br />
<br />
* Estimate the expected standard deviation from the measured ES value<br />
:: The %RSD value for 5-shots is 27.0% so:<br />
:::&nbsp;<math>\hat{s} = 1.53 \text{ inches} \cdot 0.270 = 0.413 \text{ inches} </math><br />
<br />
* Estimate the expected Standard Deviation of the average of 4 targets<br />
:&nbsp;<math>\text{SD}_{ES\ 4 \ Targets}\ = \frac{27.0\%}{\sqrt{4}} = 13.5\% </math><br />
<br />
=== Variance ===<br />
<br />
=== Efficiency ===<br />
[[File:Extreme Spread Relative Efficiency.png|300px|thumb|right|]]<br />
<br />
The efficiency graph on the right is based on the number of groups, hence total shots, to get a 10% confidence interval. <br />
The efficiency depends on <math>n</math>, but it is best about 5-7 shots. Essentially there are two competing factors. First as the number of shots increases then the midpoint of the line segment which defines the MR is, on average, closer to the COI which improves efficiency. Second as the number of shots increases then it is increasingly unlikely that the next shot will increase the MR which decreases efficiency. The product of these two factors thus peaks at about 5-7 shots.<br />
<br />
Notice too that the figure utilizes fractional groups to define a smoother curve. In reality, especially for small samples, using all the shots would be important. Thus for ammunition which is packaged as 20 cartridges per box, then using 8 shot groups leaves 4 cartridges unused which is 20% "inefficient." <br />
<br />
This result is also assuming no fliers. If 5-7 shots is likely to give groups with multiple fliers, then less shots per group might be better.<br />
<br />
So what is the optimal number of shots per group? '''''It depends...'''''<br />
<br />
=== Outlier Tests ===<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - A discussion of the different cases for projectile dispersion<br />
<br />
Other measurements practical for range use are: <br />
<br />
* [[Covering Circle Radius]] - about same precision as Extreme Spread if Rayleigh distributed<br />
* [[Diagonal]] - somewhat better precision than Extreme Spread if Rayleigh distributed<br />
* [[Figure of Merit]] - somewhat better precision than Extreme Spread if Rayleigh distributed</div>Herbhttp://ballistipedia.com/index.php?title=Extreme_Spread&diff=1264Extreme Spread2015-06-16T15:14:38Z<p>Herb: /* Efficiency */</p>
<hr />
<div> {|align=right<br />
|__TOC__ <br />
|}<br />
= Experimental Summary =<br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /><br />
All of the (''h'',''v'') positions do not need to be known so a ragged hole will suffice. <br />
|-<br />
| Assumptions<br />
|<br />
* Ideally the shots would have a circular pattern (i.e. the shots would have the Rayleigh Distribution).<br />
** <math>h \sim \mathcal{N}(\bar{h},\sigma_h^2), v \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to calculate the COI, nor fit <math>\sigma</math> to calculate the Extreme Spread.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Identify two holes, <math>i, j</math> which are the farthest apart and measure <math>ES</math>.<br />
&nbsp;<math>ES = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}</math><br />
|-<br />
| Experimental Measure<br />
| <math>ES</math><br />
|}<br />
<br />
== Given ==<br />
<br />
The requirements for this test are very basic. Just a target with <math>n</math> shots, and some measuring device. Assuming an Extreme spread of under 6 inches then a vernier caliper is used. A measurement is possible to a few thousandths of an inch which is vastly more precision than is usually required. From longer distance a ruler, or perhaps a tape measure.<br />
<br />
== Assumptions ==<br />
<br />
None are needed to make measurement. However some points are worth considering.<br />
<br />
* The same ES measurement could result from a vertical group to a round group. If the shooting process can vary that much then the ES measurement won't give any indication of the change. <br />
<br />
:: If the shot patterns aren't "fairly" round, then using the measurement makes little sense. For instance if muzzle velocity variations are severe, then the vertical range will dominate the ES measurement. Muzzle velocity variations would correlate better with vertical range than with ES. <br />
<br />
* Making assumptions about the dispersion will enable theoretical predictions about the ES measurement. It must be realized that the theoretical solution, assuming the Rayleigh distribution and using Monte Carlo simulation, isn't some arbitrary goal, it is the best case scenario.<br />
<br />
== Data transformation ==<br />
<br />
The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart and measure the distance between them. If the target has a ragged hole it can be a bit tricky, but the edges of the hole should have enough curvature to make shot location possible.<br />
<br />
If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper (cheap is fine!) would be used to measure the distance from the outside edges of the holes, then the bullet caliber subtracted to get a c-t-c measurement. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] A cheap ($10-$20) vernier caliper works fine. There is no need for a $2,000 one that measures to 1/10,000<sup>th</sup> of an inch and has National Bureau of Standards calibration. The vernier caliper is nice for the c-t-c measurement because the knife edges will be parallel and won't obscure the edges of the bullet hole. Thus it is easy to accurately place both of the knife edges on a tangent to the curved bullet holes. <br />
|}<br />
<br />
If using a computer then the center location would be a matter programming. For example a mouse might be used simply to point out the holes, or to drop a dot at the center of the hole, or to drag a circle over the hole. The computer would then make the c-t-c measurement.<br />
<br />
== Experimental Measure ==<br />
<br />
No calculation needs to be done to get the measurement. The single physical measurement is the data sought for the target.<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical Evaluations =<br />
<br />
== Dispersion Follows Rayleigh Distribution ==<br />
<br />
Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>ES</math> Distribution of <math>n</math> shots<br />
|-<br />
| Parameters Needed<br />
| <br />
|-<br />
| <math>PDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| <math>CDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Mode of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Mode increases as number of shots increases. <br />
|-<br />
| Median of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases.<br />
|-<br />
| Mean of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases<br />
|-<br />
| Variance<br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Efficiency<br />
| depends on <math>n</math>, best about 5-7 shots<br />
|-<br />
| (h,v) for all points?<br />
| yes for simulation. <br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric about mean as the number of shots increases. <br />
|}<br />
<br />
<br />
=== Parameters Needed ===<br />
<br />
=== PDF ===<br />
<br />
=== CDF ===<br />
<br />
=== Mode, Median, Mean, Standard Deviation, %Rel Std Dev ===<br />
<br />
Since the distribution is positively skewed: <br />
<blockquote>Mean > Median > Mode</blockquote><br />
<br />
"Normality Error"<br />
As sort of a crude indication of normality let's use the value:<br />
<br />
"Normality Error" = <math> \frac{\frac{CDF(5) - CDF(95)}{2} - Mean}{Mean} {\cdot 100}</math><br />
<br />
So we measure half the distance between the 5<sup>th</sup> percentile and the 95<sup>th</sup> percentile to determine where the Mean should be if the distribution was symmetrical, and determine the % error based on the actual value of the mean. <br />
* + value means positively skewed, <br />
* - value means negatively skewed. <br />
<br />
The point of the "Normality Error" is to give the reader a quinsy feeling about using Student's T-Test for groups with few shots, or the average of a small number of targets. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical ES Values from Monte Carlo Simulation Distribution<br />
|-<br />
! number of shots<br />
! Mode<br />
! Median<br />
! Mean<br />
! "Normality Error"<br />
! Standard<br />
Deviation<br />
! %Rel Std Dev<br />
|-<br />
| 2<br />
| <br />
|<br />
| 1.772<br />
| <br />
| 0.932<br />
| 52.6%<br />
|-<br />
| 3<br />
|<br />
|<br />
| 2.406<br />
| 4.95%<br />
| 0.887<br />
| 36.9%<br />
|-<br />
| 4<br />
|<br />
|<br />
| 2.787<br />
| <br />
| 0.856<br />
| 30.7%<br />
|-<br />
| 5<br />
| <br />
|<br />
| 3.066<br />
| 3.06%<br />
| 0.828<br />
| 27.0%<br />
|-<br />
| 6<br />
| <br />
| <br />
| 3.277<br />
| <br />
| 0.806<br />
|<br />
|-<br />
| 7<br />
| <br />
| <br />
| 3.443<br />
| <br />
| 0.783<br />
|<br />
|-<br />
| 9<br />
|<br />
|<br />
| 3.710<br />
| <br />
| 0.754<br />
|<br />
|-<br />
| 10<br />
| <br />
|<br />
| 3.813<br />
|<br />
| 0.745<br />
|<br />
|-<br />
| 20<br />
|<br />
| <br />
|<br />
| <br />
|<br />
|<br />
|-<br />
| 30<br />
| <br />
|<br />
| 4.788<br />
| 1.63%<br />
| 0.745<br />
| 15.6%<br />
|}<br />
<br />
The tabular values can be used in a number of ways:<br />
<br />
'''Estimate a 95% confidence Interval for Given 2-shot groups based on one ES measuremnt'''<br />
<br />
So a 2-shot group has been measured. If the measured value is accepted as the true value, what would the standard deviation of multiple 2-shot groups be? <br />
<br />
This is another example to warn the reader. Just because you can calculate a standard deviation doesn't mean that a Student's T Test will work. A typical 95% confidence Interval for an individual ES measurement is <math>\pm 1.96 \sigma</math> and for a 2-shot group that is:<br />
:&nbsp;<math>\pm 1.96 \cdot 52.6\% = \pm 103.1\%</math> of the measurement<br />
so the lower confidence interval would be at '''-3.1% !!!''' The nonsensical result is because the distribution is skewed. A negative extreme spread measurement is impossible. It isn't the standard deviation that is wrong, it is the assumption that the confidence interval would be <math>\pm 1.96 \sigma</math> that is the problem. Since the distribution is skewed, the low side of the confidence interval at the 2.5 percentile is at ?? and the high side of the confidence interval at the 97.5 percentile is at +??. <br />
<br />
At 5 shots the T-test is reasonable, and at 10 shots pretty good. <br />
<br />
'''Given ES of one 5-shot group is 1.53 inches'''<br />
<br />
* Estimate ES values for different group sizes.<br />
:: A 3-shot group would be given by measured size times ratios of the Means from the table<br />
:::&nbsp;<math>1.53 \frac{2.406}{3.066} = 1.20</math> inches<br />
:: A 10-shot group would be given by measured size times ratios of Means from the table<br />
:::&nbsp;<math>1.53 \frac{3.813}{3.066} = 1.90</math> inches<br />
<br />
* Estimate the expected standard deviation from the measured ES value<br />
:: The %RSD value for 5-shots is 27.0% so:<br />
:::&nbsp;<math>\hat{s} = 1.53 \text{ inches} \cdot 0.270 = 0.413 \text{ inches} </math><br />
<br />
* Estimate the expected Standard Deviation of the average of 4 targets<br />
:&nbsp;<math>\text{SD}_{ES\ 4 \ Targets}\ = \frac{27.0\%}{\sqrt{4}} = 13.5\% </math><br />
<br />
=== Variance ===<br />
<br />
=== Efficiency ===<br />
[[File:Extreme Spread Relative Efficiency.png|300px|thumb|right|]]<br />
<br />
The efficiency graph on the right is based on the number of groups, hence total shots, to get a 10% confidence interval. <br />
The efficiency depends on <math>n</math>, but it is best about 5-7 shots. Essentially there are two competing factors. First as the number of shots increases then the midpoint of the line segment which defines the MR is, on average, closer to the COI which improves efficiency. Second as the number of shots increases then it is increasingly unlikely that the next shot will increase the MR which decreases efficiency. The product of these two factors thus peaks at about 5-7 shots.<br />
<br />
Notice too that the figure utilizes fractional groups to define a smoother curve. In reality, especially for small samples, using all the shots would be important. Thus for ammunition which is packaged as 20 cartridges per box, then using 8 shot groups leaves 4 cartridges unused which is 20% "inefficient." <br />
<br />
This result is also assuming no fliers. If 5-7 shots is likely to give groups with multiple fliers, then less shots per group might be better.<br />
<br />
So what is the optimal number of shots per group? '''''It depends...'''''<br />
<br />
=== Outlier Tests ===<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - A discussion of the different cases for projectile dispersion<br />
<br />
Other measurements practical for range use are: <br />
<br />
* [[Covering Circle Radius]] - about same precision as Extreme Spread if Rayleigh distributed<br />
* [[Diagonal]] - somewhat better precision than Extreme Spread if Rayleigh distributed<br />
* [[Figure of Merit]] - somewhat better precision than Extreme Spread if Rayleigh distributed</div>Herbhttp://ballistipedia.com/index.php?title=Extreme_Spread&diff=1263Extreme Spread2015-06-16T14:50:30Z<p>Herb: /* Experimental Measure */</p>
<hr />
<div> {|align=right<br />
|__TOC__ <br />
|}<br />
= Experimental Summary =<br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /><br />
All of the (''h'',''v'') positions do not need to be known so a ragged hole will suffice. <br />
|-<br />
| Assumptions<br />
|<br />
* Ideally the shots would have a circular pattern (i.e. the shots would have the Rayleigh Distribution).<br />
** <math>h \sim \mathcal{N}(\bar{h},\sigma_h^2), v \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to calculate the COI, nor fit <math>\sigma</math> to calculate the Extreme Spread.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Identify two holes, <math>i, j</math> which are the farthest apart and measure <math>ES</math>.<br />
&nbsp;<math>ES = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}</math><br />
|-<br />
| Experimental Measure<br />
| <math>ES</math><br />
|}<br />
<br />
== Given ==<br />
<br />
The requirements for this test are very basic. Just a target with <math>n</math> shots, and some measuring device. Assuming an Extreme spread of under 6 inches then a vernier caliper is used. A measurement is possible to a few thousandths of an inch which is vastly more precision than is usually required. From longer distance a ruler, or perhaps a tape measure.<br />
<br />
== Assumptions ==<br />
<br />
None are needed to make measurement. However some points are worth considering.<br />
<br />
* The same ES measurement could result from a vertical group to a round group. If the shooting process can vary that much then the ES measurement won't give any indication of the change. <br />
<br />
:: If the shot patterns aren't "fairly" round, then using the measurement makes little sense. For instance if muzzle velocity variations are severe, then the vertical range will dominate the ES measurement. Muzzle velocity variations would correlate better with vertical range than with ES. <br />
<br />
* Making assumptions about the dispersion will enable theoretical predictions about the ES measurement. It must be realized that the theoretical solution, assuming the Rayleigh distribution and using Monte Carlo simulation, isn't some arbitrary goal, it is the best case scenario.<br />
<br />
== Data transformation ==<br />
<br />
The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart and measure the distance between them. If the target has a ragged hole it can be a bit tricky, but the edges of the hole should have enough curvature to make shot location possible.<br />
<br />
If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper (cheap is fine!) would be used to measure the distance from the outside edges of the holes, then the bullet caliber subtracted to get a c-t-c measurement. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] A cheap ($10-$20) vernier caliper works fine. There is no need for a $2,000 one that measures to 1/10,000<sup>th</sup> of an inch and has National Bureau of Standards calibration. The vernier caliper is nice for the c-t-c measurement because the knife edges will be parallel and won't obscure the edges of the bullet hole. Thus it is easy to accurately place both of the knife edges on a tangent to the curved bullet holes. <br />
|}<br />
<br />
If using a computer then the center location would be a matter programming. For example a mouse might be used simply to point out the holes, or to drop a dot at the center of the hole, or to drag a circle over the hole. The computer would then make the c-t-c measurement.<br />
<br />
== Experimental Measure ==<br />
<br />
No calculation needs to be done to get the measurement. The single physical measurement is the data sought for the target.<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical Evaluations =<br />
<br />
== Dispersion Follows Rayleigh Distribution ==<br />
<br />
Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>ES</math> Distribution of <math>n</math> shots<br />
|-<br />
| Parameters Needed<br />
| <br />
|-<br />
| <math>PDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| <math>CDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Mode of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Mode increases as number of shots increases. <br />
|-<br />
| Median of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases.<br />
|-<br />
| Mean of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases<br />
|-<br />
| Variance<br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Efficiency<br />
| depends on <math>n</math>, best about 5-7 shots<br />
|-<br />
| (h,v) for all points?<br />
| yes for simulation. <br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric about mean as the number of shots increases. <br />
|}<br />
<br />
<br />
=== Parameters Needed ===<br />
<br />
=== PDF ===<br />
<br />
=== CDF ===<br />
<br />
=== Mode, Median, Mean, Standard Deviation, %Rel Std Dev ===<br />
<br />
Since the distribution is positively skewed: <br />
<blockquote>Mean > Median > Mode</blockquote><br />
<br />
"Normality Error"<br />
As sort of a crude indication of normality let's use the value:<br />
<br />
"Normality Error" = <math> \frac{\frac{CDF(5) - CDF(95)}{2} - Mean}{Mean} {\cdot 100}</math><br />
<br />
So we measure half the distance between the 5<sup>th</sup> percentile and the 95<sup>th</sup> percentile to determine where the Mean should be if the distribution was symmetrical, and determine the % error based on the actual value of the mean. <br />
* + value means positively skewed, <br />
* - value means negatively skewed. <br />
<br />
The point of the "Normality Error" is to give the reader a quinsy feeling about using Student's T-Test for groups with few shots, or the average of a small number of targets. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical ES Values from Monte Carlo Simulation Distribution<br />
|-<br />
! number of shots<br />
! Mode<br />
! Median<br />
! Mean<br />
! "Normality Error"<br />
! Standard<br />
Deviation<br />
! %Rel Std Dev<br />
|-<br />
| 2<br />
| <br />
|<br />
| 1.772<br />
| <br />
| 0.932<br />
| 52.6%<br />
|-<br />
| 3<br />
|<br />
|<br />
| 2.406<br />
| 4.95%<br />
| 0.887<br />
| 36.9%<br />
|-<br />
| 4<br />
|<br />
|<br />
| 2.787<br />
| <br />
| 0.856<br />
| 30.7%<br />
|-<br />
| 5<br />
| <br />
|<br />
| 3.066<br />
| 3.06%<br />
| 0.828<br />
| 27.0%<br />
|-<br />
| 6<br />
| <br />
| <br />
| 3.277<br />
| <br />
| 0.806<br />
|<br />
|-<br />
| 7<br />
| <br />
| <br />
| 3.443<br />
| <br />
| 0.783<br />
|<br />
|-<br />
| 9<br />
|<br />
|<br />
| 3.710<br />
| <br />
| 0.754<br />
|<br />
|-<br />
| 10<br />
| <br />
|<br />
| 3.813<br />
|<br />
| 0.745<br />
|<br />
|-<br />
| 20<br />
|<br />
| <br />
|<br />
| <br />
|<br />
|<br />
|-<br />
| 30<br />
| <br />
|<br />
| 4.788<br />
| 1.63%<br />
| 0.745<br />
| 15.6%<br />
|}<br />
<br />
The tabular values can be used in a number of ways:<br />
<br />
'''Estimate a 95% confidence Interval for Given 2-shot groups based on one ES measuremnt'''<br />
<br />
So a 2-shot group has been measured. If the measured value is accepted as the true value, what would the standard deviation of multiple 2-shot groups be? <br />
<br />
This is another example to warn the reader. Just because you can calculate a standard deviation doesn't mean that a Student's T Test will work. A typical 95% confidence Interval for an individual ES measurement is <math>\pm 1.96 \sigma</math> and for a 2-shot group that is:<br />
:&nbsp;<math>\pm 1.96 \cdot 52.6\% = \pm 103.1\%</math> of the measurement<br />
so the lower confidence interval would be at '''-3.1% !!!''' The nonsensical result is because the distribution is skewed. A negative extreme spread measurement is impossible. It isn't the standard deviation that is wrong, it is the assumption that the confidence interval would be <math>\pm 1.96 \sigma</math> that is the problem. Since the distribution is skewed, the low side of the confidence interval at the 2.5 percentile is at ?? and the high side of the confidence interval at the 97.5 percentile is at +??. <br />
<br />
At 5 shots the T-test is reasonable, and at 10 shots pretty good. <br />
<br />
'''Given ES of one 5-shot group is 1.53 inches'''<br />
<br />
* Estimate ES values for different group sizes.<br />
:: A 3-shot group would be given by measured size times ratios of the Means from the table<br />
:::&nbsp;<math>1.53 \frac{2.406}{3.066} = 1.20</math> inches<br />
:: A 10-shot group would be given by measured size times ratios of Means from the table<br />
:::&nbsp;<math>1.53 \frac{3.813}{3.066} = 1.90</math> inches<br />
<br />
* Estimate the expected standard deviation from the measured ES value<br />
:: The %RSD value for 5-shots is 27.0% so:<br />
:::&nbsp;<math>\hat{s} = 1.53 \text{ inches} \cdot 0.270 = 0.413 \text{ inches} </math><br />
<br />
* Estimate the expected Standard Deviation of the average of 4 targets<br />
:&nbsp;<math>\text{SD}_{ES\ 4 \ Targets}\ = \frac{27.0\%}{\sqrt{4}} = 13.5\% </math><br />
<br />
=== Variance ===<br />
<br />
=== Efficiency ===<br />
[[File:Extreme Spread Relative Efficiency.png|300px|thumb|right|]]<br />
<br />
The efficiency depends on <math>n</math>, but it is best about 5-7 shots. Essentially there are two competing factors. First as the number of shots increases then the midpoint of the line segment which defines the MR is, on average, closer to the COI which improves efficiency. Second as the number of shots increases then it is increasingly unlikely that the next shot will increase the MR which decreases efficiency. The product of these two factors thus peaks at about 5-7 shots.<br />
<br />
Notice too that the figure is assuming one group of shots. If 8 shots were used then four groups of 2-shots per group could have been made, or one group of 8-shots per group. Using 2-shots per group though requires four targets. If ammunition comes 20 cartridges per box then using 8 shot groups leaves 4 cartridges unused. <br />
<br />
This result is assuming no fliers. If 5-7 shots is likely to give groups with multiple fliers, then less shots per group might be better.<br />
<br />
So what is the optimal number of shots per group? '''''It depends...'''''<br />
<br />
=== Outlier Tests ===<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - A discussion of the different cases for projectile dispersion<br />
<br />
Other measurements practical for range use are: <br />
<br />
* [[Covering Circle Radius]] - about same precision as Extreme Spread if Rayleigh distributed<br />
* [[Diagonal]] - somewhat better precision than Extreme Spread if Rayleigh distributed<br />
* [[Figure of Merit]] - somewhat better precision than Extreme Spread if Rayleigh distributed</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1262Herb References2015-06-15T23:24:09Z<p>Herb: added ref links & added type to links</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| (Abstract @ http://www.jstor.org/stable/2282775)]<br />
<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 (Abstract @ http://www.jstor.org/stable/1266181)]<br />
<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 (PDF @ http://handle.dtic.mil/100.2/ADA059117)]<br />
<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333. [http://www.jstor.org/stable/2346911?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2346911?seq=1#page_scan_tab_contents)] [http://www.robertnz.net/pdf/lc_chisq.pdf (PDF @ http://www.robertnz.net/pdf/lc_chisq.pdf)]<br />
<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf (PDF @ http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf)]<br />
<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA476368 (PDF @ http://handle.dtic.mil/100.2/ADA476368)]<br />
<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA476368 (PDF @ http://handle.dtic.mil/100.2/ADA476368)] <br />
