Difference between revisions of "User:Herb"

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[[Dispersion Assumptions]]
 
  
 
[[MediaWiki:Sidebar]]
 
[[MediaWiki:Sidebar]]
  
[[Figure of Merit]]
+
[http://ballistipedia.com/index.php?title=Special:AllPages| All Pages]
  
[[Sighting a Weapon]]
+
=My notion of sidebar=
  
[[Mean Radius]]
+
* [http://ballistipedia.com/index.php?title=What_is_Precision%3F What is precision?]
 +
* [[Projectile Dispersion Classifications]]
 +
* [[Measuring Precision]]
 +
* [[Herb_References]]
 +
* Examples
  
 +
 +
 +
= Measures =
 +
 +
* Circular Error Probable (CEP)
 +
* Covering Circle Radius (CCR)
 +
* Diagonal (D)
 +
* Elliptical Error Probable (EEP)
 +
* [[Extreme Spread]]
 +
* [[Figure of Merit]]
 +
* Horizontal and Vertical Variances
 +
* [[Mean Radius]]
 +
* Rayleigh Distribution Mode (RDM)
 +
* Radial Standard Deviation (RSD)
 +
 +
= Wiki pages I created =
 +
 +
[[Covering Circle Radius versus Extreme Spread]] - should be pretty good.
 +
 +
[[Data Transformations to Rayleigh Distribution]]
 +
 +
[[Derivation of the Rayleigh Distribution Equation | Mathematical Formulas and Derivations]]
 +
 +
[[Projectile Dispersion Classifications]] - getting close...
 +
 +
[[Error Propagation]]
 +
 +
[[Extreme Spread]] * measure
 +
 +
[[Figure of Merit]] * measure
 +
 +
[[Fliers vs. Outliers]]
 +
 +
[[Leslie 1993]] - notion ok, disagree with content on page.
 +
 +
[[Measuring Precision]] - this is fairly solid.
 +
 +
[[Mean Radius]] * measure
 +
 +
[[Sighting a Weapon]] ** needs work
 +
 +
[[Stringing]] seems mostly ok. Fuzzy on how to handle inter/exterior ballastics.
 +
 +
[[What is ρ in the Bivariate Normal distribution?]] think this pretty good.
 +
 +
----
 +
 +
 +
Interrelationship of the Range Measurements
 +
* Range
 +
* Studentized Range
 +
** Covering Circle
 +
** Diagonal
 +
** ES
 +
** FOM
 +
** ES
 +
 +
[[Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD]]
  