<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. [http://www.jstor.org/stable/2003026 (READ @ http://www.jstor.org/stable/2003026)] [http://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0129116-8/ (PDF @ http://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0129116-8/)]<br />
<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. [http://www.jstor.org/stable/2004054 (READ @ http://www.jstor.org/stable/2004054)]<br />
<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory. [http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0270739 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0270739)]<br />
<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862. [http://www.sciencedirect.com/science/article/pii/S0167947309004381 (Abstract @ http://www.sciencedirect.com/science/article/pii/S0167947309004381)]<br />
<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. [http://handle.dtic.mil/100.2/ADA172498 (PDF @ http://handle.dtic.mil/100.2/ADA172498)]<br />
<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. [http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257)]<br />
<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339. [http://www.jstor.org/stable/2347721?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2347721?seq=1#page_scan_tab_contents)] [http://www.robertnz.net/pdf/lc_chisq.pdf (PDF @ http://www.robertnz.net/pdf/lc_chisq.pdf)]<br />
<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309. [http://www.jstor.org/stable/2347778?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2347778?seq=1#page_scan_tab_contents)] <br />
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* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. [http://earth-info.nga.mil/GandG/publications/tr96.pdf (PDF @ http://earth-info.nga.mil/GandG/publications/tr96.pdf]<br />
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* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. [https://projecteuclid.org/euclid.aoms/1177731316 (PDF @ https://projecteuclid.org/euclid.aoms/1177731316)]<br />
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* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. [http://www.jstor.org/stable/167752 (Abstract @ http://www.jstor.org/stable/167752)]<br />
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* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| (Cached private monograph)]]. <br />
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* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. [http://projecteuclid.org/euclid.aoms/1177703747 (PDF @ http://projecteuclid.org/euclid.aoms/1177703747)]<br />
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* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. [http://www.jstor.org/stable/2281595 (Abstract @ http://www.jstor.org/stable/2281595)]<br />
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* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684 (PDF @ http://projecteuclid.org/euclid.aoms/1177705684)]<br />
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* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. [http://www.jstor.org/stable/2283768 (Abstract @ http://www.jstor.org/stable/2283768)]<br />
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* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. [http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf (PDF @ http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf)]<br />
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* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. [https://archive.org/details/bstj26-2-318 (PDF @ https://archive.org/details/bstj26-2-318)]<br />
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* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. [http://www.jstor.org/stable/2332763 (Read @ http://www.jstor.org/stable/2332763)]<br />
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* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. [http://www.jstor.org/stable/2282450 (Abstract @ http://www.jstor.org/stable/2282450)]<br />
<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached PDF)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
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* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856. [http://www.sciencedirect.com/science/article/pii/S0167947308005653 (Abstract @ http://www.sciencedirect.com/science/article/pii/S0167947308005653)] [http://www4.stat.ncsu.edu/~hzhang2/paper/chisq.pdf (PDF @ http://www4.stat.ncsu.edu/~hzhang2/paper/chisq.pdf)]<br />
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* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. (Version 1.01) [http://statshooting.com/papers/measuring-cep-mcmillan2008.pdf (PDF @ http://statshooting.com/papers/measuring-cep-mcmillan2008.pdf)]<br />
<br />
* <div id="Molon2006">Molon (2006). The Trouble With 3-Shot Groups. [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 (Webpage @ http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218)] (ID of poster??)<br />
<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. [http://www.jstor.org/stable/2282503 (Abstract @ http://www.jstor.org/stable/2282503)]<br />
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* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. [http://www.jstor.org/stable/2281596 (Abstract @ http://www.jstor.org/stable/2281596)]<br />
<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. [http://handle.dtic.mil/100.2/ADA199190 (PDF @ http://handle.dtic.mil/100.2/ADA199190)]<br />
<br />
* <div id="Nuttall1975a"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. Report: AD0779846 Naval Underwater Systems Center, New London Laboratory, New London. Connecticut [http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0779846 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0779846)]<br />
<br />
* <div id="Nuttall1975b"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96. [http://dx.doi.org/10.1109/TIT.1975.1055327 (Abstract @ http://dx.doi.org/10.1109/TIT.1975.1055327)]<br />
<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4784312 (Abstract @ http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4784312)] [http://dx.doi.org/10.1049/el.2009.0828 (Erratum Notice, Electronics Letters, 45 (8), 432. @ http://dx.doi.org/10.1049/el.2009.0828)]<br />
<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. [http://www.jstor.org/stable/2332542 (Read @ http://www.jstor.org/stable/2332542)]<br />
<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. [http://www.jstor.org/stable/2333533 (Read @ http://www.jstor.org/stable/2333533)]<br />
<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. {http://handle.dtic.mil/100.2/ADA248105 (PDF @ http://handle.dtic.mil/100.2/ADA248105)]<br />
<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024. [http://dx.doi.org/10.1109/7.220962 (Abstract @ http://dx.doi.org/10.1109/7.220962)]<br />
<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. [http://www.rand.org/pubs/reports/2008/R234.pdf (PDF @ http://www.rand.org/pubs/reports/2008/R234.pdf)]<br />
<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. [http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf (PDF @ http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf)]<br />
<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98. [http://www.jstor.org/stable/2346884 (Read @ http://www.jstor.org/stable/2346884)]<br />
<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
<br />
* <div id="Siddiqui1961"></div>Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(Cached PDF)]]<br />
<br />
* <div id="Siddiqui1964"></div>Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(Cached PDF)]]<br />
<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. [http://www.jstor.org/stable/25052751 (Abstract @ http://www.jstor.org/stable/25052751)]<br />
<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. [http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html (HTML Fulltext @ http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html)]<br />
<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its estimation. Metrika, 15 (1), 149–163. [http://link.springer.com/article/10.1007%2FBF02613568 (Abstract @ http://link.springer.com/article/10.1007%2FBF02613568)]<br />
<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. [http://www.jstor.org/stable/2290205 (Abstract @ http://www.jstor.org/stable/2290205)]<br />
<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA153828 (PDF @ http://handle.dtic.mil/100.2/ADA153828)]<br />
<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. [http://handle.dtic.mil/100.2/ADA153828 (PDF @ http://handle.dtic.mil/100.2/ADA153828)]<br />
<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Memorandum Rept. ADA006586, Army Ballistic Research Lab, Aberdeen Proving Ground [http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf (PDF @ http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf)]<br />
<br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. [http://dx.doi.org/10.1002/nav.3800220407 Abstract @ http://dx.doi.org/10.1002/nav.3800220407] [https://archive.org/details/navalresearchlog2241975offi (PDF of Naval Logistics Quarterly issue @ https://archive.org/details/navalresearchlog2241975offi0]<br />
<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. [http://handle.dtic.mil/100.2/AD0759989 (PDF @ http://handle.dtic.mil/100.2/AD0759989)]<br />
<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. [http://handle.dtic.mil/100.2/ADA266528 (PDF @ http://handle.dtic.mil/100.2/ADA266528)]<br />
<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960. [http://www.scientific.net/AMM.341-342.955 (Abstract @ http://www.scientific.net/AMM.341-342.955)]<br />
<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228. [http://www.scientific.net/AMR.765-767.2224 (Abstract @ http://www.scientific.net/AMR.765-767.2224)]<br />
<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. [http://handle.dtic.mil/100.2/ADA324337 (PDF @ http://handle.dtic.mil/100.2/ADA324337)]<br />
<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. [http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797 (Abstract @ http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797)]<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
Siddiqui had a two part series on the Rayleigh distribution ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).<br />
<br />
=== Derivation ===<br />
<br />
== Sample Range ==<br />
[[#Harter1960b | Harter (1964) ]] gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentized range. <br />
<br />
= Reference Data =<br />
<br />
* [[File:Confidence Interval Convergence.xlsx]]: Shows how precision confidence intervals shrink as sample size increases.<br />
<br />
* [[File:Sigma1RangeStatistics.xls]]: Simulated median, 50%, 80%, and 95% quantiles, plus first four sample moments, for shot groups containing 2 to 100 shots, of: Extreme Spread, Diagonal, Figure of Merit.<br />
<br />
* [[File:SymmetricBivariateSigma1.xls]]: Monte Carlo simulation results validating the [[Closed Form Precision]] math.<br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1261Herb References2015-06-15T03:47:13Z<p>Herb: updated link in McMillan ref</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| Abstract @ http://www.jstor.org/stable/2282775]<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 Abstract @ http://www.jstor.org/stable/1266181]<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 PDF @ http://handle.dtic.mil/100.2/ADA059117]<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684 PDF @ http://projecteuclid.org/euclid.aoms/1177705684 ]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. (Version 1.01) [http://statshooting.com/papers/measuring-cep-mcmillan2008.pdf PDF @ http://statshooting.com/papers/measuring-cep-mcmillan2008.pdf]<br />
* <div id="Molon2006">Molon (2006). The Trouble With 3-Shot Groups. [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 webpage @ http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218] (ID of poster??)<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Siddiqui1961"></div>Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(cached PDF)]]<br />
* <div id="Siddiqui1964"></div>Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(cached PDF)]]<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Memorandum Rept. ADA006586, Army Ballistic Research Lab, Aberdeen Proving Ground [http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf pdf @ http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf]<br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. [http://dx.doi.org/10.1002/nav.3800220407 Abstract @ http://dx.doi.org/10.1002/nav.3800220407] [https://archive.org/details/navalresearchlog2241975offi pdf of Naval Logistics Quarterly issue @ https://archive.org/details/navalresearchlog2241975offi]<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
Siddiqui had a two part series on the Rayleigh distribution ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).<br />
<br />
=== Derivation ===<br />
<br />
== Sample Range ==<br />
[[#Harter1960b | Harter (1964) ]] gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentized range. <br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1260Herb References2015-06-15T03:32:02Z<p>Herb: fixed two grubbs entries</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| Abstract @ http://www.jstor.org/stable/2282775]<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 Abstract @ http://www.jstor.org/stable/1266181]<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 PDF @ http://handle.dtic.mil/100.2/ADA059117]<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684 PDF @ http://projecteuclid.org/euclid.aoms/1177705684 ]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Molon2006">Molon (2006). The Trouble With 3-Shot Groups. [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 webpage @ http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218] (ID of poster??)<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Siddiqui1961"></div>Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(cached PDF)]]<br />
* <div id="Siddiqui1964"></div>Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(cached PDF)]]<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Memorandum Rept. ADA006586, Army Ballistic Research Lab, Aberdeen Proving Ground [http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf pdf @ http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf]<br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. [http://dx.doi.org/10.1002/nav.3800220407 Abstract @ http://dx.doi.org/10.1002/nav.3800220407] [https://archive.org/details/navalresearchlog2241975offi pdf of Naval Logistics Quarterly issue @ https://archive.org/details/navalresearchlog2241975offi]<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
Siddiqui had a two part series on the Rayleigh distribution ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).<br />
<br />
=== Derivation ===<br />
<br />
== Sample Range ==<br />
[[#Harter1960b | Harter (1964) ]] gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentized range. <br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1259Herb References2015-06-15T02:51:46Z<p>Herb: /* Sample Range */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| Abstract @ http://www.jstor.org/stable/2282775]<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 Abstract @ http://www.jstor.org/stable/1266181]<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 PDF @ http://handle.dtic.mil/100.2/ADA059117]<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684 PDF @ http://projecteuclid.org/euclid.aoms/1177705684 ]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Molon2006">Molon (2006). The Trouble With 3-Shot Groups. [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 webpage @ http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218] (ID of poster??)<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Siddiqui1961"></div>Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(cached PDF)]]<br />
* <div id="Siddiqui1964"></div>Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(cached PDF)]]<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf <br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
Siddiqui had a two part series on the Rayleigh distribution ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).<br />
<br />
=== Derivation ===<br />
<br />
== Sample Range ==<br />
[[#Harter1960b | Harter (1964) ]] gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentized range. <br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1258Herb References2015-06-15T01:23:43Z<p>Herb: </p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| Abstract @ http://www.jstor.org/stable/2282775]<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 Abstract @ http://www.jstor.org/stable/1266181]<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 PDF @ http://handle.dtic.mil/100.2/ADA059117]<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684 PDF @ http://projecteuclid.org/euclid.aoms/1177705684 ]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Molon2006">Molon (2006). The Trouble With 3-Shot Groups. [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 webpage @ http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218] (ID of poster??)<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Siddiqui1961"></div>Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(cached PDF)]]<br />
* <div id="Siddiqui1964"></div>Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(cached PDF)]]<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf <br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
Siddiqui had a two part series on the Rayleigh distribution ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).<br />
<br />
=== Derivation ===<br />
<br />
== Sample Range ==<br />
[[#Harter1960b | Harter (1964) ]] gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentship range. <br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1257Herb References2015-06-14T23:17:11Z<p>Herb: /* References */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. [http://www.jstor.org/stable/2282775| Abstract @ http://www.jstor.org/stable/2282775]<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. [http://www.jstor.org/stable/1266181 Abstract @ http://www.jstor.org/stable/1266181]<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. [http://handle.dtic.mil/100.2/ADA059117 PDF @ http://handle.dtic.mil/100.2/ADA059117]<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf <br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
=== Derivation ===<br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Molon (2006). [[Prior_Art#Molon.2C_2006.2C_The_Trouble_With_3-Shot_Groups|'''The Trouble With 3-Shot Groups''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].<br />
<br />
* Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|'''(cached PDF)''']]<br />
<br />
* Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|'''(cached PDF)''']]</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1256Herb References2015-06-14T22:50:53Z<p>Herb: </p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics. <br />
<br />
* <div id="Blischke1966"></div>[Blischke1966]<br />
: Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf <br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
=== Derivation ===<br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Molon (2006). [[Prior_Art#Molon.2C_2006.2C_The_Trouble_With_3-Shot_Groups|'''The Trouble With 3-Shot Groups''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].<br />
<br />
* Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|'''(cached PDF)''']]<br />
<br />
* Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|'''(cached PDF)''']]</div>Herbhttp://ballistipedia.com/index.php?title=Mean_Radius&diff=1255Mean Radius2015-06-14T21:45:01Z<p>Herb: </p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Mean Radius<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
= Experimental Summary =<br />
<br />
yada yada <br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* The dispersion of shot <math>i</math> follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius.<br />
** <math>h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to fit <math>\sigma</math> to calculate the Mean Radius.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Measure positions <math>(h_i, v_i)</math> for each shot, <math>i</math>.<br />
|-<br />
| Experimental Measure<br />
| <math>MR = \overline{r_n}</math> method<br />
Preliminary Cartesian Calculations<br />
* <math>\bar{h} = \frac{1}{n} \sum_{i=1}^n h_i </math><br />
* <math>\bar{v} = \frac{1}{n} \sum_{i=1}^n v_i </math><br />
Shot impact positions converted from Cartesian Coordinates<br />
* <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br /><br />
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).<br />
<br />
<math>\overline{r_n}</math> - the average radius of ''n'' shots<br />
<br />
<math>\overline{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
<hr /><br />
<math>MR = f(\Re)</math> method<br />
|-<br />
| Outlier Tests<br />
|<br />
|}<br />
<br />
== Given ==<br />
<br />
== Assumptions ==<br />
<br />
== Data transformation ==<br />
<br />
== Experimental Measure ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>r_i</math> Distribution =<br />
<br />
Distribution for a single shot as a function of r. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\Re</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
|-<br />
| <math>PDF_{r(1)}(r; \Re)</math><br />
| <math>\frac {r}{\Re^2} \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \Re)</math><br />
| <math> 1 - \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\Re</math><br />
|-<br />
| Median of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)}</math><br />
| <math>\frac{(4-\pi)}{2}\Re^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>\overline{r(n)}</math> Distribution =<br />
Given:<br />
* <math>n</math> shots were taken on a target<br />
* The average mean radius, <math>\overline{r(n)}</math>, was calculated<br />
* The Rayleigh shape parameter <math>\Re</math> for an individual shot is known.<br />
then using <math>r</math> as a variable, the properties of the distribution of the average mean radius for <math>n</math> shots is investigated in this section. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>\bar{r_n}</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>n</math> - n of shots in sample<br />
<math>\Re</math> - Rayleigh shape parameter from individual shot distribution <br />
|-<br />
| <math>PDF(\bar{r_n}; n, \Re)</math><br />
| <math>\frac{\Gamma(n,2\Re^2)}{n}</math><br><br />
where <math>\Gamma(n,2\Re^2)</math> is the Gamma Distribution<br />
|-<br />
| <math>CDF(r; n, \Re)</math><br />
| <br />
|-<br />
| Mode of PDF)<br />
| <math>\bar{r_n}</math><br />
|-<br />
| Median of PDF<br />
| no closed form, but <math>\approx 1.177\bar{r_n}</math><br />
|-<br />
| Mean of PDF<br />
| <math>\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}</math><br />
|-<br />
| Variance<br />
| <br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|- <br />
| NSPG Invariant<br />
| Yes<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
<br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
== CDF ==<br />
yada yada <br />
<br />
== Mode, Median, Mean ==<br />
yada yada <br />
<br />
== Outlier Tests ==<br />
yada yada<br />
<br />
<br />
= Theoretical <math>r_{\Re}</math> Distribution =<br />
<br />
Distribution for MR where MR calculated from <math>\Re</math><br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\Re</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
|-<br />
| <math>PDF_{r(1)}(r; \Re)</math><br />
| <math>\frac {r}{\Re^2} \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \Re)</math><br />
| <math> 1 - \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\Re</math><br />
|-<br />
| Median of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)}</math><br />
| <math>\frac{(4-\pi)}{2}\Re^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= ''Studentized'' Mean Radius =<br />
<br />
'''need table for this...''' <br />
<br />
== Outlier Tests ==<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - Discussion of other models for shot dispersion<br />
<br />
<!--<br />
[[Data Transformations to Rayleigh Distribution]] - Methods to transform non-conforming data to Rayleigh Distribution<br />
--></div>Herbhttp://ballistipedia.com/index.php?title=Leslie_1993&diff=1254Leslie 19932015-06-14T21:18:01Z<p>Herb: </p>
<hr />
<div>= Leslie, 1993, ''Is "Group Size" the Best Measure of Accuracy?'' =<br />
[[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|''Is "Group Size" the Best Measure of Accuracy?'', John "Jack" E. Leslie III, 1993]]. <br />
<br />
'''ABSTRACT:'''<br />
<br />
Compares the following measures as a function of the number of shots per group.<br />
* Extreme Spread: Maximum distance between any two shots in group. <br />
* Figure of Merit (FoM): Average of the maximum horizontal group spread and the maximum vertical group spread. This uses only 2-4 data points depending on the group. Like Diagonal, FoM becomes more efficient than Extreme Spread for larger group sizes.<br />