 
---
 
---
 
Carnac the Magnificent
 
Carnac the Magnificent
 +
 +
ab initio
 +
----
 +
 +
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→∞:
 +
 +
&nbsp;<math>Z_k = \frac{X_k−kb}{b\sqrt{k}}</math>
 +
 +
----
 +
= Measurements =
 +
 +
{| class="wikitable" class="wikitable" style="font-size:&nbsp;"
 +
|-
 +
|
 +
# Circular Error Probable - CEP(50)
 +
# Experimental Summary
 +
## Given
 +
## Assumptions
 +
## Data transformation
 +
## Experimental Measure
 +
## Outlier Tests
 +
# Theoretical ES Distribution
 +
## Circular Dispersion
 +
### Parameters Needed
 +
### PDF
 +
### CDF
 +
### Mode, Median, Mean, Standard Deviation, %RSD
 +
### Sample Variance and Its distribution
 +
### Outlier Tests
 +
## Errors caused by Orthogonal Elliptical Dispersion
 +
### Parameters Needed
 +
### PDF
 +
### CDF
 +
### Mode, Median, Mean, Standard Deviation, %RSD
 +
### Sample Variance and Its distribution
 +
### Outlier Tests
 +
## Errors caused by Nonorthogonal Elliptical Dispersion
 +
### Parameters Needed
 +
### PDF
 +
### CDF
 +
### Mode, Median, Mean, Standard Deviation, %RSD
 +
### Sample Variance and Its distribution
 +
### Outlier Tests
 +
# See Also
 +
|
 +
# Circular Error Probable - CEP(50)
 +
# Experimental Summary
 +
## Given
 +
## Assumptions
 +
## Data transformation
 +
# CEP(50) Using Ranking
 +
## Value
 +
## Confidence Interval
 +
## Outlier Tests
 +
# CEP(50) Using Rayleigh distribution
 +
## Value
 +
## Confidence Interval
 +
## Outlier Tests
 +
|}
 +
 +
 +
 +
 +
# Elliptical Error Probable
 +
# Experimental Summary
 +
## Given
 +
## Assumptions
 +
## Data transformation
 +
## Experimental Measure
 +
## Outlier Tests
 +
# Theoretical ES Distribution
 +
## Dispersion by Rayleigh Distribution
 +
## Dispersion by Orthogonal Elliptical Distribution
 +
### Parameters Needed
 +
### PDF
 +
### CDF
 +
### Mode, Median, Mean, Standard Deviation, %RSD
 +
### Sample Variance and Its distribution
 +
### Outlier Tests
 +
## Dispersion by Hoyt Distribution
 +
### Parameters Needed
 +
### PDF
 +
### CDF
 +
### Mode, Median, Mean, Standard Deviation, %RSD
 +
### Sample Variance and Its distribution
 +
### Outlier Tests
 +
# See Also
 +
 +
 +
 +
----
 +
 +
  "The difference between theory and practice is larger in
 +
  practice than in theory."
 +
 +
In theory there is no difference between theory and practice. But, in practice, there is.
 +
 +
 
----
 
----
  
Line 73: Line 232:
 
| No
 
| No
 
|}
 
|}
 
 
 
----
 
 
In target shooting a "flier" is a shot that flies wide of the target, or too far from the other shots on the target. Every shooter experiences fliers.  The shooter may, or may not know the cause of the flier.
 
 
== Definitions ==
 
 
In using statistics to analyze target precision it is very necessary to differentiate between "fliers" and "outliers."
 
 
'''Definitions:'''
 
 
; Flier
 
: A shot that is atypical of some shooting process(es) for some reason. The shooter may or may not be aware of the atypical factor.
 
 
; Outlier
 
: A shot that is at an extreme value for the distribution. In general the shot would be outside of some set confidence interval for the distribution.
 
 
Thus there is a subtle but significant difference between the two terms as they are used on this website. The gist is that there is no way to mathematically model a flier since how its dispersion is distributed is unknown. However outliers can be mathematically modeled.
 
 
== Fliers ==
 
 
A flyer might have a known cause before the target is examined, for example:
 
* Benchrest-level shooters traditionally discard the first round(s) after cleaning a barrel as a "fouling" shot(s).  The friction difference between a clean and a fouled bore are enough to significantly alter the point of impact.
 
* A shooter may "call a flier" if he knows he committed an error that is not characteristic of his shooting.
 
* If the shots are being chronographed, then the shooter might "call a flier" on any shot that chronographs outside of the 95% confidence interval around the mean muzzle velocity.
 
 
However a flier (or fliers) might have an unknown cause, and might not be suspected until the shot pattern on the target is observed.
 
* If the shots are not being chronographed, but exhibit significant vertical stringing, then the shooter would suspect excessive muzzle velocity variation. So the shooter would need to design an experiment to test for that process difference. 
 
* A projectile might be off-balance in the distribution of mass, or in its aerodynamic characteristics.
 
 
The salient point is that some objective evidence of process variation must exist to be able to label a shot a flier. If the only evidence is the position of the holes in the target, then a shot can't be labeled a flier. The only way to analyze the target when just the relative holes positions are known is through the consideration of outliers.
 