* Mean Radius: Average distance to center of group for all shots.<br />
* Radial Standard Deviation: Sqrt (Horizontal Variance + Vertical Variance).<br />
<br />
Found military using RSD and Mean Radius as early a 1918.<br />
<br />
His Monte Carlo analysis shows sample RSD to be most efficient predictor of precision, followed closely by Mean Radius. I.e., they can distinguish between loads of different inherent precision more accurately and using fewer sample shots than the other measures.<br />
<br />
'''Ballistipedia Notes:''' <br />
<ol><br />
<li> In discussing Extreme spread measurement Leslie makes the following statement: ''Also, by only using data from two shots within the group, it ignores the data represented by the other, more likely to be repeated, shots.''<br />
<br />
<p>This is the right notion, but not quite correct from the point of view of statistics. From a statistical point of view there is the sample size, the number of shots in a group, and an "effective" sample size also known as the degrees of freedom. On average the ES will increase as the number of shots in a group increases. But with each increase in average ES, the next shot is less likely to increase the size of the ES. Thus subsequent shots don't increase the degrees of freedom by 1, but only by a fraction, and that fraction gets smaller and smaller as the number of shots increases. </p> <br />
<br />
<p>The point would apply to the Diagonal and the FOM measurements as well.</p><br />
<li> Leslie, like Grubbs, estimates MR by sampling the mean of radii. This is less efficient than using [[Closed_Form_Precision#Mean_Radius_.28MR.29|the MR computed from the Rayleigh parameter fitted to the sample]]. The latter process is equally and maximally efficient for all invariant measures that are products of the Rayleigh distribution parameter <math>\Re</math>.<br />
<li> Leslie notes "The RSD was the most accurate measure I examined for determining the tightest grouping load."<br />
: But of course, since the simulation is based on the Rayleigh model for which the RSD measurement is essentially the fitted parameter. <br />
<li> Leslie compares "load differences" to get a notion of the relative performance of the statistics. Take the analysis with a grain of salt. There are additional considerations to such an analysis. First, different "loads" would probably have different COI's as well as different dispersions. Second, since 5-shot groups are about optimal for ES, then 20 total shots should be shot as four 5-shot groups since the average of 4 groups would have a much better precision than the ES of one 20-shot group. Third, this analysis ignores fliers. <br />
</ol></div>Herbhttp://ballistipedia.com/index.php?title=Leslie_1993&diff=1253Leslie 19932015-06-14T21:17:11Z<p>Herb: /* Leslie, 1993, Is "Group Size" the Best Measure of Accuracy? */</p>
<hr />
<div>= Leslie, 1993, ''Is "Group Size" the Best Measure of Accuracy?'' =<br />
[[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|''Is "Group Size" the Best Measure of Accuracy?'', John "Jack" E. Leslie III, 1993]]. <br />
<br />
Compares the following measures as a function of the number of shots per group.<br />
* Extreme Spread: Maximum distance between any two shots in group. <br />
* Figure of Merit (FoM): Average of the maximum horizontal group spread and the maximum vertical group spread. This uses only 2-4 data points depending on the group. Like Diagonal, FoM becomes more efficient than Extreme Spread for larger group sizes.<br />
* Mean Radius: Average distance to center of group for all shots.<br />
* Radial Standard Deviation: Sqrt (Horizontal Variance + Vertical Variance).<br />
<br />
Found military using RSD and Mean Radius as early a 1918.<br />
<br />
His Monte Carlo analysis shows sample RSD to be most efficient predictor of precision, followed closely by Mean Radius. I.e., they can distinguish between loads of different inherent precision more accurately and using fewer sample shots than the other measures.<br />
<br />
'''Ballistipedia Notes:''' <br />
<ol><br />
<li> In discussing Extreme spread measurement Leslie makes the following statement: ''Also, by only using data from two shots within the group, it ignores the data represented by the other, more likely to be repeated, shots.''<br />
<br />
<p>This is the right notion, but not quite correct from the point of view of statistics. From a statistical point of view there is the sample size, the number of shots in a group, and an "effective" sample size also known as the degrees of freedom. On average the ES will increase as the number of shots in a group increases. But with each increase in average ES, the next shot is less likely to increase the size of the ES. Thus subsequent shots don't increase the degrees of freedom by 1, but only by a fraction, and that fraction gets smaller and smaller as the number of shots increases. </p> <br />
<br />
<p>The point would apply to the Diagonal and the FOM measurements as well.</p><br />
<li> Leslie, like Grubbs, estimates MR by sampling the mean of radii. This is less efficient than using [[Closed_Form_Precision#Mean_Radius_.28MR.29|the MR computed from the Rayleigh parameter fitted to the sample]]. The latter process is equally and maximally efficient for all invariant measures that are products of the Rayleigh distribution parameter <math>\Re</math>.<br />
<li> Leslie notes "The RSD was the most accurate measure I examined for determining the tightest grouping load."<br />
: But of course, since the simulation is based on the Rayleigh model for which the RSD measurement is essentially the fitted parameter. <br />
<li> Leslie compares "load differences" to get a notion of the relative performance of the statistics. Take the analysis with a grain of salt. There are additional considerations to such an analysis. First, different "loads" would probably have different COI's as well as different dispersions. Second, since 5-shot groups are about optimal for ES, then 20 total shots should be shot as four 5-shot groups since the average of 4 groups would have a much better precision than the ES of one 20-shot group. Third, this analysis ignores fliers. <br />
</ol></div>Herbhttp://ballistipedia.com/index.php?title=Leslie_1993&diff=1252Leslie 19932015-06-14T20:55:49Z<p>Herb: </p>
<hr />
<div>= Leslie, 1993, ''Is "Group Size" the Best Measure of Accuracy?'' =<br />
[[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|''Is "Group Size" the Best Measure of Accuracy?'', John "Jack" E. Leslie III, 1993]]. <br />
<br />
Compares the following measures as a function of the number of shots per group.<br />
* Extreme Spread: Maximum distance between any two shots in group. <br />
* Figure of Merit (FoM): Average of the maximum horizontal group spread and the maximum vertical group spread. This uses only 2-4 data points depending on the group. Like Diagonal, FoM becomes more efficient than Extreme Spread for larger group sizes.<br />
* Mean Radius: Average distance to center of group for all shots.<br />
* Radial Standard Deviation: Sqrt (Horizontal Variance + Vertical Variance).<br />
<br />
Found military using RSD and Mean Radius as early a 1918.<br />
<br />
His Monte Carlo analysis shows sample RSD to be most efficient predictor of precision, followed closely by Mean Radius. I.e., they can distinguish between loads of different inherent precision more accurately and using fewer sample shots than the other measures.<br />
<br />
'''Notes:''' <br />
<ol><br />
<li> In discussing Extreme spread measurement Leslie makes the following statement:<br />
<blockquote>''Also, by only using data from two shots within the group, it ignores the data represented by the other, more likely to be repeated, shots.''</blockquote><br />
:: This is the right notion, but not quite correct from the point of view of statistics. From a statistical point of view there is the sample size, the number of shots in a group, and an "effective" sample size also known as the degrees of freedom. On average the ES will increase as the number of shots in a group increases. But with each increase in average ES, the next shot is less likely to increase the size for a random particular group. Thus subsequent shots don't increase the degrees of freedom by 1, but only by a fraction, and the fraction gets smaller and smaller as the number of shots increases. <br />
:: The point would apply to the Diagonal and the FOM measurements as well. <br />
<li> Leslie, like Grubbs, estimates MR by sampling the mean of radii. This is less efficient than using the Rayleigh estimator on the radii, and than [[Closed_Form_Precision#Mean_Radius_.28MR.29|computing MR based on the sample Rayleigh parameter]]. The latter process is equally and maximally efficient for all invariant measures that are products of the Rayleigh distribution parameter <math>\Re</math> .<br />
<li> Leslie compares "load differences" to get a notion of the relative performance of the statistics. Take the analysis with a grain of salt. There are additional considerations to such an analysis. First, different "loads" would probably have different COI's as well as different dispersions. Second, since 5-shot groups are about optimal for ES, then 20 total shots should be shot as four 5-shot groups. The average of 4 groups would have a much better precision than the ES of one 20-shot group. Third, this analysis ignores fliers. <br />
</ol></div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1251Herb References2015-06-14T19:21:56Z<p>Herb: /* References */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
= References =<br />
<br />
* <div id="Blischke1966"></div>[Blischke1966]<br />
: Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf <br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed publications is not uniformly high. The relevant publications may be roughly categorized into different groups:<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
=== Derivation ===<br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Molon (2006). [[Prior_Art#Molon.2C_2006.2C_The_Trouble_With_3-Shot_Groups|'''The Trouble With 3-Shot Groups''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].<br />
<br />
* Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|'''(cached PDF)''']]<br />
<br />
* Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|'''(cached PDF)''']]</div>Herbhttp://ballistipedia.com/index.php?title=Herb_References&diff=1250Herb References2015-06-14T19:14:03Z<p>Herb: </p>
<hr />
<div> {|align=right<br />
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|}<br />
= References =<br />
<br />
* <div id="Blischke1966"></div>[Blischke1966] Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
* <div id="Chew1962"></div>Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
* <div id="Culpepper1978"></div>Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117<br />
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2<br />
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.<br />
* <div id="DiDonato1988"></div>DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf<br />
* <div id="DiDonato2004"></div>DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato2007"></div>DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368<br />
* <div id="DiDonato1961a"></div>DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026<br />
* <div id="DiDonato1961b"></div>DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="DiDonato1962a"></div>DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054<br />
* <div id="DiDonato1962b"></div>DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.<br />
* <div id="Duchesne2010"></div>Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.<br />
* <div id="Elder1986"></div>Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498<br />
* <div id="Ethridge1983"></div>Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.<br />
* <div id="Evans1985"></div>Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257<br />
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.<br />
* <div id="Farebrother1990"></div>Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.<br />
* <div id="Greenwalt1962"></div>Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf<br />
* <div id="Grubbs1944"></div>Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316<br />
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752<br />
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]]. <br />
* <div id="Guenther1964"></div>Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747<br />
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595<br />
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684]<br />
* <div id="Hogg1967"></div>Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768<br />
* <div id="Hoover1984"></div>Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf<br />
* <div id="Hoyt1947"></div>Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. https://archive.org/details/bstj26-2-318<br />
* <div id="Imhof1961"></div>Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763<br />
* <div id="Kamat1962"></div>Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450<br />
* <div id="Leslie_1993"></div>Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached copy)]] [[Leslie_1993 | (Ballistipedia Notes)]]<br />
* <div id="Liu2009"></div>Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.<br />
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/<br />
* <div id="Moranda1959"></div>Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503<br />
* <div id="Moranda1960"></div>Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596<br />
* <div id="Nelson1988"></div>Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190<br />
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.<br />
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828<br />
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542<br />
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533<br />
* <div id="Pesapane1977"></div>Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.<br />
* <div id="Puhek1992"></div>Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105<br />
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.<br />
* <div id="RAND1952"></div>RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf<br />
* <div id="Sathe1991"></div>Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.<br />
* <div id="Saxena2005"></div>Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf<br />
* <div id="Sheil1977"></div>Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.<br />
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.<br />
* <div id="Singh1992"></div>Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751<br />
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html<br />
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.<br />
* <div id="Spall1992"></div>Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. URL http://www.jstor.org/stable/2290205<br />
* <div id="Taub1983a"></div>Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taub1983b"></div>Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828<br />
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf <br />
* <div id="Taylor1975b"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407<br />
* <div id="Thomas1973"></div>Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989<br />
* <div id="Tongue1993"></div>Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528<br />
* <div id="Wang2013a"></div>Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.<br />
* <div id="Wang2013b"></div>Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.<br />
* <div id="Wang2014"></div>Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.<br />
* <div id="Williams1997"></div>Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337<br />
* <div id="Zhang2012"></div>Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586.<br />
<br />
= Groups of Publications =<br />
<br />
== CEP ==<br />
<br />
The literature on the [[Circular_Error_Probable|circular error probable (CEP)]] is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.<br />
<br />
The following list is by no means intended to be complete. Beware that the quality of the listed publications is not uniformly high. The relevant publications may be roughly categorized into different groups:<br />
<br />
=== Develop CEP Estimator ===<br />
<br />
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).<br />
<br />
=== Comparing CEP Estimators ===<br />
<br />
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).<br />
<br />
=== CEP in polar Coordinates ===<br />
<br />
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|1947]]) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].<br />
<br />
=== CEP with Bias ===<br />
<br />
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|Evans et al. (1985)]] develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.<br />
<br />
=== Hoyt Distribution Algorithms ===<br />
<br />
<div id="algos"></div>A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].<br />
<br />
=== Spherical Error Probable ===<br />
<br />
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.<br />
<br />
== Extreme Spread ==<br />
<br />
=== Monte Carlo Simulation ===<br />
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].<br />
<br />
=== Sampling Problems ===<br />
<br />
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.<br />
<br />
=== Advocating Conversion From ===<br />
<br />
== Rayleigh Distribution ==<br />
<br />
=== Derivation ===<br />
<br />
----<br />
<br />
* Bookstaber, David (2014). [http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].<br />
<br />
* Danielson, Brent J. (2005). [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2005). [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Hogema, Jeroen (2006). [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Kolbe, Geoffrey (2010). [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Molon (2006). [[Prior_Art#Molon.2C_2006.2C_The_Trouble_With_3-Shot_Groups|'''The Trouble With 3-Shot Groups''' &ndash; ''detailed in Prior Art'']].<br />
<br />
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].<br />
<br />
* Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|'''(cached PDF)''']]<br />
<br />
* Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (''Summarizes and extends Siddiqui, 1961.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|'''(cached PDF)''']]</div>Herbhttp://ballistipedia.com/index.php?title=User:Herb&diff=1249User:Herb2015-06-14T16:48:36Z<p>Herb: </p>
<hr />
<div><br />
[[MediaWiki:Sidebar]]<br />
<br />
[http://ballistipedia.com/index.php?title=Special:AllPages| All Pages]<br />
<br />
=My notion of sidebar=<br />
<br />
* [http://ballistipedia.com/index.php?title=What_is_Precision%3F What is precision?]<br />
* [[Projectile Dispersion Classifications]]<br />
* [[Measuring Precision]]<br />
* [[Herb_References]]<br />
* Examples<br />
<br />
<br />
<br />
= Measures =<br />
<br />
* Circular Error Probable (CEP)<br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Elliptical Error Probable (EEP)<br />
* [[Extreme Spread]]<br />
* [[Figure of Merit]]<br />
* Horizontal and Vertical Variances<br />
* [[Mean Radius]]<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
= Wiki pages I created =<br />
<br />
[[Covering Circle Radius versus Extreme Spread]] - should be pretty good. <br />
<br />
[[Data Transformations to Rayleigh Distribution]]<br />
<br />
[[Derivation of the Rayleigh Distribution Equation | Mathematical Formulas and Derivations]]<br />
<br />
[[Projectile Dispersion Classifications]] - getting close...<br />
<br />
[[Error Propagation]]<br />
<br />
[[Extreme Spread]] * measure<br />
<br />
[[Figure of Merit]] * measure<br />
<br />
[[Fliers vs. Outliers]]<br />
<br />
[[Leslie 1993]] - notion ok, disagree with content on page. <br />
<br />
[[Measuring Precision]] - this is fairly solid. <br />
<br />
[[Mean Radius]] * measure<br />
<br />
[[Sighting a Weapon]] ** needs work<br />
<br />
[[Stringing]] seems mostly ok. Fuzzy on how to handle inter/exterior ballastics. <br />
<br />
[[What is ρ in the Bivariate Normal distribution?]] think this pretty good. <br />
<br />
----<br />
<br />
<br />
Interrelationship of the Range Measurements<br />
* Range<br />
* Studentized Range<br />
** Covering Circle<br />
** Diagonal<br />
** ES<br />
** FOM<br />
** ES<br />
<br />
[[Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD]]<br />
<br />
---<br />
Carnac the Magnificent<br />
<br />
ab initio<br />
----<br />
<br />
Suppose that Xk has the gamma distribution with shape parameter k∈(0,∞) and fixed scale parameter b∈(0,∞). Then the distribution of the standardized variable below converges to the standard normal distribution as k→∞:<br />
<br />
&nbsp;<math>Z_k = \frac{X_k−kb}{b\sqrt{k}}</math><br />
<br />
----<br />
= Measurements =<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Circular Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Orthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Nonorthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
# CEP(50) Using Ranking<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
# CEP(50) Using Rayleigh distribution<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
|}<br />
<br />
<br />
<br />
<br />
# Elliptical Error Probable<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Dispersion by Rayleigh Distribution<br />
## Dispersion by Orthogonal Elliptical Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Dispersion by Hoyt Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
<br />
<br />
<br />
----<br />
<br />
"The difference between theory and practice is larger in<br />
practice than in theory."<br />
<br />
In theory there is no difference between theory and practice. But, in practice, there is.<br />
<br />
<br />
----<br />
<br />
sighting shot distribution<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* Origin at <math>(r,\theta) = (0,0)</math><br />
* Rayleigh Distribution for Shots<br />
** <math>\sigma_h = \sigma_v</math><br />
**<math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
* With conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius<br />
* No Flyers<br />
|-<br />
| Data Pretreatment<br />
| Shot impact positions converted from Cartesian Coordinates (''h'', ''v'') to Polar Coordinates <math>(r,\theta)</math><br />
* Origin translated from Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
|-<br />
| Experimental Measure<br />
| <math>\bar{r_n}</math> - the average radius of ''n'' shots<br />
<math>\bar{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
&nbsp;&nbsp;&nbsp; where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
|-<br />
| <math>PDF_{r_0}(r; n, \sigma)</math><br />
| <math>\frac{nr}{\sigma^2}e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| <math>CDF_{r_0}(r; n, \sigma)</math><br />
| <math>1 - e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| Mode of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}</math><br />
|-<br />
| Median of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{ln{(4)}}</math><br />
|-<br />
| Mean of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| <br />
|- <br />
| NSPG Invariant<br />
| No<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
= master ref page =<br />
<br />
I like the structure of this wiki page. You can look at the "groups of papers" then jump to a specific paper and use the browser back button to go back to the group. <br />
<br />
Could we make this the "master" reference page?<br />
<br />
(1) Move references to top of page <br />
(2) put TOC that floats to right<br />
(3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model)<br />
(4) Each level 1 heading would have various "groups" of papers. <br />
(5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page<br />
<br />
how I'd redo references so as to provide some that was "linkable" and could be "named"<br />
<br />
So '''Blischke_Halpin_1966''' could be name of wiki page and a "named" link within the page. thus reference in a wiki page would be something like:<br />
: yada yada yada (Blischke_Halpin_1966) yada yada yada <br />
the link would jump to the "master" page of references to that entry. <br />
<br />
; Blischke_Halpin_1966<br />
:Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
; Chew_Boyce_1962<br />
:Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
: Culpepper_1978<br />
;Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117</div>Herbhttp://ballistipedia.com/index.php?title=Mean_Radius&diff=1248Mean Radius2015-06-14T16:03:10Z<p>Herb: /* Theoretical \overline{r(n)} Distribution */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Mean Radius<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
= Experimental Summary =<br />
<br />
yada yada <br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* The dispersion of shot <math>i</math> follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius.<br />
** <math>h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to fit <math>\sigma</math> to calculate the Mean Radius.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Measure positions <math>(h_i, v_i)</math> for each shot, <math>i</math>.<br />
|-<br />
| Experimental Measure<br />
| Preliminary Cartesian Calculations<br />
* <math>\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2 </math><br />
* <math>\bar{v} = \frac{1}{n-1} \sum_{i=1}^n v_i^2 </math><br />
Shot impact positions converted from Cartesian Coordinates<br />
* <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br /><br />
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).<br />
<br />
<math>\overline{r_n}</math> - the average radius of ''n'' shots<br />
<br />
<math>\overline{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
|-<br />
| Outlier Tests<br />
|<br />
|}<br />
<br />
== Given ==<br />
<br />
== Assumptions ==<br />
<br />
== Data transformation ==<br />
<br />
== Experimental Measure ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>r(1)</math> Distribution =<br />
<br />
Distribution for a single shot as a function of r. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\Re</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
|-<br />
| <math>PDF_{r(1)}(r; \Re)</math><br />
| <math>\frac {r}{\Re^2} \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \Re)</math><br />
| <math> 1 - \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\Re</math><br />
|-<br />
| Median of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)}</math><br />
| <math>\frac{(4-\pi)}{2}\Re^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>\overline{r(n)}</math> Distribution =<br />
Given:<br />
* <math>n</math> shots were taken on a target<br />
* The average mean radius, <math>\overline{r(n)}</math>, was calculated<br />
* The Rayleigh shape parameter <math>\Re</math> for an individual shot is known.<br />
then using <math>r</math> as a variable, the properties of the distribution of the average mean radius for <math>n</math> shots is investigated in this section. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>\bar{r_n}</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>n</math> - n of shots in sample<br />
<math>\Re</math> - Rayleigh shape parameter from individual shot distribution <br />
|-<br />
| <math>PDF(\bar{r_n}; n, \Re)</math><br />
| <math>\frac{\Gamma(n,2\Re^2)}{n}</math><br><br />
where <math>\Gamma(n,2\Re^2)</math> is the Gamma Distribution<br />
|-<br />
| <math>CDF(r; n, \Re)</math><br />
| <br />
|-<br />
| Mode of PDF)<br />
| <math>\bar{r_n}</math><br />
|-<br />
| Median of PDF<br />
| no closed form, but <math>\approx 1.177\bar{r_n}</math><br />
|-<br />
| Mean of PDF<br />
| <math>\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}</math><br />
|-<br />
| Variance<br />
| <br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|- <br />
| NSPG Invariant<br />
| Yes<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
<br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
== CDF ==<br />
yada yada <br />
<br />
== Mode, Median, Mean ==<br />
yada yada <br />
<br />
== Outlier Tests ==<br />
yada yada<br />
<br />
= ''Studentized'' Mean Radius =<br />
<br />
'''need table for this...''' <br />
<br />
== Outlier Tests ==<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - Discussion of other models for shot dispersion<br />
<br />
<!--<br />
[[Data Transformations to Rayleigh Distribution]] - Methods to transform non-conforming data to Rayleigh Distribution<br />
--></div>Herbhttp://ballistipedia.com/index.php?title=Mean_Radius&diff=1247Mean Radius2015-06-14T15:59:21Z<p>Herb: /* Theoretical r(1) Distribution */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Mean Radius<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
= Experimental Summary =<br />
<br />
yada yada <br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* The dispersion of shot <math>i</math> follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius.<br />
** <math>h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to fit <math>\sigma</math> to calculate the Mean Radius.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Measure positions <math>(h_i, v_i)</math> for each shot, <math>i</math>.<br />
|-<br />
| Experimental Measure<br />
| Preliminary Cartesian Calculations<br />
* <math>\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2 </math><br />
* <math>\bar{v} = \frac{1}{n-1} \sum_{i=1}^n v_i^2 </math><br />
Shot impact positions converted from Cartesian Coordinates<br />
* <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br /><br />
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).<br />
<br />
<math>\overline{r_n}</math> - the average radius of ''n'' shots<br />
<br />
<math>\overline{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
|-<br />
| Outlier Tests<br />
|<br />
|}<br />
<br />
== Given ==<br />
<br />
== Assumptions ==<br />
<br />
== Data transformation ==<br />
<br />
== Experimental Measure ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>r(1)</math> Distribution =<br />
<br />
Distribution for a single shot as a function of r. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\Re</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
|-<br />
| <math>PDF_{r(1)}(r; \Re)</math><br />
| <math>\frac {r}{\Re^2} \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \Re)</math><br />
| <math> 1 - \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\Re</math><br />
|-<br />
| Median of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)}</math><br />
| <math>\Re\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)}</math><br />
| <math>\frac{(4-\pi)}{2}\Re^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>\overline{r(n)}</math> Distribution =<br />
Given:<br />
* <math>n</math> shots were taken on a target<br />
* The average mean radius, <math>\overline{r(n)}</math>, was calculated<br />
* The Rayleigh shape parameter <math>\sigma</math> for an individual shot is known.<br />
then using <math>r</math> as a variable, the properties of the distribution of the average mean radius for <math>n</math> shots is investigated in this section. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>\bar{r_n}</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>n</math> - n of shots in sample<br />
<math>\sigma</math> - Rayleigh shape parameter from individual shot distribution <br />
|-<br />
| <math>PDF(\bar{r_n}; n, \sigma)</math><br />
| <math>\frac{\Gamma(n,2\sigma^2)}{n}</math><br><br />
where <math>\Gamma(n,2\sigma^2)</math> is the Gamma Distribution<br />
|-<br />
| <math>CDF(r; n, \sigma)</math><br />
| <br />
|-<br />
| Mode of PDF)<br />
| <math>\bar{r_n}</math><br />
|-<br />
| Median of PDF<br />
| no closed form, but <math>\approx 1.177\bar{r_n}</math><br />
|-<br />
| Mean of PDF<br />
| <math>\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}</math><br />
|-<br />
| Variance<br />
| <br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|- <br />
| NSPG Invariant<br />
| Yes<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
<br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
== CDF ==<br />
yada yada <br />
<br />
== Mode, Median, Mean ==<br />
yada yada <br />
<br />
== Outlier Tests ==<br />
yada yada<br />
<br />
= ''Studentized'' Mean Radius =<br />
<br />
'''need table for this...''' <br />
<br />
== Outlier Tests ==<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - Discussion of other models for shot dispersion<br />
<br />
<!--<br />
[[Data Transformations to Rayleigh Distribution]] - Methods to transform non-conforming data to Rayleigh Distribution<br />
--></div>Herbhttp://ballistipedia.com/index.php?title=Mean_Radius&diff=1246Mean Radius2015-06-14T15:55:02Z<p>Herb: /* Experimental Summary */</p>
<hr />
<div> {|align=right<br />
|__TOC__<br />
|}<br />
Mean Radius<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
= Experimental Summary =<br />
<br />
yada yada <br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* The dispersion of shot <math>i</math> follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius.<br />
** <math>h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to fit <math>\sigma</math> to calculate the Mean Radius.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Measure positions <math>(h_i, v_i)</math> for each shot, <math>i</math>.<br />
|-<br />
| Experimental Measure<br />
| Preliminary Cartesian Calculations<br />
* <math>\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2 </math><br />
* <math>\bar{v} = \frac{1}{n-1} \sum_{i=1}^n v_i^2 </math><br />
Shot impact positions converted from Cartesian Coordinates<br />
* <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br /><br />
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).<br />
<br />
<math>\overline{r_n}</math> - the average radius of ''n'' shots<br />
<br />
<math>\overline{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
|-<br />
| Outlier Tests<br />
|<br />
|}<br />
<br />
== Given ==<br />
<br />
== Assumptions ==<br />
<br />
== Data transformation ==<br />
<br />
== Experimental Measure ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>r(1)</math> Distribution =<br />
<br />
Distribution for a single shot as a function of r. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>r(1)</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>\sigma</math> - Rayleigh shape parameter fit to experimental shot distribution <br />
|-<br />
| <math>PDF_{r(1)}(r; \sigma)</math><br />
| <math>\frac {r}{\sigma^2} \exp\Big \{-\frac {r^2}{2\sigma^2} \Big\}</math><br />
|-<br />
| <math>CDF_{r(1)}(r; \sigma)</math><br />
| <math> 1 - \exp\Big \{-\frac {r^2}{2\sigma^2} \Big\}</math><br />
|-<br />
| Mode of <math>PDF_{r(1)</math><br />
| <math>\sigma</math><br />
|-<br />
| Median of <math>PDF_{r(1)</math><br />
| <math>\sigma\sqrt{\ln{4}}</math><br />
|-<br />
| Mean of <math>PDF_{r(1)</math><br />
| <math>\sigma\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| Variance of <math>PDF_{r(1)</math><br />
| <math>\frac{(4-\pi)}{2}\sigma^2</math><br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
<br />
== CDF ==<br />
<br />
== Mode, Median, Mean ==<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical <math>\overline{r(n)}</math> Distribution =<br />
Given:<br />
* <math>n</math> shots were taken on a target<br />
* The average mean radius, <math>\overline{r(n)}</math>, was calculated<br />
* The Rayleigh shape parameter <math>\sigma</math> for an individual shot is known.<br />
then using <math>r</math> as a variable, the properties of the distribution of the average mean radius for <math>n</math> shots is investigated in this section. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>\bar{r_n}</math> Distribution<br />
|-<br />
| Parameters Needed<br />
| <math>n</math> - n of shots in sample<br />
<math>\sigma</math> - Rayleigh shape parameter from individual shot distribution <br />
|-<br />
| <math>PDF(\bar{r_n}; n, \sigma)</math><br />
| <math>\frac{\Gamma(n,2\sigma^2)}{n}</math><br><br />
where <math>\Gamma(n,2\sigma^2)</math> is the Gamma Distribution<br />
|-<br />
| <math>CDF(r; n, \sigma)</math><br />
| <br />
|-<br />
| Mode of PDF)<br />
| <math>\bar{r_n}</math><br />
|-<br />
| Median of PDF<br />
| no closed form, but <math>\approx 1.177\bar{r_n}</math><br />
|-<br />
| Mean of PDF<br />
| <math>\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}</math><br />
|-<br />
| Variance<br />
| <br />
|-<br />
| Variance Distribution<br />
|<br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| No, skewed to larger values. <br />
More symmetric as number of shots increases. <br />
|- <br />
| NSPG Invariant<br />
| Yes<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
== Parameters Needed ==<br />
yada yada <br />
<br />
== Variance and Its distribution ==<br />
yada yada <br />
<br />
== PDF ==<br />
yada yada <br />
== CDF ==<br />
yada yada <br />
<br />
== Mode, Median, Mean ==<br />
yada yada <br />
<br />
== Outlier Tests ==<br />
yada yada<br />
<br />
= ''Studentized'' Mean Radius =<br />
<br />
'''need table for this...''' <br />
<br />
== Outlier Tests ==<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - Discussion of other models for shot dispersion<br />
<br />
<!--<br />
[[Data Transformations to Rayleigh Distribution]] - Methods to transform non-conforming data to Rayleigh Distribution<br />
--></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1245Measuring Precision2015-06-14T15:30:50Z<p>Herb: /* Which Measure is Best? */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that covers proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set. <br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA.]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough precise experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] "The difference between theory and practice is larger in practice than in theory."<br />
|}<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1244Measuring Precision2015-06-14T15:29:07Z<p>Herb: /* Dispersion Measures about COI */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that covers proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set. <br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA.]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1243Measuring Precision2015-06-14T15:23:59Z<p>Herb: /* Diagonal (D) */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that covers proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set. <br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA.]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1242Measuring Precision2015-06-14T15:23:18Z<p>Herb: /* Extreme Spread (ES) */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that covers proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set. <br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
<br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA.]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=Measuring_Precision&diff=1241Measuring Precision2015-06-14T15:20:56Z<p>Herb: /* Which Measure is Best? */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
<br />
The following text considers weapons systems precision as demonstrated by the projectile impact points on a two dimensional target. In order to have a consistent point of view the target is assumed to be be mounted as if it is a target at a rifle or pistol range. So the line of fire is assumed to be perpendicular to the target, and the target's axes are vertical and horizontal. <br />
<br />
= Precision Units =<br />
<br />
When we talk about shooting precision we are referring to a measure of the dispersion about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle. <br />
<br />
''Linear distance'' typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards. Thus a linear distance measure should change in direct proportion to the distance. <br />
<br />
''[[Angular Size]]'' is another common unit and is the angle at the tip of the ''cone of fire'' since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.<br />
<br />
== Linear Distance ==<br />
<br />
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches. <br />
<br />
=== Mil ===<br />
<br />
The other common linear unit is the '''mil''', which simply means thousandth. For example, '''at 100 yards a mil is 100 yards / 1000 = 3.6"'''. <br />
<br />
'''Note:''' Some confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s little harm in interchanging ''mil'', ''mrad'', ''milrad'', and ''milliradian''.<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;milliradian = 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards<br />
<!--<br />
Note also: Even '''mil''' is encumbered by some historical ambiguity. For example,<br />
western militaries going back at least a century used an angular unit for artillery<br />
calculations that divided the circle into 6400 "mils," which persists the "NATO mil."<br />
<br />
[http://en.wikipedia.org/wiki/Angular_mil#Definitions_of_the_angular_mil]<br />
--><br />
== Angular Size ==<br />
<br />
The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the <br />
muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases. <br />
<br />
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade <br />
faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and <br />
a three foot group at 1000 yards. <br />
<br />
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the <br />
projectile from the muzzle results in an projectile instability which is damped by air resistance. In this <br />
case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a <br />
longer distance is larger, but not geometrically larger. Note however that if you were using an angular <br />
measure, then the group size would be smaller at 100 yards than 50 yards. <br />
<br />
=== Minute Of Arc ===<br />
<br />
One of two popular angular units used by shooters is '''MOA''', though there is some ambiguity in this term.<br />
From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. <br />
Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. <br />
<br />
'''At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047"'''. <br />
<br />
=== Shooter's Minute of Angle===<br />
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to <br />
1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or '''SMOA'''.<br />
<br />
== Conversions between measuring units==<br />
<br />
See [[Angular Size]] wiki page for detailed illustrations and conversion formulas.<br />
<br />
= Variant and Invariant Target Measures =<br />
<br />
We will define the following measures as ''invariant target measures''. The expected value of the measure does not change as more shots are made on the same target, rather more shots means a more precise measurement. The "cost" of a more precise measurement though is that the position of each shot on the target must be known. <br />
* Circular Error Probable (CEP)<br />
* Elliptical Error Probable (EEP)<br />
* Horizontal and Vertical Variances<br />
* Mean Radius (MR)<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
We will define the following measures as ''variant target measures''. The value of the measure increases as more shots are made on the target. Using Extreme Spread as an example, 5 shots have been taken on the target. The 6th shot can't make the ES of the other five shots smaller, only larger. Hence invariant measures increase randomly with sample size. <br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Extreme Spread (ES)<br />
* Figure of Merit (FOM)<br />
<br />
The last method is the string method. Since it measures from the center of the target it conflates an accuracy measurement with a precision measurement. <br />
<br />
== Degrees of Freedom ==<br />
<br />
Before leaving the discussion of variant and invariant target measures, one more esoteric concept needs discussion. The nature of statistics is to depend on the sample size. Usually the integer count of the sample size is identical to the statistical measure of the sample size known as the ''degrees of freedom.'' But the degrees of freedom does not have to be an integer quantity. It can, and for some statistical tests often does, non-integer values. In essence the degrees of freedom give you an efficiency measure of the sampling. If the sampling is 100% efficient then each additional sample not only adds 1 to the sample size, but it also adds 1 to the degrees of freedom. <br />
<br />
* So for the invariant target measures, each shot in the group increases the sample size by 1 and the degrees of freedom by 1. <br />
<br />
* For the variant target measures the situation is a bit different. Each shot does increase the sample size by 1. However each shot doesn't increase the degrees of freedom by 1. Rather each shot increases the degrees of freedom by a small amount. As more shots are fired the ES measurement does, on average, get randomly larger, but it becomes less and less probable that the next shot will increase the ES. Thus each shot adds a smaller and smaller amount to the overall degrees of freedom.<br />
<br />
= Measurement Robustness =<br />
<br />
Robustness is a straightforward notion conceptually, but mathematically fuzzy. The notion is that a robust measure would be tolerant of an outlier or of a difference in the probability distribution. <br />
<br />
The average is not a robust statistic. A single very large value would greatly perturb the average. However the median would be a robust statistic. A single large value, no matter how large, wouldn't change the median much. <br />
<br />
As with the mean, the standard deviation isn't a robust statistic. A single very large value would change the standard deviation significantly. Thus the width of the 25% quartile to the 75% quartile would be more robust. <br />
<br />
A number of robustness scales have been proposed, but such scales would depend on the variations that would be considered to be usual and what sorts of unusual variations the robustness of the estimator was designed to protect against. <br />
<br />
It is possible to take a standard statistic such as the mean and make the measurement more robust against a single large value by testing for outliers, or by trimming the data. Ideally such treatments would be done in such a way that the data treatment wouldn't introduce bias into the measurement. For instance if the data were normally distributed, then trimming the largest 10% of the measurements would bias the average low. Trimming both the highest and lowest 10% however would not bias the measurement.<br />
<br />
= Dispersion Measures about COI =<br />
[[File:raleigh.jpg|365px|thumb|right| Shots dispersed about the COI. A circular dispersion is the Rayleigh distribution.]]<br />
<br />
Different measures have been used to characterize the precision of bullet holes in a sample target. The measures detailed below are popular. Some are easier to calculate than others, and thus would be suitable for range use. Others require the (''h,v'') positions of each shot and considerable calculations. Such measurements would more amiable to analysis with a calculator or computer. <br />
<br />
Measures of precision are directly, or indirectly, measures about the COI. Such measures do not depend at all upon the different between the COI and the POA which is the accuracy of the shooting.<br />
<br />
The implicit goal of a parametric measurement is related to process control. In general it is ideal if there is a single number that gives a measure of the overall process quality. Then that number would allow you to make a judgement to decide if a process change results in poorer quality, the same quality, or better quality. <br />
<br />
{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]]: '''!! CAREFUL !!''' An old adage: '''A fool with a tool is still a fool.''' <br /><br />
The measures and statistical analyses on this wiki will provide the shooter with tools to achieve some process control of shooting. The nature of reducing a two-dimensional pattern into a single number is not without risk. The proper use of statistics requires constant vigilance to insure that the techniques used are sound and that assumptions upon which those techniques are based are valid. <br />
|}<br />
<br />
In the following sections on the various measures assume that:<br />
# We are looking at a target reflecting ''n'' shots<br />
# We are able to determine the center coordinates ''h'' and ''v'' as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target (aka Ragged Hole Problem).<br />
# Appropriate consideration has been made to the underlying assumptions about shot dispersion. Unless otherwise noted the measurement assumes that a large number of shots (i.e. the population of possible shots) would have a circular (or nearly so) shot distribution. <br />
# Fliers are not present. <br />
<br />
For mathematical symbols and symbols see the [http://ballistipedia.com/index.php?title=Glossary#Mathematical_Notation Glossary].<br />
<br />
The following headings for each measure are linked to a more detailed discussion of that measure.<br />
<br />
=== [[Circular Error Probable]] (CEP) ===<br />
[[File:SCAR17 150gr 100yd.png|365px|thumb|right|Precision Measures diagrammed on a 10-shot 100-yard group. Data in [[Media:SCAR17_150gr_100yd.xls]]]] <br />
CEP(p), for <math>0 \leq p \leq 1</math>, is the radius of the smallest circle, centered at the COI, that covers proportion ''p'' of the shot group. When ''p'' is not indicated it is assumed to be CEP(0.5), which is the ''median shot radius'' (50% radius).<br />
<br />
CEP is a robust estimator in that the median value wouldn't change much if one extreme value flier was in the measured set. <br />
<br />
=== Covering Circle Radius (CCR) ===<br />
<br />
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will <br />
pass through at least the two shots used for the extreme spread measure (in which case CCR = (Extreme Spread)/2 ) or at most it will pass <br />
through three outside shots. Thus in general the CCR will be at least as large as ES and typically a bit larger.<br />
<br />
The CCR(50) measurement were based on the median value then it would be a robust estimator. If it is calculated by fitting the Rayleigh distribution shape parameter to the data then it is not a robust estimator.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Diagonal (D) ===<br />