 
== Outliers ==
 
 
'''But not every outlier is a flier.'''  Unbounded distributions have been accepted as models for the shooting process, and so outliers are part of both the model and real world, and that our model can correctly account for them if they are part of the modeled process.  Granted, if I had a rail gun on an indoor range and had triple-checked every component of every round I sent downrange I may not accept an unbounded normal distribution as a model of my shot dispersion.  But once we allow for outdoor conditions and normal ammunition, not to mention a shooter operating the gun, then in the normal course of events we will get outliers, and they ''are'' representative of the underlying normally-distributed process.
 
 
It is not unreasonable to accept a model that says 1 round in 100 is going to miss the target entirely.  If we are recording statistically significant samples and using robust estimators then including such outliers will not ruin our estimates.  And in a way our metrics for "statistical significance" will tell us whether an outlier is valid.  E.g., if in my first three shots after sighting in one shot nicks the edge of the target backer then I know right away I need more samples because so far my confidence interval is wider than my target!  If I take another 20 shots and they cluster into a single hole then perhaps I can decide whether to exclude the outlier as a "flier" or incorporate it as a sample from the "true" model of my precision.
 
 
Ideally maybe we do want to clip our unbounded distribution models, or maybe we want to overlay our shot distribution model with a Poisson dispersion model that allows us to exclude samples that may be due to a defective round, wind gust, etc.  But practically we are already pushing the bounds on the sample size needed just to determine covariance for a general bivariate normal model, so adding a fourth parameter to the [[Precision_Models#Models_of_Dispersion|models of dispersion]] may be a bridge too far.
 
 
== Examples ==
 
 
To perhaps belabor the difference between fliers and outliers consider the following examples.
 
 
'''Example 1''' - Ten shots are shot at a paper target with ten bulls-eyes.  The cartridge cases are lined after each is shot. After shooting it is discovered that nine of the shots are ammunition type-A  and one is type-B. Shot 7 is the type-B ammunition shot. 
 
 
:''Shot 7 is a flier and just ignored in the measurement(s). Note here that it doesn't matter where the shot hit. The only reason to allow type-A ammunition and type-B ammunition to mix would be if the two types were comparatively tested and found to have the same performance. Here performance doesn't just mean the same precision since the two types of ammunition could have the same precision, but have different average POI positions.''
 
 
'''Example 2''' - Ten shots are shot at a target (single bulls-eye). After shooting it is discovered that nine of the shots are ammunition type-A  and one is type-B. It is unknown which shot used the type-B ammunition. There is one shot which is wide of the group of the other nine.
 
 
:''In this situation the shot with the type-B ammunition is a really flier since it isn't of the same type as the other shots. Since it is unknown which shot used the type-B ammunition, it is invalid to just throw out "worst" shot and assumed it was the shot with type-B ammunition. The shot with the type-B ammunition may in fact be the closet shot to the center of the group!''
 
 
:''The wide shot can only be labeled as an "outlier" if it falls outside of some set confidence interval. Ideally the confidence interval for acceptance should be decided upon before the experiment, and then data outside of the confidence interval would be properly rejected.''
 
  
:''So here some ad hoc judgment may be required. The best option is probably to throw out the group/measurement entirely. This would be especially true if using the measurement Extreme Spread. However if we're using the mean radius measurement then the one Type-B shot probably won't perturb the mean radius measurement too much. Thus for the mean radius measurement the solution to the predicament might be to consider the confidence interval about the measurement to decide if the wide shot should be thrown out, and use the resulting 9 or 10 shot measurement. Such a group could be used to estimate the sample size needed to get a mean radius measurement of specific precision.''
+
= master ref page =
  
'''Example 3''' - Ten shots are shot at a target (single bulls-eye). There is one shot which is wide of the group of the other nine. The shooter has no idea why there is one wide shot.
+
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.  
  
:''In this case the wide shot would be an outlier if it was rejected based on some confidence limit.''
+
Could we make this the "master" reference page?
  