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note <br />
that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be <br />
determined by two to four points depending on the pattern of shots within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
where <math>(h_{max} - h_{min})</math> and <math>(v_{max} - v_{min})</math> are the observed horizontal and vertical ranges respectively.<br />
<br />
The D measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical variance, the difference is essentially is "How is the ''average'' of both axes to be calculated?" The Diagonal uses square root of the horizontal and vertical ranges squared. The FOM is averaging the horizontal and vertical ranges as:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
=== Elliptical Error Probable (EEP) ===<br />
[[File:Hoyt.jpg|250px|thumb|right| Hoyt Distribution - Shots dispersed about COI in an elliptical pattern which has its major axis at an angle to the coordinate axes.]]<br />
The EEP(p) is analogous to the Circular Error Probable (CEP), in that covers proportion ''p'' of the shot group with <math>0 \leq p \le 1</math>, the ellipse being centered about the COI. When ''p'' is not indicated it is assumed to be EEP(0.5). Elliptical Error Probable assumes that the shots follow the Hoyt distribution, so the calculations would be flexible enough to calculate <math>s_h, s_v,</math> and <math>\rho</math>.<br />
<br />
The general notion is that the ratios of the major and minor axes should equal the ratios of the variances. There are numerous ways to do the actual calculation of course. <br />
<br />
The EEP is the only measurement considered which is appropriate for a non-circular distribution. In a computer program sophisticated enough to handle the calculation of the EEP, the CEP could be programmed as a simpler special case.<br />
<br />
The EEP(50) measurement were based on the median values then it would be a robust estimator. If it is calculated by fitting the Hoyt distribution to the data then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br clear=both><br />
<br />
=== [[Extreme Spread]] (ES) ===<br />
The ''Extreme Spread'' is is the largest center-to-center distance between any two points, ''i'' and ''j'', in the group. <br />
Formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>ES = \max \sqrt{(h_i - h_j)^2 + (v_i - v_j)^2}</math> <br />
<br />
The ES measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
Statisticians have used the terms ''extreme spread'' and ''bivariant range'' for this measure. Shooters typically call this measure the ''Extreme Spread'' or ''group size''.<br />
<br />
'''Note:''' Be careful with with the phrase ''extreme spread''. Shooters will often refer to the range of values from a chronograph as the ''extreme spread''. Context should allow an easy determination of the correct meaning of the phrase.<br />
<br />
'''See Also:''' [[Covering Circle Radius versus Extreme Spread]] - A discussion of the difference and interrelationship between the Covering Circle Radius and the Extreme Spread measurements.<br />
<br />
=== Figure of Merit (FOM) ===<br />
<br />
The Figure of Merit is the average range of the width and height of the group. The FOM may be determined by <br />
two to four points depending on the pattern within the group. <br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>FOM = \frac{(h_{max} - h_{min}) + (v_{max} - v_{min})}{2}</math><br />
<br />
The FOM measurement is a not a robust estimator since it depends on the extreme shot values. <br />
<br />
The FOM of merit and Diagonal both really assume a underlying circular distribution. In the case that there is a small difference in the horizontal and vertical range, the difference is essentially is "How is the ''average'' of both ranges to be calculated?"<br />
<br />
The FOM is averaging the horizontal and vertical ranges. The Diagonal uses square root of the horizontal and vertical ranges squared. <br /><br />
<math>D = \sqrt{(h_{max} - h_{min})^2 + (v_{max} - v_{min})^2)}</math><br /><br />
<br />
=== Horizontal and Vertical Variances ===<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}</math><br />
<br />
Often these will be given as standard deviations, which is just the square root of variance.<br />
<br />
The variances are not robust estimators since they weight the extreme shot values more heavily. <br />
<br />
=== [[Mean Radius]] (MR) ===<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>\bar{r} = \sum_{i=1}^n r_i / n</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
<br />
The MR measurement is not a robust estimator since one large extreme value could change the value significantly. <br />
<br />
As we will see in [[Closed Form Precision]], the Mean Radius is typically only 6% larger than the Circular <br />
Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be <br />
interchanged in casual usage.<br />
<br />
=== Rayleigh Distribution Mode (RDM) ===<br />
<br />
The ''mode'' is the value at the peak of a distribution. Thus Rayleigh Distribution Mode (RDM) is the peak value of the Rayleigh distribution. Given that the shots follow the Rayleigh distribution, then an alternate measurement was desired other than the Radial Standard Deviation. Using the mode value of the Rayleigh distribution seems like a logical choice since:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>RDM = \Re</math><br />
<br />
The RDM measure is directly proportional to the RSD measurement, so it has exactly the same relative standard deviation though the actual value will be a little smaller. Since the Rayleigh distribution is skewed towards larger values:<br />
<br />
&nbsp;&nbsp;&nbsp;<math>\text{Mean} ( = \Re \sqrt{\frac{\pi}{2}} \approx 1.253 \Re ) \gt \text{Median} ( = \Re \sqrt{\ln{4}} \approx 1.177 \Re) > Mode ( = \Re)</math><br /><br />
<br />
Since <math>\Re</math> isn't calculated as the second moment about some mean, it seems more natural to think of the fitted parameter <math>\Re</math> for the Rayleigh distribution as a special radius, than it does to think of it as "standard deviation" of some sort.<br />
<br />
If the RDM measurement is determined from a measurement of the actual peak of a measured distribution then it would be a robust estimator. If it was calculated based on fitting an assumed distribution then it is not a robust estimator. Overall it is unlikely that this measurement would be made in the manner so that it is robust since that would require an extraordinary amount of experimental data. <br />
<br />
=== Radial Standard Deviation (RSD) ===<br />
<br />
The Radial Standard Deviation (RSD) is typically defined as <math>\sqrt{\sigma_h^2 + \sigma_v^2}</math> in the literature. It is proportional to the constant coefficient, the Rayleigh shape parameter <math>\Re</math>, in the Rayleigh distribution equation, and has therefore served as a useful reference to that constant. <br />
<br />
&nbsp;&nbsp;&nbsp;<math>RSD = \Re \sqrt{2}</math><br />
<br />
The RSD is also not the same as the standard deviation of the mean radius which is given by the formula:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \sqrt{ \frac{\sum_{i=1}^n r_i^2}{n-1}}</math> where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br /><br />
<br />
In terms of the Rayleigh shape parameter <math>\Re</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>\sigma_{MR} = \Re \sqrt{\frac{4 - \pi}{2}}</math><br />
<br />
In spite of the rather sexy name there is nothing special about the RSD. Since the Raleigh distribution has a single constant to be fitted, any of its expressions which is directly proportional to to <math>\Re</math> would have the same relative error (i.e. error as a %) as the fitted constant <math>\Re</math>. <br />
<br />
Using a "standard deviation" as a measure also lacks a certain intuitive feel. The other measures are all in linear units (or angular equivalents), so it would be nice if the fitted <math>\Re</math> coefficient were used in a measurement that was linear too. Thus the main discussion of fitting the <math>\Re</math> coefficient will be moved to the ''Rayleigh Distribution Mode (RDM)'' measurement where <math>\Re = RDM</math>. <br />
<br />
In order to avoid confusion with this measure that is both biased and misnamed, reference to RSD will be minimized.<br />
<br />
The RSD measurement is a not a robust estimator since it is fitting an assumed distribution to the experimental data in such a way that extreme shot values are weighted more heavily.<br />
<br />
= Other Measures =<br />
<br />
== Dispersion Measures From POA ==<br />
<br />
=== String Length (SL) Method ===<br />
<br />
[[File:Rice.jpg|250px|thumb|right| Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA.]]<br />
<br />
This is what could be called an "old-fashion" method for measuring a shooters skill. There are variations of the method as well. <br />
<br />
The gist is that a end of a string is held at the center of the target which would be the POA for the shooter. At first the end of the string is placed at the center of the target and then the string is played out to the center of a hole on the target. The string is pinched over the center of that hole, and the pinched section is moved to the center of the target. The process is repeated until all holes have been measured. Then the total length of the string is the shooter's ''string length''. Thus this version of the measurement isn't just measuring precision but accuracy as well.<br />
<br />
Formula:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>SL =\sum_{i=1}^n \sqrt{(h_i - h_{POA})^2 + (v_i - v_{POA})^2}</math><br />
<br />
Assuming that the shot dispersion around the COI followed the Rayleigh distribution, then the individual string segments measured to the POA would follow the Rice Distribution.<br />
<br />
The SL measurement is a not a robust estimator since it depends on the absence of extreme shot values.<br clear=both /><br />
<br />
= Comparative Summary of measures =<br />
<br />
:{| class="wikitable" <br />
! Dispersion<br />Measure<br />
! Measure:<br />Invariant<br />&nbsp;&nbsp;- or -<br />Variant<br />
! Robust<br />Estimator<br />
! Shot Pattern:<br />Circular<br />&nbsp;&nbsp;- or -<br />Ellipitical<br />
! Dispersion<br /> Class<br />
! Accuracy<br />&nbsp;&nbsp;- or -<br />Precision<br />
|-<br />
| [[#Circular_Error_Probable_.28CEP.29| Circular Error Probable (CEP)]]<br />
| Invariant<br />
| Maybe<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Covering_Circle_Radius_.28CCR.29| Covering Circle Radius (CCR)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Diagonal_.28D.29| Diagonal (D)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Elliptical_Error_Probable_.28EEP.29| Elliptical Error Probable (EEP)]]<br />
| Invariant<br />
| Unlikely<br />
| Elliptical<br />
| Hoyt<br />
| Precision<br />
|-<br />
| [[#Extreme_Spread_.28ES.29| Extreme Spread (ES)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Figure_of_Merit_.28FOM.29| Figure of Merit (FOM)]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Horizontal_and_Vertical_Variances| Horiz. and Vert. Variances]]<br />
| Invariant<br />
| No<br />
| Elliptical<br />
| Orthogonal Elliptical<br />&nbsp;&nbsp;- or -<br />Hoyt<br />
| Precision<br />
|-<br />
| [[#Mean_Radius_.28MR.29| Mean Radius (MR)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Rayleigh_Distribution_Mode_.28RDM.29| Rayleigh Distribution Mode (RDM)]]<br />
| Invariant<br />
| Unlikely<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#Radial_Standard_Deviation_.28RSD.29| Radial Standard Deviation (RSD)]]<br />
| Invariant<br />
| No<br />
| Circular<br />
| Rayleigh<br />
| Precision<br />
|-<br />
| [[#String_Length_.28SL.29_Method| String Length]]<br />
| Variant<br />
| No<br />
| Circular<br />
| Rice<br />
| SL = F(Precision, Accuracy)<br />
|}<br />
<br />
= Which Measure is Best? =<br />
<br />
[[Precision Models]] discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no ''universally best measurement''. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "fliers" in the data. In practice the effect of neither of these factors is known. <br />
<br />
The lack of an absolute truth may be mitigated with an expectation of making reasonable assumptions and picking a mathematical model that is ''good enough''. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate normal distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate normal distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit. <br />
<br />
Shooters use the term ''flyer'' to denote the statistical term ''outlier''. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed. <br />
<br />
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate)<br />
as the gold standard given the situation that the underlying assumptions about shot dispersion and the<br />
lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use<br />
of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless percentage. The efficiency of various measures can be thus compared by using the ratios of the relative standard deviations squared. <br />
<br />
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough experimental data to infer that the theory is incorrect. Also most of the measures are not [[http://en.wikipedia.org/wiki/Robust_statistics robust estimators]].<br />
<br />
<br /><br />
<hr /><br />
<p style="text-align:right"><B>Next:</B> [[Precision Models]]</p></div>Herbhttp://ballistipedia.com/index.php?title=User:Herb&diff=1240User:Herb2015-06-13T22:08:24Z<p>Herb: /* Measures */</p>
<hr />
<div><br />
[[MediaWiki:Sidebar]]<br />
<br />
[http://ballistipedia.com/index.php?title=Special:AllPages| All Pages]<br />
<br />
=My notion of sidebar=<br />
<br />
* [http://ballistipedia.com/index.php?title=What_is_Precision%3F What is precision?]<br />
* [[Projectile Dispersion Classifications]]<br />
* [[Measuring Precision]]<br />
* [[Herb_References]]<br />
* Examples<br />
<br />
<br />
<br />
= Measures =<br />
<br />
* Circular Error Probable (CEP)<br />
* Covering Circle Radius (CCR)<br />
* Diagonal (D)<br />
* Elliptical Error Probable (EEP)<br />
* [[Extreme Spread]]<br />
* [[Figure of Merit]]<br />
* Horizontal and Vertical Variances<br />
* [[Mean Radius]]<br />
* Rayleigh Distribution Mode (RDM)<br />
* Radial Standard Deviation (RSD)<br />
<br />
= Wiki pages I created =<br />
<br />
[[Covering Circle Radius versus Extreme Spread]] - should be pretty good. <br />
<br />
[[Data Transformations to Rayleigh Distribution]]<br />
<br />
[[Derivation of the Rayleigh Distribution Equation | Mathematical Formulas and Derivations]]<br />
<br />
[[Projectile Dispersion Classifications]] - getting close...<br />
<br />
[[Error Propagation]]<br />
<br />
[[Extreme Spread]] * measure<br />
<br />
[[Figure of Merit]] * measure<br />
<br />
[[Fliers vs. Outliers]]<br />
<br />
[[Leslie 1993]] - notion ok, disagree with content on page. <br />
<br />
[[Measuring Precision]] - this is fairly solid. <br />
<br />
[[Mean Radius]] * measure<br />
<br />
[[Sighting a Weapon]] ** needs work<br />
<br />
[[Stringing]] seems mostly ok. Fuzzy on how to handle inter/exterior ballastics. <br />
<br />
[[What is ρ in the Bivariate Normal distribution?]] think this pretty good. <br />
<br />
----<br />
<br />
<br />
Interrelationship of the Range Measurements<br />
* Range<br />
* Studentized Range<br />
** Covering Circle<br />
** Diagonal<br />
** ES<br />
** FOM<br />
** ES<br />
<br />
[[Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD]]<br />
<br />
---<br />
Carnac the Magnificent<br />
----<br />
<br />
Suppose that Xk has the gamma distribution with shape parameter k∈(0,∞) and fixed scale parameter b∈(0,∞). Then the distribution of the standardized variable below converges to the standard normal distribution as k→∞:<br />
<br />
&nbsp;<math>Z_k = \frac{X_k−kb}{b\sqrt{k}}</math><br />
<br />
----<br />
= Measurements =<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Circular Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Orthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Nonorthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
# CEP(50) Using Ranking<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
# CEP(50) Using Rayleigh distribution<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
|}<br />
<br />
<br />
<br />
<br />
# Elliptical Error Probable<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Dispersion by Rayleigh Distribution<br />
## Dispersion by Orthogonal Elliptical Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Dispersion by Hoyt Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
<br />
<br />
<br />
----<br />
<br />
"The difference between theory and practice is larger in<br />
practice than in theory."<br />
<br />
In theory there is no difference between theory and practice. But, in practice, there is.<br />
<br />
<br />
----<br />
<br />
sighting shot distribution<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* Origin at <math>(r,\theta) = (0,0)</math><br />
* Rayleigh Distribution for Shots<br />
** <math>\sigma_h = \sigma_v</math><br />
**<math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
* With conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius<br />
* No Flyers<br />
|-<br />
| Data Pretreatment<br />
| Shot impact positions converted from Cartesian Coordinates (''h'', ''v'') to Polar Coordinates <math>(r,\theta)</math><br />
* Origin translated from Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
|-<br />
| Experimental Measure<br />
| <math>\bar{r_n}</math> - the average radius of ''n'' shots<br />
<math>\bar{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
&nbsp;&nbsp;&nbsp; where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
|-<br />
| <math>PDF_{r_0}(r; n, \sigma)</math><br />
| <math>\frac{nr}{\sigma^2}e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| <math>CDF_{r_0}(r; n, \sigma)</math><br />
| <math>1 - e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| Mode of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}</math><br />
|-<br />
| Median of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{ln{(4)}}</math><br />
|-<br />
| Mean of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| <br />
|- <br />
| NSPG Invariant<br />
| No<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
= master ref page =<br />
<br />
I like the structure of this wiki page. You can look at the "groups of papers" then jump to a specific paper and use the browser back button to go back to the group. <br />
<br />
Could we make this the "master" reference page?<br />
<br />
(1) Move references to top of page <br />
(2) put TOC that floats to right<br />
(3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model)<br />
(4) Each level 1 heading would have various "groups" of papers. <br />
(5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page<br />
<br />
how I'd redo references so as to provide some that was "linkable" and could be "named"<br />
<br />
So '''Blischke_Halpin_1966''' could be name of wiki page and a "named" link within the page. thus reference in a wiki page would be something like:<br />
: yada yada yada (Blischke_Halpin_1966) yada yada yada <br />
the link would jump to the "master" page of references to that entry. <br />
<br />
; Blischke_Halpin_1966<br />
:Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
; Chew_Boyce_1962<br />
:Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
: Culpepper_1978<br />
;Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117</div>Herbhttp://ballistipedia.com/index.php?title=User:Herb&diff=1239User:Herb2015-06-13T22:06:51Z<p>Herb: /* Measures */</p>
<hr />
<div><br />
[[MediaWiki:Sidebar]]<br />
<br />
[http://ballistipedia.com/index.php?title=Special:AllPages| All Pages]<br />
<br />
=My notion of sidebar=<br />
<br />
* [http://ballistipedia.com/index.php?title=What_is_Precision%3F What is precision?]<br />
* [[Projectile Dispersion Classifications]]<br />
* [[Measuring Precision]]<br />
* [[Herb_References]]<br />
* Examples<br />
<br />
<br />
<br />
= Measures =<br />
<br />
Circular Error Probable (CEP)<br />
Covering Circle Radius (CCR)<br />
Diagonal (D)<br />
Elliptical Error Probable (EEP)<br />
[[Extreme Spread]]<br />
[[Figure of Merit]]<br />
Horizontal and Vertical Variances<br />
[[Mean Radius]]<br />
Rayleigh Distribution Mode (RDM)<br />
Radial Standard Deviation (RSD)<br />
<br />
= Wiki pages I created =<br />
<br />
[[Covering Circle Radius versus Extreme Spread]] - should be pretty good. <br />
<br />
[[Data Transformations to Rayleigh Distribution]]<br />
<br />
[[Derivation of the Rayleigh Distribution Equation | Mathematical Formulas and Derivations]]<br />
<br />
[[Projectile Dispersion Classifications]] - getting close...<br />
<br />
[[Error Propagation]]<br />
<br />
[[Extreme Spread]] * measure<br />
<br />
[[Figure of Merit]] * measure<br />
<br />
[[Fliers vs. Outliers]]<br />
<br />
[[Leslie 1993]] - notion ok, disagree with content on page. <br />
<br />
[[Measuring Precision]] - this is fairly solid. <br />
<br />
[[Mean Radius]] * measure<br />
<br />
[[Sighting a Weapon]] ** needs work<br />
<br />
[[Stringing]] seems mostly ok. Fuzzy on how to handle inter/exterior ballastics. <br />
<br />
[[What is ρ in the Bivariate Normal distribution?]] think this pretty good. <br />
<br />
----<br />
<br />
<br />
Interrelationship of the Range Measurements<br />
* Range<br />
* Studentized Range<br />
** Covering Circle<br />
** Diagonal<br />
** ES<br />
** FOM<br />
** ES<br />
<br />
[[Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD]]<br />
<br />
---<br />
Carnac the Magnificent<br />
----<br />
<br />
Suppose that Xk has the gamma distribution with shape parameter k∈(0,∞) and fixed scale parameter b∈(0,∞). Then the distribution of the standardized variable below converges to the standard normal distribution as k→∞:<br />
<br />
&nbsp;<math>Z_k = \frac{X_k−kb}{b\sqrt{k}}</math><br />
<br />
----<br />
= Measurements =<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Circular Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Orthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Nonorthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
# CEP(50) Using Ranking<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
# CEP(50) Using Rayleigh distribution<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
|}<br />
<br />
<br />
<br />
<br />
# Elliptical Error Probable<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Dispersion by Rayleigh Distribution<br />
## Dispersion by Orthogonal Elliptical Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Dispersion by Hoyt Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
<br />
<br />
<br />
----<br />
<br />
"The difference between theory and practice is larger in<br />
practice than in theory."<br />
<br />
In theory there is no difference between theory and practice. But, in practice, there is.<br />
<br />
<br />
----<br />
<br />
sighting shot distribution<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* Origin at <math>(r,\theta) = (0,0)</math><br />
* Rayleigh Distribution for Shots<br />
** <math>\sigma_h = \sigma_v</math><br />
**<math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
* With conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius<br />
* No Flyers<br />
|-<br />
| Data Pretreatment<br />
| Shot impact positions converted from Cartesian Coordinates (''h'', ''v'') to Polar Coordinates <math>(r,\theta)</math><br />
* Origin translated from Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
|-<br />
| Experimental Measure<br />
| <math>\bar{r_n}</math> - the average radius of ''n'' shots<br />
<math>\bar{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
&nbsp;&nbsp;&nbsp; where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
|-<br />
| <math>PDF_{r_0}(r; n, \sigma)</math><br />
| <math>\frac{nr}{\sigma^2}e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| <math>CDF_{r_0}(r; n, \sigma)</math><br />
| <math>1 - e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| Mode of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}</math><br />
|-<br />
| Median of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{ln{(4)}}</math><br />
|-<br />
| Mean of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| <br />
|- <br />
| NSPG Invariant<br />
| No<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
= master ref page =<br />
<br />
I like the structure of this wiki page. You can look at the "groups of papers" then jump to a specific paper and use the browser back button to go back to the group. <br />
<br />
Could we make this the "master" reference page?<br />
<br />
(1) Move references to top of page <br />
(2) put TOC that floats to right<br />
(3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model)<br />
(4) Each level 1 heading would have various "groups" of papers. <br />
(5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page<br />
<br />
how I'd redo references so as to provide some that was "linkable" and could be "named"<br />
<br />
So '''Blischke_Halpin_1966''' could be name of wiki page and a "named" link within the page. thus reference in a wiki page would be something like:<br />
: yada yada yada (Blischke_Halpin_1966) yada yada yada <br />
the link would jump to the "master" page of references to that entry. <br />
<br />
; Blischke_Halpin_1966<br />
:Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
; Chew_Boyce_1962<br />
:Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
: Culpepper_1978<br />
;Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117</div>Herbhttp://ballistipedia.com/index.php?title=Extreme_Spread&diff=1238Extreme Spread2015-06-13T21:49:56Z<p>Herb: </p>
<hr />
<div> {|align=right<br />
|__TOC__ <br />
|}<br />
= Experimental Summary =<br />
<br />
{| class="wikitable" <br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /><br />
All of the (''h'',''v'') positions do not need to be known so a ragged hole will suffice. <br />
|-<br />
| Assumptions<br />
|<br />
* Ideally the shots would have a circular pattern (i.e. the shots would have the Rayleigh Distribution).<br />
** <math>h \sim \mathcal{N}(\bar{h},\sigma_h^2), v \sim \mathcal{N}(\bar{v},\sigma_v^2)</math><br />
** Horizontal and vertical dispersion are independent. <br />
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
** <math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
:: '''Note:''' It is not necessary to calculate the COI, nor fit <math>\sigma</math> to calculate the Extreme Spread.<br />
* No Fliers<br />
|-<br />
| Data transformation<br />
| Identify two holes, <math>i, j</math> which are the farthest apart and measure <math>ES</math>.<br />
&nbsp;<math>ES = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}</math><br />
|-<br />
| Experimental Measure<br />
| <math>ES</math><br />
|}<br />
<br />
== Given ==<br />
<br />
The requirements for this test are very basic. Just a target with <math>n</math> shots, and some measuring device. Assuming an Extreme spread of under 6 inches then a vernier caliper is used. A measurement is possible to a few thousandths of an inch which is vastly more precision than is usually required. From longer distance a ruler, or perhaps a tape measure.<br />
<br />
== Assumptions ==<br />
<br />
None are needed to make measurement. However some points are worth considering.<br />
<br />
* The same ES measurement could result from a vertical group to a round group. If the shooting process can vary that much then the ES measurement won't give any indication of the change. <br />
<br />
:: If the shot patterns aren't "fairly" round, then using the measurement makes little sense. For instance if muzzle velocity variations are severe, then the vertical range will dominate the ES measurement. Muzzle velocity variations would correlate better with vertical range than with ES. <br />
<br />
* Making assumptions about the dispersion will enable theoretical predictions about the ES measurement. It must be realized that the theoretical solution, assuming the Rayleigh distribution and using Monte Carlo simulation, isn't some arbitrary goal, it is the best case scenario.<br />
<br />
== Data transformation ==<br />
<br />
The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart and measure the distance between them. If the target has a ragged hole it can be a bit tricky, but the edges of the hole should have enough curvature to make shot location possible.<br />
<br />
If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper (cheap is fine!) would be used to measure the distance from the outside edges of the holes, then the bullet caliber subtracted to get a c-t-c measurement. <br />
<br />
:{| class="wikitable" <br />
| [[File:Bullseye.jpg|50px]] A cheap ($10-$20) vernier caliper works fine. There is no need for a $2,000 one that measures to 1/10,000<sup>th</sup> of an inch and has National Bureau of Standards calibration. The vernier caliper is nice for the c-t-c measurement because the knife edges will be parallel and won't obscure the edges of the bullet hole. Thus it is easy to accurately place both of the knife edges on a tangent to the curved bullet holes. <br />
|}<br />
<br />
If using a computer then the center location would be a matter programming. For example a mouse might be used simply to point out the holes, or to drop a dot at the center of the hole, or to drag a circle over the hole. The computer would then make the c-t-c measurement.<br />
<br />
== Experimental Measure ==<br />
<br />
No calculation needs to be done to get the measurement. The physical measurement is the data sought.<br />
<br />
== Outlier Tests ==<br />
<br />
= Theoretical Evaluations =<br />
<br />
== Dispersion Follows Rayleigh Distribution ==<br />
<br />
Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates. <br />
<br />
{| class="wikitable" <br />
|+ Theoretical <math>ES</math> Distribution of <math>n</math> shots<br />
|-<br />
| Parameters Needed<br />
| <br />
|-<br />
| <math>PDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| <math>CDF_{ES}(r; n)</math><br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Mode of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Mode increases as number of shots increases. <br />
|-<br />
| Median of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases.<br />
|-<br />
| Mean of <math>PDF_{ES}</math><br />
| depends on <math>n</math>, Median increases as number of shots increases<br />
|-<br />
| Variance<br />
| no deterministic solution, must be simulated via Monte Carlo<br />
|-<br />
| Efficiency<br />
| depends on <math>n</math>, best about 5-7 shots<br />
|-<br />
| (h,v) for all points?<br />
| yes for simulation. <br />
|- <br />
| Symmetric about Mean?<br />
| No, skewed to larger values. <br />
More symmetric about mean as the number of shots increases. <br />
|}<br />
<br />
<br />
=== Parameters Needed ===<br />
<br />
=== PDF ===<br />
<br />
=== CDF ===<br />
<br />
=== Mode, Median, Mean, Standard Deviation, %Rel Std Dev ===<br />
<br />
Since the distribution is positively skewed: <br />
<blockquote>Mean > Median > Mode</blockquote><br />
<br />
"Normality Error"<br />
As sort of a crude indication of normality let's use the value:<br />
<br />
"Normality Error" = <math> \frac{\frac{CDF(5) - CDF(95)}{2} - Mean}{Mean} {\cdot 100}</math><br />
<br />
So we measure half the distance between the 5<sup>th</sup> percentile and the 95<sup>th</sup> percentile to determine where the Mean should be if the distribution was symmetrical, and determine the % error based on the actual value of the mean. <br />
* + value means positively skewed, <br />
* - value means negatively skewed. <br />
<br />
The point of the "Normality Error" is to give the reader a quinsy feeling about using Student's T-Test for groups with few shots, or the average of a small number of targets. <br />
<br />
<br />
{| class="wikitable" <br />
|+ Theoretical ES Values from Monte Carlo Simulation Distribution<br />
|-<br />
! number of shots<br />
! Mode<br />
! Median<br />
! Mean<br />
! "Normality Error"<br />
! Standard<br />
Deviation<br />
! %Rel Std Dev<br />
|-<br />
| 2<br />
| <br />
|<br />
| 1.772<br />
| <br />
| 0.932<br />
| 52.6%<br />
|-<br />
| 3<br />
|<br />
|<br />
| 2.406<br />
| 4.95%<br />
| 0.887<br />
| 36.9%<br />
|-<br />
| 4<br />
|<br />
|<br />
| 2.787<br />
| <br />
| 0.856<br />
| 30.7%<br />
|-<br />
| 5<br />
| <br />
|<br />
| 3.066<br />
| 3.06%<br />
| 0.828<br />
| 27.0%<br />
|-<br />
| 6<br />
| <br />
| <br />
| 3.277<br />
| <br />
| 0.806<br />
|<br />
|-<br />
| 7<br />
| <br />
| <br />
| 3.443<br />
| <br />
| 0.783<br />
|<br />
|-<br />
| 9<br />
|<br />
|<br />
| 3.710<br />
| <br />
| 0.754<br />
|<br />
|-<br />
| 10<br />
| <br />
|<br />
| 3.813<br />
|<br />
| 0.745<br />
|<br />
|-<br />
| 20<br />
|<br />
| <br />
|<br />
| <br />
|<br />
|<br />
|-<br />
| 30<br />
| <br />
|<br />
| 4.788<br />
| 1.63%<br />
| 0.745<br />
| 15.6%<br />
|}<br />
<br />
The tabular values can be used in a number of ways:<br />
<br />
'''Estimate a 95% confidence Interval for Given 2-shot groups based on one ES measuremnt'''<br />
<br />
So a 2-shot group has been measured. If the measured value is accepted as the true value, what would the standard deviation of multiple 2-shot groups be? <br />
<br />
This is another example to warn the reader. Just because you can calculate a standard deviation doesn't mean that a Student's T Test will work. A typical 95% confidence Interval for an individual ES measurement is <math>\pm 1.96 \sigma</math> and for a 2-shot group that is:<br />
:&nbsp;<math>\pm 1.96 \cdot 52.6\% = \pm 103.1\%</math> of the measurement<br />
so the lower confidence interval would be at '''-3.1% !!!''' The nonsensical result is because the distribution is skewed. A negative extreme spread measurement is impossible. It isn't the standard deviation that is wrong, it is the assumption that the confidence interval would be <math>\pm 1.96 \sigma</math> that is the problem. Since the distribution is skewed, the low side of the confidence interval at the 2.5 percentile is at ?? and the high side of the confidence interval at the 97.5 percentile is at +??. <br />
<br />
At 5 shots the T-test is reasonable, and at 10 shots pretty good. <br />
<br />
'''Given ES of one 5-shot group is 1.53 inches'''<br />
<br />
* Estimate ES values for different group sizes.<br />
:: A 3-shot group would be given by measured size times ratios of the Means from the table<br />
:::&nbsp;<math>1.53 \frac{2.406}{3.066} = 1.20</math> inches<br />
:: A 10-shot group would be given by measured size times ratios of Means from the table<br />
:::&nbsp;<math>1.53 \frac{3.813}{3.066} = 1.90</math> inches<br />
<br />
* Estimate the expected standard deviation from the measured ES value<br />
:: The %RSD value for 5-shots is 27.0% so:<br />
:::&nbsp;<math>\hat{s} = 1.53 \text{ inches} \cdot 0.270 = 0.413 \text{ inches} </math><br />
<br />
* Estimate the expected Standard Deviation of the average of 4 targets<br />
:&nbsp;<math>\text{SD}_{ES\ 4 \ Targets}\ = \frac{27.0\%}{\sqrt{4}} = 13.5\% </math><br />
<br />
=== Variance ===<br />
<br />
=== Efficiency ===<br />
[[File:Extreme Spread Relative Efficiency.png|300px|thumb|right|]]<br />
<br />
The efficiency depends on <math>n</math>, but it is best about 5-7 shots. Essentially there are two competing factors. First as the number of shots increases then the midpoint of the line segment which defines the MR is, on average, closer to the COI which improves efficiency. Second as the number of shots increases then it is increasingly unlikely that the next shot will increase the MR which decreases efficiency. The product of these two factors thus peaks at about 5-7 shots.<br />
<br />
Notice too that the figure is assuming one group of shots. If 8 shots were used then four groups of 2-shots per group could have been made, or one group of 8-shots per group. Using 2-shots per group though requires four targets. If ammunition comes 20 cartridges per box then using 8 shot groups leaves 4 cartridges unused. <br />
<br />
This result is assuming no fliers. If 5-7 shots is likely to give groups with multiple fliers, then less shots per group might be better.<br />
<br />
So what is the optimal number of shots per group? '''''It depends...'''''<br />
<br />
=== Outlier Tests ===<br />
<br />
= See Also =<br />
<br />
[[Projectile Dispersion Classifications]] - A discussion of the different cases for projectile dispersion<br />
<br />
Other measurements practical for range use are: <br />
<br />
* [[Covering Circle Radius]] - about same precision as Extreme Spread if Rayleigh distributed<br />
* [[Diagonal]] - somewhat better precision than Extreme Spread if Rayleigh distributed<br />
* [[Figure of Merit]] - somewhat better precision than Extreme Spread if Rayleigh distributed</div>Herbhttp://ballistipedia.com/index.php?title=User:Herb&diff=1237User:Herb2015-06-13T21:43:28Z<p>Herb: /* Wiki pages I created */</p>
<hr />
<div><br />
[[MediaWiki:Sidebar]]<br />
<br />
[http://ballistipedia.com/index.php?title=Special:AllPages| All Pages]<br />
<br />
=My notion of sidebar=<br />
<br />
* [http://ballistipedia.com/index.php?title=What_is_Precision%3F What is precision?]<br />
* [[Projectile Dispersion Classifications]]<br />
* [[Measuring Precision]]<br />
* [[Herb_References]]<br />
* Examples<br />
<br />
<br />
<br />
= Measures =<br />
<br />
[[Extreme Spread]]<br />
<br />
[[Figure of Merit]]<br />
<br />
[[Mean Radius]]<br />
<br />
= Wiki pages I created =<br />
<br />
[[Covering Circle Radius versus Extreme Spread]] - should be pretty good. <br />
<br />
[[Data Transformations to Rayleigh Distribution]]<br />
<br />
[[Derivation of the Rayleigh Distribution Equation | Mathematical Formulas and Derivations]]<br />
<br />
[[Projectile Dispersion Classifications]] - getting close...<br />
<br />
[[Error Propagation]]<br />
<br />
[[Extreme Spread]] * measure<br />
<br />
[[Figure of Merit]] * measure<br />
<br />
[[Fliers vs. Outliers]]<br />
<br />
[[Leslie 1993]] - notion ok, disagree with content on page. <br />
<br />
[[Measuring Precision]] - this is fairly solid. <br />
<br />
[[Mean Radius]] * measure<br />
<br />
[[Sighting a Weapon]] ** needs work<br />
<br />
[[Stringing]] seems mostly ok. Fuzzy on how to handle inter/exterior ballastics. <br />
<br />
[[What is ρ in the Bivariate Normal distribution?]] think this pretty good. <br />
<br />
----<br />
<br />
<br />
Interrelationship of the Range Measurements<br />
* Range<br />
* Studentized Range<br />
** Covering Circle<br />
** Diagonal<br />
** ES<br />
** FOM<br />
** ES<br />
<br />
[[Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD]]<br />
<br />
---<br />
Carnac the Magnificent<br />
----<br />
<br />
Suppose that Xk has the gamma distribution with shape parameter k∈(0,∞) and fixed scale parameter b∈(0,∞). Then the distribution of the standardized variable below converges to the standard normal distribution as k→∞:<br />
<br />
&nbsp;<math>Z_k = \frac{X_k−kb}{b\sqrt{k}}</math><br />
<br />
----<br />
= Measurements =<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Circular Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Orthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Nonorthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
# CEP(50) Using Ranking<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
# CEP(50) Using Rayleigh distribution<br />
## Value<br />
## Confidence Interval<br />
## Outlier Tests<br />
|}<br />
<br />
<br />
<br />
<br />
# Elliptical Error Probable<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Dispersion by Rayleigh Distribution<br />
## Dispersion by Orthogonal Elliptical Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Dispersion by Hoyt Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
<br />
<br />
<br />
----<br />
<br />
"The difference between theory and practice is larger in<br />
practice than in theory."<br />
<br />
In theory there is no difference between theory and practice. But, in practice, there is.<br />
<br />
<br />
----<br />
<br />
sighting shot distribution<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* Origin at <math>(r,\theta) = (0,0)</math><br />
* Rayleigh Distribution for Shots<br />
** <math>\sigma_h = \sigma_v</math><br />
**<math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
* With conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius<br />
* No Flyers<br />
|-<br />
| Data Pretreatment<br />
| Shot impact positions converted from Cartesian Coordinates (''h'', ''v'') to Polar Coordinates <math>(r,\theta)</math><br />
* Origin translated from Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
|-<br />
| Experimental Measure<br />
| <math>\bar{r_n}</math> - the average radius of ''n'' shots<br />
<math>\bar{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
&nbsp;&nbsp;&nbsp; where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
|-<br />
| <math>PDF_{r_0}(r; n, \sigma)</math><br />
| <math>\frac{nr}{\sigma^2}e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| <math>CDF_{r_0}(r; n, \sigma)</math><br />
| <math>1 - e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| Mode of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}</math><br />
|-<br />
| Median of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{ln{(4)}}</math><br />
|-<br />
| Mean of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| <br />
|- <br />
| NSPG Invariant<br />
| No<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
= master ref page =<br />
<br />
I like the structure of this wiki page. You can look at the "groups of papers" then jump to a specific paper and use the browser back button to go back to the group. <br />
<br />
Could we make this the "master" reference page?<br />
<br />
(1) Move references to top of page <br />
(2) put TOC that floats to right<br />
(3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model)<br />
(4) Each level 1 heading would have various "groups" of papers. <br />
(5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page<br />
<br />
how I'd redo references so as to provide some that was "linkable" and could be "named"<br />
<br />
So '''Blischke_Halpin_1966''' could be name of wiki page and a "named" link within the page. thus reference in a wiki page would be something like:<br />
: yada yada yada (Blischke_Halpin_1966) yada yada yada <br />
the link would jump to the "master" page of references to that entry. <br />
<br />
; Blischke_Halpin_1966<br />
:Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
; Chew_Boyce_1962<br />
:Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
: Culpepper_1978<br />
;Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117</div>Herbhttp://ballistipedia.com/index.php?title=Derivation_of_the_Rayleigh_Distribution_Equation&diff=1236Derivation of the Rayleigh Distribution Equation2015-06-13T20:31:38Z<p>Herb: /* Derivations */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
= Mathematical Formulas and Derivations =<br />
<br />
= Bivariate Normal Distribution =<br />
Starting only with the assumptions that the horzontial and vertical measurements are normally distributed as notated by:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma_h^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma_v^2)</math><br />
<br />
then the horizontal and vertical measures follow the general bivariate normal distribution which is given by the following equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Simplification of the Bivariate Normal Distribution to the Hoyt Distribution<br />
<br />
= Correction Factors =<br />
The following three correction factors will be used throughout this statistical inference and deduction. <br />
<br />
Note that all of these correction factors are > 1, are significant for very small ''n'', and converge towards 1 as <math>n \to \infty</math>. Their values are listed for ''n'' up to 100 in [[Media:Sigma1ShotStatistics.ods]]. [[File:SymmetricBivariate.c]] uses Monte Carlo simulation to confirm that their application produces valid corrected estimates.<br />
<br />
== [http://en.wikipedia.org/wiki/Bessel%27s_correction Bessel correction factor] ==<br />
The Bessel correction removes bias in sample variance.<br />
:&nbsp; <math>c_{B}(n) = \frac{n}{n-1}</math><br />
<br />
== [http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution Gaussian correction factor] ==<br />
The Gaussian correction (sometimes called <math>c_4</math>) removes bias introduced by taking the square root of variance.<br />
:&nbsp; <math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math><br />
<br />
The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:&nbsp; <math>c_{G}(n)</math> <code>=1/EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))</code><br />
<br />
== Rayleigh correction factor ==<br />
The unbiased estimator for the Rayleigh distribution is also for <math>\sigma^2</math>. The following corrects for the concavity introduced by taking the square root to get ''σ''.<br />
:&nbsp; <math>c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}</math> <ref>[[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|''Statistical Inference for Rayleigh Distributions'', M. M. Siddiqui, 1964, p.1007]]</ref><br />
<br />
To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula: <code>=EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))</code><br />
<br />
<br />
= The Hoyt Distribution =<br />
== Derivation From the Bivariate Normal distribution ==<br />
Given the Bivariate Normal distribution as follows:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution. This translation will not affect measurements about COI, but it would of course affect measurements which are measured about POA. <br />
<br />
Given a translation to point <math>(\mu_h, \mu_v)</math> then let:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>h_* = h - \mu_h</math> &nbsp;&nbsp; and &nbsp;&nbsp; <math>v_* = v - \mu_v</math><br /><br />
<br />
Since the derivative of <math>h_*</math> with respect to <math>(h - \mu_h)</math> is 1, (and similarity for <math>v_*</math>) then no change results to the integration constant of the function. Thus <math>h_*</math> can be substituted for <math>(h - \mu_h)</math> and <math>v_*</math> for <math>(v - \mu_v)</math>. At this point the asterisk subscript is superfluous and will be dropped, giving the Hoyt distribution. <br />
&nbsp;&nbsp;&nbsp;<math> <br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v }{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
= The Rayleigh distribution =<br />
<br />
== Derivations ==<br />
<br />
=== Derivation OF Single Shot PDF From the Bivariate Normal distribution ===<br />
<br />
The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:<br />
* Horizontal and vertical dispersion are independent. <br />
* <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
* <math>\rho = 0</math><br />
* No Fliers<br />
for which the PDF for any shot, <math>i</math>, around the horizontal and vertical point <math>(\mu_h, \mu_v)</math> is given by:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(r; \sigma_{\Re}) = \frac{r}{\sigma_{\Re}^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma_{\Re}^2} <br />
\right)<br />
</math><br />
: where <math>\sigma_{\Re} = \sigma_h = \sigma_v</math> and <math>r = \sqrt{h_i - \mu_h)^2 + sqrt(v_i - \mu_v)^2}</math><br />
<br />
'''PROOF'''<br />
<br />
Using the assumptions in the first section, the distribution of an individual shot is easily simplified from the Bivariate Normal Distribution which has the equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
By substituting <math>\rho = 0</math> the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v }<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
Since <math>\sigma_h</math> and <math>\sigma_v</math> are equal, substitute <math>\sigma</math> for each, then collect terms in the exponential, after which the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
-\left[<br />
\frac{(h-\mu_h)^2 + (v-\mu_h)^2}{2\sigma^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
Letting <math>r^2 = (h-\mu_h)^2 + (v-\mu_v)^2</math> the equation becomes: <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
Now transforming to the polar coordinate system:<br /><br />
<br />
** ok, here I'm lost **<br />
<br />
and finally:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(r) =<br />
\frac{r}{\sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
=== Derivation OF Single Shot CDF from the PDF ===<br />
<br />
Given the single shot Rayleigh distribution, calculate the single shot Cumulative Distribution Function (CDF) for the Rayleigh distribution.<br />
<br />
=== Derive the Mode of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mode for the Rayleigh distribution.<br />
<br />
=== Derive the Median of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
=== Derive the Mean Radius of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
== Calculating the Rayleigh Shape factor, <math>\Re</math> ==<br />
<br />
=== Value from <math>n</math> Shots ===<br />
<br />
=== Confidence Intervals of <math>n</math> Shots ===<br />
<br />
== Experimental Uses of <math>\Re</math> ==<br />
<br />
=== Accuracy of <math>n</math> Sighting Shots ===<br />
<br />
Given the assumptions in the starting section we again substitute <math>\sigma</math> for both <math>\sigma_h</math> and <math>\sigma_v</math>. This simplifies the distributions of <math>h</math> and <math>v</math> to:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma^2)</math><br />
<br />
Now we take some number <math>n</math> of shots <math>( n \geq 1)</math>and calculate their centers <math>\bar{h}</math> and <math>\bar{v}</math> which will be normal distributions as well. <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>\bar{h} \sim \mathcal{N}(\mu_h,\sigma^2/n)</math>, and <math>\bar{v} \sim \mathcal{N}(\mu_v,\sigma^2/n)</math><br />
<br />
Let <math>r_n</math> be the distance of this sample center <math>(\bar{h}, \bar{v})</math> from the true distribution center <math>(\mu_h, \mu_v)</math> as:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>r_n = \sqrt{(\bar{h}-\mu_h)^2 + (\bar{v}-\mu_v)^2}</math><br />
<br />