:''The nasty part here is that the wide shot might, unknown to the shooter, truly be a flier. For example in the manufacture of the projectile, this particular projectile might have had its mass off-balanced outside of the normal process variations. After shooting this would of course be impossible to determine. Even if this sort of quality problem had been suspected, it would be virtually impossible to measure for a commercial cartridge. So some ammunition manufacturing problems can not be isolated by independent measurement, but rather only a nebulous judgement is possible that the "quality" of the ammunition is "poor" based on the fact that the system variance was much larger than for other ammunition types.''
+
(1) Move references to top of page
 +
(2) put TOC that floats to right
 +
(3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model)
 +
(4) Each level 1 heading would have various "groups" of papers.  
 +
(5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page
  
 +
how I'd redo references so as to provide some that was "linkable" and could be "named"
  
<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:
<hr/>
+
: yada yada yada (Blischke_Halpin_1966) yada yada yada
 +
the link would jump to the "master" page of references to that entry.
  
<small>Note on spelling: ''Flier'' vs ''flyer'' has not been well established. We use the former spelling here because [http://www.dailywritingtips.com/flier-vs-flyer/ ''flyer'' seems to be more commonly used to refer to leaflets and architectural features, as opposed to "things that fly"].</small>
+
; Blischke_Halpin_1966
 +
: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
 +
; Chew_Boyce_1962
 +
: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
 +
: Culpepper_1978
 +
;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

Latest revision as of 12:48, 14 June 2015

MediaWiki:Sidebar

All Pages

My notion of sidebar


Measures

  • Circular Error Probable (CEP)
  • Covering Circle Radius (CCR)
  • Diagonal (D)
  • Elliptical Error Probable (EEP)
  • Extreme Spread
  • Figure of Merit
  • Horizontal and Vertical Variances
  • Mean Radius
  • Rayleigh Distribution Mode (RDM)
  • Radial Standard Deviation (RSD)

Wiki pages I created

Covering Circle Radius versus Extreme Spread - should be pretty good.

Data Transformations to Rayleigh Distribution

Mathematical Formulas and Derivations

Projectile Dispersion Classifications - getting close...

Error Propagation

Extreme Spread * measure

Figure of Merit * measure

Fliers vs. Outliers

Leslie 1993 - notion ok, disagree with content on page.

Measuring Precision - this is fairly solid.

Mean Radius * measure

Sighting a Weapon ** needs work

Stringing seems mostly ok. Fuzzy on how to handle inter/exterior ballastics.

What is ρ in the Bivariate Normal distribution? think this pretty good.



Interrelationship of the Range Measurements

  • Range
  • Studentized Range
    • Covering Circle
    • Diagonal
    • ES
    • FOM
    • ES

Derivation_of_the_Rayleigh_Distribution_Equation#BND_to_1_shot_RD

--- Carnac the Magnificent

ab initio


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→∞:

 \(Z_k = \frac{X_k−kb}{b\sqrt{k}}\)


Measurements

  1. Circular Error Probable - CEP(50)
  2. Experimental Summary
    1. Given
    2. Assumptions
    3. Data transformation
    4. Experimental Measure
    5. Outlier Tests
  3. Theoretical ES Distribution
    1. Circular Dispersion
      1. Parameters Needed
      2. PDF
      3. CDF
      4. Mode, Median, Mean, Standard Deviation, %RSD
      5. Sample Variance and Its distribution
      6. Outlier Tests
    2. Errors caused by Orthogonal Elliptical Dispersion
      1. Parameters Needed
      2. PDF
      3. CDF
      4. Mode, Median, Mean, Standard Deviation, %RSD
      5. Sample Variance and Its distribution
      6. Outlier Tests
    3. Errors caused by Nonorthogonal Elliptical Dispersion
      1. Parameters Needed
      2. PDF
      3. CDF
      4. Mode, Median, Mean, Standard Deviation, %RSD
      5. Sample Variance and Its distribution
      6. Outlier Tests
  4. See Also
  1. Circular Error Probable - CEP(50)
  2. Experimental Summary
    1. Given
    2. Assumptions
    3. Data transformation
  3. CEP(50) Using Ranking
    1. Value
    2. Confidence Interval
    3. Outlier Tests
  4. CEP(50) Using Rayleigh distribution
    1. Value
    2. Confidence Interval
    3. Outlier Tests