Define random variables <math>Z_h</math> and <math>Z_v</math> as the squared ''Studentized'' horizontal and vertical errors by dividing by the respective standard deviations. Each of these variables with have a Chi-Squared Distribution with one degree of freedom.<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_h = \left(\frac{(\bar{h}-\mu_h)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{h}-\mu_h)^2 \sim \chi^2(1)</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_v = \left(\frac{(\bar{v}-\mu_v)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{v}-\mu_v)^2\sim \chi^2(1)</math><br /><br />
<br />
Define random the variable <math>W</math> which will have a Chi-Squared Distribution with two degrees of freedom as:<br /> <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>W = Z_x + Z_y =\frac n{\sigma^2}\left((\bar{v}-\mu_v)^2+(\bar{v}-\mu_v)^2\right)\sim \chi^2(2)</math><br />
<br />
Rescale the variable <math>W</math> by <math>\frac {\sigma^2}{n}</math> and denote the new variable <math>w_n</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n=\frac {\sigma^2}nW</math> and note that <math>w_n=r_n^2</math><br />
<br />
By the properties of a chi-square random variable, we have:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n \sim \text {Gamma}(k=1, \theta = 2\sigma^2/n) = \text{Exp}(2\sigma^2/n)</math><br />
<br />
so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(w_n) = \frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} w_n\Big\}</math><br />
<br />
But from above <math>r_n = \sqrt {w_n}</math>. By the change-of-variable formula we have<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n = r_n^2 \Rightarrow \frac {dw_n}{dr_n} = 2r_n</math><br /><br />
<br />
and so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> PDF(r_n) = 2r_n\frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} r_n^2\Big\} = \frac {r_n}{\alpha^2} \exp\Big \{-\frac {r_n^2}{2\alpha^2} \Big\},\;\;\alpha \equiv \sigma/\sqrt n</math><br />
<br />
So for any number of shots <math>n</math>, the expected accuracy is given by <math>r_n</math> follows a Rayleigh distribution with parameter <math>\alpha = \sigma / \sqrt{n}</math> where <math>\sigma</math> is the Rayleigh shape factor for one shot. <br />
<br />
'''Thanks to Alecos Papadopoulos for the solution.'''<br />
<br />
=== Precision of Mean Radius for <math>n</math> Shots About COI ===<br />
<br />
=== CEP Values ===</div>Herbhttp://ballistipedia.com/index.php?title=User:Herb&diff=1235User:Herb2015-06-13T20:12:37Z<p>Herb: /* Wiki pages I created */</p>
<hr />
<div><br />
[[MediaWiki:Sidebar]]<br />
<br />
[http://ballistipedia.com/index.php?title=Special:AllPages| All Pages]<br />
<br />
=My notion of sidebar=<br />
<br />
* [http://ballistipedia.com/index.php?title=What_is_Precision%3F What is precision?]<br />
* [[Projectile Dispersion Classifications]]<br />
* [[Measuring Precision]]<br />
* [[Herb_References]]<br />
* Examples<br />
<br />
<br />
<br />
= Measures =<br />
<br />
[[Extreme Spread]]<br />
<br />
[[Figure of Merit]]<br />
<br />
[[Mean Radius]]<br />
<br />
= Wiki pages I created =<br />
<br />
[[Covering Circle Radius versus Extreme Spread]] - should be pretty good. <br />
<br />
[[Data Transformations to Rayleigh Distribution]]<br />
<br />
[[Derivation of the Rayleigh Distribution Equation | Mathematical Formulas and Derivations]]<br />
<br />
[[Projectile Dispersion Classifications]] - getting close...<br />
<br />
[[Error Propagation]]<br />
<br />
[[Extreme Spread]] * measure<br />
<br />
[[Figure of Merit]] * measure<br />
<br />
[[Fliers vs. Outliers]]<br />
<br />
[[Leslie 1993]] - notion ok, disagree with content on page. <br />
<br />
[[Measuring Precision]] - this is fairly solid. <br />
<br />
[[Mean Radius]] * measure<br />
<br />
[[Sighting a Weapon]] ** needs work<br />
<br />
[[Stringing]] seems mostly ok. Fuzzy on how to handle inter/exterior ballastics. <br />
<br />
[[What is ρ in the Bivariate Normal distribution?]] think this pretty good. <br />
<br />
----<br />
<br />
<br />
Interrelationship of the Range Measurements<br />
* Range<br />
* Studentized Range<br />
** Covering Circle<br />
** Diagonal<br />
** ES<br />
** FOM<br />
** ES<br />
<br />
[[Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD]]<br />
<br />
---<br />
Carnac the Magnificent<br />
----<br />
<br />
Suppose that Xk has the gamma distribution with shape parameter k∈(0,∞) and fixed scale parameter b∈(0,∞). Then the distribution of the standardized variable below converges to the standard normal distribution as k→∞:<br />
<br />
&nbsp;<math>Z_k = \frac{X_k−kb}{b\sqrt{k}}</math><br />
<br />
----<br />
Measurements<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Circular Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Orthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Errors caused by Nonorthogonal Elliptical Dispersion<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
|<br />
# Circular Error Probable - CEP(50)<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure (CEP(50) From Ranking)<br />
## Outlier Tests<br />
# CEP(50) From Fitting to Rayleigh distribution<br />
## Parameters Needed<br />
## PDF<br />
## CDF<br />
## Mode, Median, Mean, Standard Deviation, %RSD<br />
## Sample Variance and Its distribution<br />
## Errors<br />
### Due to Orthogonal Elliptical Dispersion<br />
### Due to Hoyt Dispersion<br />
### Due to Fliers<br />
## Outlier Tests<br />
### Within a System<br />
### Between Systems<br />
|}<br />
<br />
<br />
<br />
<br />
# Elliptical Error Probable<br />
# Experimental Summary<br />
## Given<br />
## Assumptions<br />
## Data transformation<br />
## Experimental Measure<br />
## Outlier Tests<br />
# Theoretical ES Distribution<br />
## Dispersion by Rayleigh Distribution<br />
## Dispersion by Orthogonal Elliptical Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
## Dispersion by Hoyt Distribution<br />
### Parameters Needed<br />
### PDF<br />
### CDF<br />
### Mode, Median, Mean, Standard Deviation, %RSD<br />
### Sample Variance and Its distribution<br />
### Outlier Tests<br />
# See Also<br />
<br />
<br />
<br />
----<br />
<br />
"The difference between theory and practice is larger in<br />
practice than in theory."<br />
<br />
In theory there is no difference between theory and practice. But, in practice, there is.<br />
<br />
<br />
----<br />
<br />
sighting shot distribution<br />
<br />
The Mean Radius is the average distance over all shots to the groups center.<br />
<br />
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"<br />
|-<br />
! <br />
! <br />
|-<br />
| Given<br />
|<br />
* set of ''n'' shots {<math> (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) </math>}<br /> for which all of the (''h'',''v'') positions are known<br />
|-<br />
| Assumptions<br />
|<br />
* Origin at <math>(r,\theta) = (0,0)</math><br />
* Rayleigh Distribution for Shots<br />
** <math>\sigma_h = \sigma_v</math><br />
**<math>\rho = 0</math><br />
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math><br />
* With conversion from Cartesian coordinates to Polar coordinates, <math>\theta</math> will be entirely random and independent of radius<br />
* No Flyers<br />
|-<br />
| Data Pretreatment<br />
| Shot impact positions converted from Cartesian Coordinates (''h'', ''v'') to Polar Coordinates <math>(r,\theta)</math><br />
* Origin translated from Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
|-<br />
| Experimental Measure<br />
| <math>\bar{r_n}</math> - the average radius of ''n'' shots<br />
<math>\bar{r_n} = \sum_{i=1}^n r_i / n</math><br /><br />
&nbsp;&nbsp;&nbsp; where <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math><br />
|-<br />
| <math>PDF_{r_0}(r; n, \sigma)</math><br />
| <math>\frac{nr}{\sigma^2}e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| <math>CDF_{r_0}(r; n, \sigma)</math><br />
| <math>1 - e^{-nr^2/2\sigma^2}</math><br />
|-<br />
| Mode of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}</math><br />
|-<br />
| Median of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{ln{(4)}}</math><br />
|-<br />
| Mean of PDF(<math>\bar{r_n}</math>)<br />
| <math> \frac{\sigma}{\sqrt{n}}\sqrt{\frac{\pi}{2}}</math><br />
|-<br />
| (h,v) for all points?<br />
| Yes<br />
|- <br />
| Symmetric about Measure?<br />
| <br />
|- <br />
| NSPG Invariant<br />
| No<br />
|-<br />
| Robust<br />
| No<br />
|}<br />
<br />
= master ref page =<br />
<br />
I like the structure of this wiki page. You can look at the "groups of papers" then jump to a specific paper and use the browser back button to go back to the group. <br />
<br />
Could we make this the "master" reference page?<br />
<br />
(1) Move references to top of page <br />
(2) put TOC that floats to right<br />
(3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model)<br />
(4) Each level 1 heading would have various "groups" of papers. <br />
(5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page<br />
<br />
how I'd redo references so as to provide some that was "linkable" and could be "named"<br />
<br />
So '''Blischke_Halpin_1966''' could be name of wiki page and a "named" link within the page. thus reference in a wiki page would be something like:<br />
: yada yada yada (Blischke_Halpin_1966) yada yada yada <br />
the link would jump to the "master" page of references to that entry. <br />
<br />
; Blischke_Halpin_1966<br />
:Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775<br />
; Chew_Boyce_1962<br />
:Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181<br />
: Culpepper_1978<br />
;Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117</div>Herbhttp://ballistipedia.com/index.php?title=Derivation_of_the_Rayleigh_Distribution_Equation&diff=1234Derivation of the Rayleigh Distribution Equation2015-06-13T19:47:10Z<p>Herb: /* The Rayleigh distribution */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
= Mathematical Formulas and Derivations =<br />
<br />
= Bivariate Normal Distribution =<br />
Starting only with the assumptions that the horzontial and vertical measurements are normally distributed as notated by:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma_h^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma_v^2)</math><br />
<br />
then the horizontal and vertical measures follow the general bivariate normal distribution which is given by the following equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Simplification of the Bivariate Normal Distribution to the Hoyt Distribution<br />
<br />
= Correction Factors =<br />
The following three correction factors will be used throughout this statistical inference and deduction. <br />
<br />
Note that all of these correction factors are > 1, are significant for very small ''n'', and converge towards 1 as <math>n \to \infty</math>. Their values are listed for ''n'' up to 100 in [[Media:Sigma1ShotStatistics.ods]]. [[File:SymmetricBivariate.c]] uses Monte Carlo simulation to confirm that their application produces valid corrected estimates.<br />
<br />
== [http://en.wikipedia.org/wiki/Bessel%27s_correction Bessel correction factor] ==<br />
The Bessel correction removes bias in sample variance.<br />
:&nbsp; <math>c_{B}(n) = \frac{n}{n-1}</math><br />
<br />
== [http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution Gaussian correction factor] ==<br />
The Gaussian correction (sometimes called <math>c_4</math>) removes bias introduced by taking the square root of variance.<br />
:&nbsp; <math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math><br />
<br />
The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:&nbsp; <math>c_{G}(n)</math> <code>=1/EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))</code><br />
<br />
== Rayleigh correction factor ==<br />
The unbiased estimator for the Rayleigh distribution is also for <math>\sigma^2</math>. The following corrects for the concavity introduced by taking the square root to get ''σ''.<br />
:&nbsp; <math>c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}</math> <ref>[[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|''Statistical Inference for Rayleigh Distributions'', M. M. Siddiqui, 1964, p.1007]]</ref><br />
<br />
To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula: <code>=EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))</code><br />
<br />
<br />
= The Hoyt Distribution =<br />
== Derivation From the Bivariate Normal distribution ==<br />
Given the Bivariate Normal distribution as follows:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution. This translation will not affect measurements about COI, but it would of course affect measurements which are measured about POA. <br />
<br />
Given a translation to point <math>(\mu_h, \mu_v)</math> then let:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>h_* = h - \mu_h</math> &nbsp;&nbsp; and &nbsp;&nbsp; <math>v_* = v - \mu_v</math><br /><br />
<br />
Since the derivative of <math>h_*</math> with respect to <math>(h - \mu_h)</math> is 1, (and similarity for <math>v_*</math>) then no change results to the integration constant of the function. Thus <math>h_*</math> can be substituted for <math>(h - \mu_h)</math> and <math>v_*</math> for <math>(v - \mu_v)</math>. At this point the asterisk subscript is superfluous and will be dropped, giving the Hoyt distribution. <br />
&nbsp;&nbsp;&nbsp;<math> <br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v }{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
= The Rayleigh distribution =<br />
<br />
== Derivations ==<br />
<br />
=== Derivation OF Single Shot PDF From the Bivariate Normal distribution ===<br />
<br />
The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:<br />
* Horizontal and vertical dispersion are independent. <br />
* <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
* <math>\rho = 0</math><br />
* No Fliers<br />
for which the PDF for any shot, <math>i</math>, around the horizontal and vertical point <math>(\mu_h, \mu_v)</math> is given by:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(r; \sigma_{\Re}) = \frac{r}{\sigma_{\Re}^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma_{\Re}^2} <br />
\right)<br />
</math><br />
: where <math>\sigma_{\Re} = \sigma_h = \sigma_v</math> and <math>r = \sqrt{h_i - \mu_h)^2 + sqrt(v_i - \mu_v)^2}</math><br />
<br />
'''PROOF'''<br />
<br />
Using the assumptions in the first section, the distribution of an individual shot is easily simplified from the Bivariate Normal Distribution which has the equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
By substituting <math>\rho = 0</math> the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v }<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
Since <math>\sigma_h</math> and <math>\sigma_v</math> are equal, substitute <math>\sigma</math> for each, then collect terms in the exponential, after which the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
-\left[<br />
\frac{(h-\mu_h)^2 + (v-\mu_h)^2}{2\sigma^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
Letting <math>r^2 = (h-\mu_h)^2 + (v-\mu_v)^2</math> the equation becomes: <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
Now transforming to the polar coordinate system:<br /><br />
<br />
** ok, here I'm lost **<br />
<br />
and finally:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(r) =<br />
\frac{r}{\sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
=== Derivation OF Single Shot CDF from the PDF ===<br />
<br />
Given the single shot Rayleigh distribution, calculate the single shot Cumulative Distribution Function (CDF) for the Rayleigh distribution.<br />
<br />
=== Calculate the Mode of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mode for the Rayleigh distribution.<br />
<br />
=== Calculate the Median of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
=== Calculate the Mean Radius of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
== Calculating the Rayleigh Shape factor, <math>\Re</math> ==<br />
<br />
=== Value from <math>n</math> Shots ===<br />
<br />
=== Confidence Intervals of <math>n</math> Shots ===<br />
<br />
== Experimental Uses of <math>\Re</math> ==<br />
<br />
=== Accuracy of <math>n</math> Sighting Shots ===<br />
<br />
Given the assumptions in the starting section we again substitute <math>\sigma</math> for both <math>\sigma_h</math> and <math>\sigma_v</math>. This simplifies the distributions of <math>h</math> and <math>v</math> to:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma^2)</math><br />
<br />
Now we take some number <math>n</math> of shots <math>( n \geq 1)</math>and calculate their centers <math>\bar{h}</math> and <math>\bar{v}</math> which will be normal distributions as well. <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>\bar{h} \sim \mathcal{N}(\mu_h,\sigma^2/n)</math>, and <math>\bar{v} \sim \mathcal{N}(\mu_v,\sigma^2/n)</math><br />
<br />
Let <math>r_n</math> be the distance of this sample center <math>(\bar{h}, \bar{v})</math> from the true distribution center <math>(\mu_h, \mu_v)</math> as:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>r_n = \sqrt{(\bar{h}-\mu_h)^2 + (\bar{v}-\mu_v)^2}</math><br />
<br />
Define random variables <math>Z_h</math> and <math>Z_v</math> as the squared ''Studentized'' horizontal and vertical errors by dividing by the respective standard deviations. Each of these variables with have a Chi-Squared Distribution with one degree of freedom.<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_h = \left(\frac{(\bar{h}-\mu_h)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{h}-\mu_h)^2 \sim \chi^2(1)</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_v = \left(\frac{(\bar{v}-\mu_v)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{v}-\mu_v)^2\sim \chi^2(1)</math><br /><br />
<br />
Define random the variable <math>W</math> which will have a Chi-Squared Distribution with two degrees of freedom as:<br /> <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>W = Z_x + Z_y =\frac n{\sigma^2}\left((\bar{v}-\mu_v)^2+(\bar{v}-\mu_v)^2\right)\sim \chi^2(2)</math><br />
<br />
Rescale the variable <math>W</math> by <math>\frac {\sigma^2}{n}</math> and denote the new variable <math>w_n</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n=\frac {\sigma^2}nW</math> and note that <math>w_n=r_n^2</math><br />
<br />
By the properties of a chi-square random variable, we have:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n \sim \text {Gamma}(k=1, \theta = 2\sigma^2/n) = \text{Exp}(2\sigma^2/n)</math><br />
<br />
so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(w_n) = \frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} w_n\Big\}</math><br />
<br />
But from above <math>r_n = \sqrt {w_n}</math>. By the change-of-variable formula we have<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n = r_n^2 \Rightarrow \frac {dw_n}{dr_n} = 2r_n</math><br /><br />
<br />
and so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> PDF(r_n) = 2r_n\frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} r_n^2\Big\} = \frac {r_n}{\alpha^2} \exp\Big \{-\frac {r_n^2}{2\alpha^2} \Big\},\;\;\alpha \equiv \sigma/\sqrt n</math><br />
<br />
So for any number of shots <math>n</math>, the expected accuracy is given by <math>r_n</math> follows a Rayleigh distribution with parameter <math>\alpha = \sigma / \sqrt{n}</math> where <math>\sigma</math> is the Rayleigh shape factor for one shot. <br />
<br />
'''Thanks to Alecos Papadopoulos for the solution.'''<br />
<br />
=== Precision of Mean Radius for <math>n</math> Shots About COI ===<br />
<br />
=== CEP Values ===</div>Herbhttp://ballistipedia.com/index.php?title=Derivation_of_the_Rayleigh_Distribution_Equation&diff=1233Derivation of the Rayleigh Distribution Equation2015-06-13T19:43:23Z<p>Herb: /* Experimental Uses */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
= Mathematical Formulas and Derivations =<br />
<br />
= Bivariate Normal Distribution =<br />
Starting only with the assumptions that the horzontial and vertical measurements are normally distributed as notated by:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma_h^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma_v^2)</math><br />
<br />
then the horizontal and vertical measures follow the general bivariate normal distribution which is given by the following equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Simplification of the Bivariate Normal Distribution to the Hoyt Distribution<br />
<br />
= Correction Factors =<br />
The following three correction factors will be used throughout this statistical inference and deduction. <br />
<br />
Note that all of these correction factors are > 1, are significant for very small ''n'', and converge towards 1 as <math>n \to \infty</math>. Their values are listed for ''n'' up to 100 in [[Media:Sigma1ShotStatistics.ods]]. [[File:SymmetricBivariate.c]] uses Monte Carlo simulation to confirm that their application produces valid corrected estimates.<br />
<br />
== [http://en.wikipedia.org/wiki/Bessel%27s_correction Bessel correction factor] ==<br />
The Bessel correction removes bias in sample variance.<br />
:&nbsp; <math>c_{B}(n) = \frac{n}{n-1}</math><br />
<br />
== [http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution Gaussian correction factor] ==<br />
The Gaussian correction (sometimes called <math>c_4</math>) removes bias introduced by taking the square root of variance.<br />
:&nbsp; <math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math><br />
<br />
The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:&nbsp; <math>c_{G}(n)</math> <code>=1/EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))</code><br />
<br />
== Rayleigh correction factor ==<br />
The unbiased estimator for the Rayleigh distribution is also for <math>\sigma^2</math>. The following corrects for the concavity introduced by taking the square root to get ''σ''.<br />
:&nbsp; <math>c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}</math> <ref>[[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|''Statistical Inference for Rayleigh Distributions'', M. M. Siddiqui, 1964, p.1007]]</ref><br />
<br />
To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula: <code>=EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))</code><br />
<br />
<br />
= The Hoyt Distribution =<br />
== Derivation From the Bivariate Normal distribution ==<br />
Given the Bivariate Normal distribution as follows:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution. This translation will not affect measurements about COI, but it would of course affect measurements which are measured about POA. <br />
<br />
Given a translation to point <math>(\mu_h, \mu_v)</math> then let:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>h_* = h - \mu_h</math> &nbsp;&nbsp; and &nbsp;&nbsp; <math>v_* = v - \mu_v</math><br /><br />
<br />
Since the derivative of <math>h_*</math> with respect to <math>(h - \mu_h)</math> is 1, (and similarity for <math>v_*</math>) then no change results to the integration constant of the function. Thus <math>h_*</math> can be substituted for <math>(h - \mu_h)</math> and <math>v_*</math> for <math>(v - \mu_v)</math>. At this point the asterisk subscript is superfluous and will be dropped, giving the Hoyt distribution. <br />
&nbsp;&nbsp;&nbsp;<math> <br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v }{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
= The Rayleigh distribution =<br />
<br />
== Derivations ==<br />
<br />
=== Derivation OF Single Shot PDF From the Bivariate Normal distribution ===<br />
<br />
The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:<br />
* Horizontal and vertical dispersion are independent. <br />
* <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
* <math>\rho = 0</math><br />
* No Fliers<br />
for which the PDF for any shot, <math>i</math>, around the horizontal and vertical point <math>(\mu_h, \mu_v)</math> is given by:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(r; \sigma_{\Re}) = \frac{r}{\sigma_{\Re}^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma_{\Re}^2} <br />
\right)<br />
</math><br />
: where <math>\sigma_{\Re} = \sigma_h = \sigma_v</math> and <math>r = \sqrt{h_i - \mu_h)^2 + sqrt(v_i - \mu_v)^2}</math><br />
<br />
'''PROOF'''<br />
<br />
Using the assumptions in the first section, the distribution of an individual shot is easily simplified from the Bivariate Normal Distribution which has the equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
By substituting <math>\rho = 0</math> the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v }<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
Since <math>\sigma_h</math> and <math>\sigma_v</math> are equal, substitute <math>\sigma</math> for each, then collect terms in the exponential, after which the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
-\left[<br />
\frac{(h-\mu_h)^2 + (v-\mu_h)^2}{2\sigma^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
Letting <math>r^2 = (h-\mu_h)^2 + (v-\mu_v)^2</math> the equation becomes: <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
Now transforming to the polar coordinate system:<br /><br />
<br />
** ok, here I'm lost **<br />
<br />
and finally:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(r) =<br />
\frac{r}{\sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
=== Derivation OF Single Shot CDF from the PDF ===<br />
<br />
Given the single shot Rayleigh distribution, calculate the single shot Cumulative Distribution Function (CDF) for the Rayleigh distribution.<br />
<br />
=== Calculate the Mode of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mode for the Rayleigh distribution.<br />
<br />
=== Calculate the Median of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
=== Calculate the Mean Radius of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
== Rayleigh Shape factor, <math>\Re</math> ==<br />
<br />
=== Value from <math>n</math> Shots ===<br />
<br />
=== Confidence Intervals of <math>n</math> Shots ===<br />
<br />
== Experimental Uses ==<br />
<br />
=== Accuracy of <math>n</math> Sighting Shots ===<br />
<br />
Given the assumptions in the starting section we again substitute <math>\sigma</math> for both <math>\sigma_h</math> and <math>\sigma_v</math>. This simplifies the distributions of <math>h</math> and <math>v</math> to:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma^2)</math><br />
<br />
Now we take some number <math>n</math> of shots <math>( n \geq 1)</math>and calculate their centers <math>\bar{h}</math> and <math>\bar{v}</math> which will be normal distributions as well. <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>\bar{h} \sim \mathcal{N}(\mu_h,\sigma^2/n)</math>, and <math>\bar{v} \sim \mathcal{N}(\mu_v,\sigma^2/n)</math><br />
<br />
Let <math>r_n</math> be the distance of this sample center <math>(\bar{h}, \bar{v})</math> from the true distribution center <math>(\mu_h, \mu_v)</math> as:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>r_n = \sqrt{(\bar{h}-\mu_h)^2 + (\bar{v}-\mu_v)^2}</math><br />
<br />
Define random variables <math>Z_h</math> and <math>Z_v</math> as the squared ''Studentized'' horizontal and vertical errors by dividing by the respective standard deviations. Each of these variables with have a Chi-Squared Distribution with one degree of freedom.<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_h = \left(\frac{(\bar{h}-\mu_h)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{h}-\mu_h)^2 \sim \chi^2(1)</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_v = \left(\frac{(\bar{v}-\mu_v)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{v}-\mu_v)^2\sim \chi^2(1)</math><br /><br />
<br />
Define random the variable <math>W</math> which will have a Chi-Squared Distribution with two degrees of freedom as:<br /> <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>W = Z_x + Z_y =\frac n{\sigma^2}\left((\bar{v}-\mu_v)^2+(\bar{v}-\mu_v)^2\right)\sim \chi^2(2)</math><br />
<br />
Rescale the variable <math>W</math> by <math>\frac {\sigma^2}{n}</math> and denote the new variable <math>w_n</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n=\frac {\sigma^2}nW</math> and note that <math>w_n=r_n^2</math><br />
<br />
By the properties of a chi-square random variable, we have:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n \sim \text {Gamma}(k=1, \theta = 2\sigma^2/n) = \text{Exp}(2\sigma^2/n)</math><br />
<br />
so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(w_n) = \frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} w_n\Big\}</math><br />
<br />
But from above <math>r_n = \sqrt {w_n}</math>. By the change-of-variable formula we have<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n = r_n^2 \Rightarrow \frac {dw_n}{dr_n} = 2r_n</math><br /><br />
<br />
and so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> PDF(r_n) = 2r_n\frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} r_n^2\Big\} = \frac {r_n}{\alpha^2} \exp\Big \{-\frac {r_n^2}{2\alpha^2} \Big\},\;\;\alpha \equiv \sigma/\sqrt n</math><br />
<br />
So for any number of shots <math>n</math>, the expected accuracy is given by <math>r_n</math> follows a Rayleigh distribution with parameter <math>\alpha = \sigma / \sqrt{n}</math> where <math>\sigma</math> is the Rayleigh shape factor for one shot. <br />
<br />
'''Thanks to Alecos Papadopoulos for the solution.'''<br />
<br />
=== Precision of Mean Radius for <math>n</math> Shots About COI ===<br />
<br />
=== CEP Values ===</div>Herbhttp://ballistipedia.com/index.php?title=Derivation_of_the_Rayleigh_Distribution_Equation&diff=1232Derivation of the Rayleigh Distribution Equation2015-06-13T19:32:12Z<p>Herb: /* Variance from n Shots */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
= Mathematical Formulas and Derivations =<br />
<br />
= Bivariate Normal Distribution =<br />
Starting only with the assumptions that the horzontial and vertical measurements are normally distributed as notated by:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma_h^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma_v^2)</math><br />
<br />
then the horizontal and vertical measures follow the general bivariate normal distribution which is given by the following equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Simplification of the Bivariate Normal Distribution to the Hoyt Distribution<br />
<br />
= Correction Factors =<br />
The following three correction factors will be used throughout this statistical inference and deduction. <br />
<br />
Note that all of these correction factors are > 1, are significant for very small ''n'', and converge towards 1 as <math>n \to \infty</math>. Their values are listed for ''n'' up to 100 in [[Media:Sigma1ShotStatistics.ods]]. [[File:SymmetricBivariate.c]] uses Monte Carlo simulation to confirm that their application produces valid corrected estimates.<br />
<br />
== [http://en.wikipedia.org/wiki/Bessel%27s_correction Bessel correction factor] ==<br />
The Bessel correction removes bias in sample variance.<br />
:&nbsp; <math>c_{B}(n) = \frac{n}{n-1}</math><br />
<br />
== [http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution Gaussian correction factor] ==<br />
The Gaussian correction (sometimes called <math>c_4</math>) removes bias introduced by taking the square root of variance.<br />
:&nbsp; <math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math><br />
<br />
The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:&nbsp; <math>c_{G}(n)</math> <code>=1/EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))</code><br />
<br />
== Rayleigh correction factor ==<br />
The unbiased estimator for the Rayleigh distribution is also for <math>\sigma^2</math>. The following corrects for the concavity introduced by taking the square root to get ''σ''.<br />
:&nbsp; <math>c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}</math> <ref>[[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|''Statistical Inference for Rayleigh Distributions'', M. M. Siddiqui, 1964, p.1007]]</ref><br />
<br />
To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula: <code>=EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))</code><br />
<br />
<br />
= The Hoyt Distribution =<br />
== Derivation From the Bivariate Normal distribution ==<br />
Given the Bivariate Normal distribution as follows:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution. This translation will not affect measurements about COI, but it would of course affect measurements which are measured about POA. <br />
<br />
Given a translation to point <math>(\mu_h, \mu_v)</math> then let:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>h_* = h - \mu_h</math> &nbsp;&nbsp; and &nbsp;&nbsp; <math>v_* = v - \mu_v</math><br /><br />
<br />
Since the derivative of <math>h_*</math> with respect to <math>(h - \mu_h)</math> is 1, (and similarity for <math>v_*</math>) then no change results to the integration constant of the function. Thus <math>h_*</math> can be substituted for <math>(h - \mu_h)</math> and <math>v_*</math> for <math>(v - \mu_v)</math>. At this point the asterisk subscript is superfluous and will be dropped, giving the Hoyt distribution. <br />
&nbsp;&nbsp;&nbsp;<math> <br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v }{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
= The Rayleigh distribution =<br />
<br />
== Derivations ==<br />
<br />
=== Derivation OF Single Shot PDF From the Bivariate Normal distribution ===<br />
<br />
The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:<br />
* Horizontal and vertical dispersion are independent. <br />
* <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
* <math>\rho = 0</math><br />
* No Fliers<br />
for which the PDF for any shot, <math>i</math>, around the horizontal and vertical point <math>(\mu_h, \mu_v)</math> is given by:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(r; \sigma_{\Re}) = \frac{r}{\sigma_{\Re}^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma_{\Re}^2} <br />
\right)<br />
</math><br />
: where <math>\sigma_{\Re} = \sigma_h = \sigma_v</math> and <math>r = \sqrt{h_i - \mu_h)^2 + sqrt(v_i - \mu_v)^2}</math><br />
<br />
'''PROOF'''<br />
<br />
Using the assumptions in the first section, the distribution of an individual shot is easily simplified from the Bivariate Normal Distribution which has the equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
By substituting <math>\rho = 0</math> the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v }<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
Since <math>\sigma_h</math> and <math>\sigma_v</math> are equal, substitute <math>\sigma</math> for each, then collect terms in the exponential, after which the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
-\left[<br />
\frac{(h-\mu_h)^2 + (v-\mu_h)^2}{2\sigma^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
Letting <math>r^2 = (h-\mu_h)^2 + (v-\mu_v)^2</math> the equation becomes: <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
Now transforming to the polar coordinate system:<br /><br />
<br />
** ok, here I'm lost **<br />
<br />
and finally:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(r) =<br />
\frac{r}{\sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
=== Derivation OF Single Shot CDF from the PDF ===<br />
<br />
Given the single shot Rayleigh distribution, calculate the single shot Cumulative Distribution Function (CDF) for the Rayleigh distribution.<br />
<br />
=== Calculate the Mode of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mode for the Rayleigh distribution.<br />
<br />
=== Calculate the Median of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
=== Calculate the Mean Radius of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
== Rayleigh Shape factor, <math>\Re</math> ==<br />
<br />
=== Value from <math>n</math> Shots ===<br />
<br />
=== Confidence Intervals of <math>n</math> Shots ===<br />
<br />
== Experimental Uses ==<br />
<br />
=== Accuracy of <math>n</math> Sighting Shots ===<br />
<br />
Given the assumptions in the starting section we again substitute <math>\sigma</math> for both <math>\sigma_h</math> and <math>\sigma_v</math>. This simplifies the distributions of <math>h</math> and <math>v</math> to:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma^2)</math><br />
<br />
Now we take some number <math>n</math> of shots <math>( n \geq 1)</math>and calculate their centers <math>\bar{h}</math> and <math>\bar{v}</math> which will be normal distributions as well. <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>\bar{h} \sim \mathcal{N}(\mu_h,\sigma^2/n)</math>, and <math>\bar{v} \sim \mathcal{N}(\mu_v,\sigma^2/n)</math><br />
<br />
Let <math>r_n</math> be the distance of this sample center <math>(\bar{h}, \bar{v})</math> from the true distribution center <math>(\mu_h, \mu_v)</math> as:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>r_n = \sqrt{(\bar{h}-\mu_h)^2 + (\bar{v}-\mu_v)^2}</math><br />
<br />
Define random variables <math>Z_h</math> and <math>Z_v</math> as the squared ''Studentized'' horizontal and vertical errors by dividing by the respective standard deviations. Each of these variables with have a Chi-Squared Distribution with one degree of freedom.<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_h = \left(\frac{(\bar{h}-\mu_h)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{h}-\mu_h)^2 \sim \chi^2(1)</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_v = \left(\frac{(\bar{v}-\mu_v)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{v}-\mu_v)^2\sim \chi^2(1)</math><br /><br />
<br />
Define random the variable <math>W</math> which will have a Chi-Squared Distribution with two degrees of freedom as:<br /> <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>W = Z_x + Z_y =\frac n{\sigma^2}\left((\bar{v}-\mu_v)^2+(\bar{v}-\mu_v)^2\right)\sim \chi^2(2)</math><br />
<br />
Rescale the variable <math>W</math> by <math>\frac {\sigma^2}{n}</math> and denote the new variable <math>w_n</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n=\frac {\sigma^2}nW</math> and note that <math>w_n=r_n^2</math><br />
<br />
By the properties of a chi-square random variable, we have:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n \sim \text {Gamma}(k=1, \theta = 2\sigma^2/n) = \text{Exp}(2\sigma^2/n)</math><br />
<br />
so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(w_n) = \frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} w_n\Big\}</math><br />
<br />
But from above <math>r_n = \sqrt {w_n}</math>. By the change-of-variable formula we have<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n = r_n^2 \Rightarrow \frac {dw_n}{dr_n} = 2r_n</math><br /><br />
<br />
and so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> PDF(r_n) = 2r_n\frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} r_n^2\Big\} = \frac {r_n}{\alpha^2} \exp\Big \{-\frac {r_n^2}{2\alpha^2} \Big\},\;\;\alpha \equiv \sigma/\sqrt n</math><br />
<br />
So for any number of shots <math>n</math>, the expected accuracy is given by <math>r_n</math> follows a Rayleigh distribution with parameter <math>\alpha = \sigma / \sqrt{n}</math> where <math>\sigma</math> is the Rayleigh shape factor for one shot. <br />
<br />
'''Thanks to Alecos Papadopoulos for the solution.'''<br />
<br />
=== Precision of Mean Radius for <math>n</math> Shots About COI ===</div>Herbhttp://ballistipedia.com/index.php?title=Derivation_of_the_Rayleigh_Distribution_Equation&diff=1231Derivation of the Rayleigh Distribution Equation2015-06-13T19:26:49Z<p>Herb: /* Value from n Shots */</p>
<hr />
<div>{|align=right<br />
|__TOC__<br />
|}<br />
= Mathematical Formulas and Derivations =<br />
<br />
= Bivariate Normal Distribution =<br />
Starting only with the assumptions that the horzontial and vertical measurements are normally distributed as notated by:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma_h^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma_v^2)</math><br />
<br />
then the horizontal and vertical measures follow the general bivariate normal distribution which is given by the following equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
Simplification of the Bivariate Normal Distribution to the Hoyt Distribution<br />
<br />
= Correction Factors =<br />
The following three correction factors will be used throughout this statistical inference and deduction. <br />
<br />
Note that all of these correction factors are > 1, are significant for very small ''n'', and converge towards 1 as <math>n \to \infty</math>. Their values are listed for ''n'' up to 100 in [[Media:Sigma1ShotStatistics.ods]]. [[File:SymmetricBivariate.c]] uses Monte Carlo simulation to confirm that their application produces valid corrected estimates.<br />
<br />
== [http://en.wikipedia.org/wiki/Bessel%27s_correction Bessel correction factor] ==<br />
The Bessel correction removes bias in sample variance.<br />
:&nbsp; <math>c_{B}(n) = \frac{n}{n-1}</math><br />
<br />
== [http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution Gaussian correction factor] ==<br />
The Gaussian correction (sometimes called <math>c_4</math>) removes bias introduced by taking the square root of variance.<br />
:&nbsp; <math>\frac{1}{c_{G}(n)} = \sqrt{\frac{2}{n-1}}\,\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \, = \, 1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})</math><br />
<br />
The third-order approximation is adequate. The following spreadsheet formula gives a more direct calculation:&nbsp; <math>c_{G}(n)</math> <code>=1/EXP(LN(SQRT(2/(N-1))) + GAMMALN(N/2) - GAMMALN((N-1)/2))</code><br />
<br />
== Rayleigh correction factor ==<br />
The unbiased estimator for the Rayleigh distribution is also for <math>\sigma^2</math>. The following corrects for the concavity introduced by taking the square root to get ''σ''.<br />
:&nbsp; <math>c_{R}(n) = 4^n \sqrt{\frac{n}{\pi}} \frac{ N!(N-1)!} {(2N)!}</math> <ref>[[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|''Statistical Inference for Rayleigh Distributions'', M. M. Siddiqui, 1964, p.1007]]</ref><br />
<br />
To avoid overflows this is better calculated using log-gammas, as in the following spreadsheet formula: <code>=EXP(LN(SQRT(N/PI())) + N*LN(4) + GAMMALN(N+1) + GAMMALN(N) - GAMMALN(2N+1))</code><br />
<br />
<br />
= The Hoyt Distribution =<br />
== Derivation From the Bivariate Normal distribution ==<br />
Given the Bivariate Normal distribution as follows:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
a simple translation of the Cartesian Coordinate System converts the Bivariate Normal distribution to the Hoyt distribution. This translation will not affect measurements about COI, but it would of course affect measurements which are measured about POA. <br />
<br />
Given a translation to point <math>(\mu_h, \mu_v)</math> then let:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;<math>h_* = h - \mu_h</math> &nbsp;&nbsp; and &nbsp;&nbsp; <math>v_* = v - \mu_v</math><br /><br />
<br />
Since the derivative of <math>h_*</math> with respect to <math>(h - \mu_h)</math> is 1, (and similarity for <math>v_*</math>) then no change results to the integration constant of the function. Thus <math>h_*</math> can be substituted for <math>(h - \mu_h)</math> and <math>v_*</math> for <math>(v - \mu_v)</math>. At this point the asterisk subscript is superfluous and will be dropped, giving the Hoyt distribution. <br />
&nbsp;&nbsp;&nbsp;<math> <br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{h^2}{\sigma_h^2} +<br />
\frac{v^2}{\sigma_v^2} -<br />
\frac{2\rho h v }{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
= The Rayleigh distribution =<br />
<br />
== Derivations ==<br />
<br />
=== Derivation OF Single Shot PDF From the Bivariate Normal distribution ===<br />
<br />
The Rayleigh Distribution makes the following simplifying assumptions to the general bivariate normal distribution:<br />
* Horizontal and vertical dispersion are independent. <br />
* <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)<br />
* <math>\rho = 0</math><br />
* No Fliers<br />
for which the PDF for any shot, <math>i</math>, around the horizontal and vertical point <math>(\mu_h, \mu_v)</math> is given by:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(r; \sigma_{\Re}) = \frac{r}{\sigma_{\Re}^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma_{\Re}^2} <br />
\right)<br />
</math><br />
: where <math>\sigma_{\Re} = \sigma_h = \sigma_v</math> and <math>r = \sqrt{h_i - \mu_h)^2 + sqrt(v_i - \mu_v)^2}</math><br />
<br />
'''PROOF'''<br />
<br />
Using the assumptions in the first section, the distribution of an individual shot is easily simplified from the Bivariate Normal Distribution which has the equation:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}<br />
\exp\left(<br />
-\frac{1}{2(1-\rho^2)}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} -<br />
\frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v}<br />
\right]<br />
\right)<br />
</math><br />
<br />
By substituting <math>\rho = 0</math> the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma_h \sigma_v }<br />
\exp\left(<br />
-\frac{1}{2}\left[<br />
\frac{(h-\mu_h)^2}{\sigma_h^2} +<br />
\frac{(v-\mu_v)^2}{\sigma_v^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
<br />
Since <math>\sigma_h</math> and <math>\sigma_v</math> are equal, substitute <math>\sigma</math> for each, then collect terms in the exponential, after which the equation reduces to:<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
-\left[<br />
\frac{(h-\mu_h)^2 + (v-\mu_h)^2}{2\sigma^2} <br />
\right]<br />
\right)<br />
</math><br />
<br />
Letting <math>r^2 = (h-\mu_h)^2 + (v-\mu_v)^2</math> the equation becomes: <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(h,v) =<br />
\frac{1}{2 \pi \sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
Now transforming to the polar coordinate system:<br /><br />
<br />
** ok, here I'm lost **<br />
<br />
and finally:<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;<math><br />
f(r) =<br />
\frac{r}{\sigma^2 }<br />
\exp\left(<br />
- \frac{r^2}{2\sigma^2} <br />
\right)<br />
</math><br />
<br />
=== Derivation OF Single Shot CDF from the PDF ===<br />
<br />
Given the single shot Rayleigh distribution, calculate the single shot Cumulative Distribution Function (CDF) for the Rayleigh distribution.<br />
<br />
=== Calculate the Mode of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mode for the Rayleigh distribution.<br />
<br />
=== Calculate the Median of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
=== Calculate the Mean Radius of the Rayleigh distribution from its PDF ===<br />
Given the Rayleigh distribution, calculate the mean for the Rayleigh distribution.<br />
<br />
== Rayleigh Shape factor, <math>\Re</math> ==<br />
<br />
=== Value from <math>n</math> Shots ===<br />
<br />
=== Variance from <math>n</math> Shots ===<br />
<br />
== Experimental Uses ==<br />
<br />
=== Accuracy of <math>n</math> Sighting Shots ===<br />
<br />
Given the assumptions in the starting section we again substitute <math>\sigma</math> for both <math>\sigma_h</math> and <math>\sigma_v</math>. This simplifies the distributions of <math>h</math> and <math>v</math> to:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>h \sim \mathcal{N}(\mu_h,\sigma^2)</math>, and <math>v \sim \mathcal{N}(\mu_v,\sigma^2)</math><br />
<br />
Now we take some number <math>n</math> of shots <math>( n \geq 1)</math>and calculate their centers <math>\bar{h}</math> and <math>\bar{v}</math> which will be normal distributions as well. <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp; <math>\bar{h} \sim \mathcal{N}(\mu_h,\sigma^2/n)</math>, and <math>\bar{v} \sim \mathcal{N}(\mu_v,\sigma^2/n)</math><br />
<br />
Let <math>r_n</math> be the distance of this sample center <math>(\bar{h}, \bar{v})</math> from the true distribution center <math>(\mu_h, \mu_v)</math> as:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>r_n = \sqrt{(\bar{h}-\mu_h)^2 + (\bar{v}-\mu_v)^2}</math><br />
<br />
Define random variables <math>Z_h</math> and <math>Z_v</math> as the squared ''Studentized'' horizontal and vertical errors by dividing by the respective standard deviations. Each of these variables with have a Chi-Squared Distribution with one degree of freedom.<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_h = \left(\frac{(\bar{h}-\mu_h)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{h}-\mu_h)^2 \sim \chi^2(1)</math><br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>Z_v = \left(\frac{(\bar{v}-\mu_v)}{\sigma/\sqrt n}\right)^2 = \frac n{\sigma^2}(\bar{v}-\mu_v)^2\sim \chi^2(1)</math><br /><br />
<br />
Define random the variable <math>W</math> which will have a Chi-Squared Distribution with two degrees of freedom as:<br /> <br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>W = Z_x + Z_y =\frac n{\sigma^2}\left((\bar{v}-\mu_v)^2+(\bar{v}-\mu_v)^2\right)\sim \chi^2(2)</math><br />
<br />
Rescale the variable <math>W</math> by <math>\frac {\sigma^2}{n}</math> and denote the new variable <math>w_n</math>:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n=\frac {\sigma^2}nW</math> and note that <math>w_n=r_n^2</math><br />
<br />
By the properties of a chi-square random variable, we have:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n \sim \text {Gamma}(k=1, \theta = 2\sigma^2/n) = \text{Exp}(2\sigma^2/n)</math><br />
<br />
so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>PDF(w_n) = \frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} w_n\Big\}</math><br />
<br />
But from above <math>r_n = \sqrt {w_n}</math>. By the change-of-variable formula we have<br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math>w_n = r_n^2 \Rightarrow \frac {dw_n}{dr_n} = 2r_n</math><br /><br />
<br />
and so:<br /><br />
<br />
&nbsp;&nbsp;&nbsp;&nbsp;<math> PDF(r_n) = 2r_n\frac {n}{2\sigma^2}\cdot \exp\Big \{-\frac {n}{2\sigma^2} r_n^2\Big\} = \frac {r_n}{\alpha^2} \exp\Big \{-\frac {r_n^2}{2\alpha^2} \Big\},\;\;\alpha \equiv \sigma/\sqrt n</math><br />
<br />
So for any number of shots <math>n</math>, the expected accuracy is given by <math>r_n</math> follows a Rayleigh distribution with parameter <math>\alpha = \sigma / \sqrt{n}</math> where <math>\sigma</math> is the Rayleigh shape factor for one shot. <br />
<br />
'''Thanks to Alecos Papadopoulos for the solution.'''<br />
<br />
=== Precision of Mean Radius for <math>n</math> Shots About COI ===</div>Herb