  1. Elliptical Error Probable
  2. Experimental Summary
    1. Given
    2. Assumptions
    3. Data transformation
    4. Experimental Measure
    5. Outlier Tests
  3. Theoretical ES Distribution
    1. Dispersion by Rayleigh Distribution
    2. Dispersion by Orthogonal Elliptical Distribution
      1. Parameters Needed
      2. PDF
      3. CDF
      4. Mode, Median, Mean, Standard Deviation, %RSD
      5. Sample Variance and Its distribution
      6. Outlier Tests
    3. Dispersion by Hoyt Distribution
      1. Parameters Needed
      2. PDF
      3. CDF
      4. Mode, Median, Mean, Standard Deviation, %RSD
      5. Sample Variance and Its distribution
      6. Outlier Tests
  4. See Also



  "The difference between theory and practice is larger in
  practice than in theory."
In theory there is no difference between theory and practice. But, in practice, there is.



sighting shot distribution

The Mean Radius is the average distance over all shots to the groups center.

Given
  • set of n shots {\( (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) \)}
    for which all of the (h,v) positions are known
Assumptions
  • Origin at \((r,\theta) = (0,0)\)
  • Rayleigh Distribution for Shots
    • \(\sigma_h = \sigma_v\)
    • \(\rho = 0\)
    • \(PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}\)
  • With conversion from Cartesian coordinates to Polar coordinates, \(\theta\) will be entirely random and independent of radius
  • No Flyers
Data Pretreatment Shot impact positions converted from Cartesian Coordinates (h, v) to Polar Coordinates \((r,\theta)\)
  • Origin translated from Cartesian Coordinate (\(\bar{h}, \bar{v}\)) to Polar Coordinate \((r = 0, \theta = 0)\)
Experimental Measure \(\bar{r_n}\) - the average radius of n shots

\(\bar{r_n} = \sum_{i=1}^n r_i / n\)
    where \(r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}\)

\(PDF_{r_0}(r; n, \sigma)\) \(\frac{nr}{\sigma^2}e^{-nr^2/2\sigma^2}\)
\(CDF_{r_0}(r; n, \sigma)\) \(1 - e^{-nr^2/2\sigma^2}\)
Mode of PDF(\(\bar{r_n}\)) \( \frac{\sigma}{\sqrt{n}}\)
Median of PDF(\(\bar{r_n}\)) \( \frac{\sigma}{\sqrt{n}}\sqrt{ln{(4)}}\)
Mean of PDF(\(\bar{r_n}\)) \( \frac{\sigma}{\sqrt{n}}\sqrt{\frac{\pi}{2}}\)
(h,v) for all points? Yes
Symmetric about Measure?
NSPG Invariant No
Robust No

master ref page

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.

Could we make this the "master" reference page?

(1) Move references to top of page (2) put TOC that floats to right (3) Have level 1 headings for various topics (eg CEP Literature, EEP Literature, ES, Rayleigh Model, Hoyt Model) (4) Each level 1 heading would have various "groups" of papers. (5) From some paper that we want to discuss create an off page link for that paper. (eg comments on "prior Art" page

how I'd redo references so as to provide some that was "linkable" and could be "named"

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:

: yada yada yada (Blischke_Halpin_1966) yada yada yada 

the link would jump to the "master" page of references to that entry.

Blischke_Halpin_1966
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
Chew_Boyce_1962
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
Culpepper_1978
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