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| − | = | + | = Modeling Dispersion = |
| − | + | Before considering the mathematical models that will be used for the actual actual calculations, let's consider the intrinsic character of how shots are dispersed. 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>. Since we are interested in shot dispersion on a two-dimensional target we will assume that the two dimensional analog of the Normal distribution, the [http://en.wikipedia.org/wiki/Bivariate_normal_distribution Bivariate Normal Distribution], describes, at least approximately, the dispersion of a gunshots about their center point, <math>(\bar{h}</math> and <math>\bar{v}</math>. The bivariate normal distribution also has separate parameters for the standard deviation in each dimension, <math>\sigma_h and \sigma_v</math>, as well as a correlation parameter ''ρ''. The full bivariate normal distribution is thus:<br > | |
| + | <math> | ||
| + | f(h,v) = | ||
| + | \frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}} | ||
| + | \exp\left( | ||
| + | -\frac{1}{2(1-\rho^2)}\left[ | ||
| + | \frac{(h-\bar{h})^2}{\sigma_h^2} + | ||
| + | \frac{(v-\bar{v})^2}{\sigma_v^2} - | ||
| + | \frac{2\rho(h-\bar{h})(v-\bar{v})}{\sigma_h \sigma_v} | ||
| + | \right] | ||
| + | \right) | ||
| + | </math> | ||
| + | |||
| + | For the overall equation note that the following restrictions apply:<br /> | ||
| + | <math>-1 ≤ \rho ≤ 1</math><br /> | ||
| + | <math> \sigma_h>0 </math> and <math> \sigma_v>0 </math> | ||
| + | |||
| + | Since we are primarily interested in the dispersion component, the overall assumption is that weapon is properly sighted so that the center of impact is the same as the point of aim. In practice this can be achieved with a simple translation of the horizontal and vertical coordinates from absolute values to values relative to the average point of impact. Therefore the terms controlled by [[FAQ#How_many_shots_do_I_need_to_sight_in.3F|sighting in the gun]] <math>(\bar{h} and \bar{v}</math> drop out in the simplification of the dispersion equation. | ||
| + | |||
| + | Looking at the overall equation two different mutually exclusive simplifications can be made: | ||
| + | |||
| + | * '''Either''' <math>\sigma_h = \sigma_v</math> (equal variances) '''or''' <math>\sigma_h \neq \sigma_v</math> (unequal variances). | ||
| + | : 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. | ||
| + | |||
| + | * '''Either''' <math>\rho = 0</math> (uncorrelated) '''or''' <math>\rho \neq 0</math> (correlated). | ||
| + | |||
| + | The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation. Before discussing the individual cases consider the following. To the extent that either <math>\sigma_h \neq \sigma_v</math> or <math>\rho \neq 0</math> then elliptical shot groups will result instead of circular shot groups. If the shot groups are not round then we have two options. We can either use a mathematical model for analysis that allows for stringing, or we can scale the raw data to remove the differences. | ||
| + | |||
| + | There isn't any theoretical ballistic requirement that requires correlation between the horizontal and vertical dispersion of gunshots. Therefore, most statistical measures implicitly assume <math>\rho = 0</math>. In general if <math>\rho \neq 0</math>, then there would be an elliptical shaped group with the major axis oriented at some angle to the horizontal or vertical axis. | ||
| + | |||
| + | We do know that targets can often exhibit vertical or horizontal stringing as evidenced by an elliptical shaped group along the vertical or horizontal axis respectively. Obviously in such cases <math>\sigma_h \neq \sigma_v</math>. | ||
| + | |||
| + | :(1) The primary source of horizontal stringing is crosswind. | ||
| + | ::If we measure the wind while shooting we can bound and remove a “wind correction” term from that axis. E.g., "Suppose the orthogonal component of wind is ranging at random from 0-10mph during the shooting. Given lag-time ''t'' this will expand the no-wind horizontal dispersion at the target by <math>\sigma_w</math>."<ref>Wind deflection is a function of the ballistic curve and distance, but can be expressed as a simple product of the cross-wind velocity and lag time. For more information on the "lag rule" see Bryan Litz, ''Applied Ballistics for Long Range Shooting, 2<sup>nd</sup> Edition'' (2011) A4; or Robert McCoy, ''Modern Exterior Ballistics, 2<sup>nd</sup> Edition'' (2012) 7.27.</ref> Since variances are additive we could adjust <math>\sigma_h</math> via the equation <math>{\sigma'}_h^2 = \sigma_h^2 - \sigma<sub>wind</sub>^2</math>. | ||
| + | :(2) The primary source of vertical stringing is muzzle velocity. | ||
| + | :: We can actually measure with a chronograph and then correct for that source of variation. E.g., If standard deviation of muzzle velocity is <math>\s_{mv}</math> then, given the bullet's ballistic model for the given target distance, the vertical spread attributable to that is some <math>\s_v</math>. Here too we can remove this known source of dispersion from our samples via the equation <math>s_v^2 = \s_v^2 - f(\sigma_{mv}^2</math>. This adjustment is shown in several of the examples: | ||
| + | ::* [[22LR CCI 40gr HV 40-shot 100-yard Example]] | ||
| + | ::* [[300BLK Subsonic 20-shot 100-yard Example]] | ||
| + | |||
| + | = Four Special Cases for Dispersion = | ||
| + | |||
| + | Neglecting flyers, and assuming perfect aim, the overall assumptions about shot dispersion result in four cases for statistical analysis. | ||
| + | |||
| + | Note that in the cases below the assumptions use ''approximately equal to'' <math>(\approx)</math>) rather than ''strictly equal to'' (=). 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. There is absolutely no method by which the true population values for <math>\sigma_h</math> and <math>\sigma_v</math> can be determined. We can only get estimates for the factors <math>\mu_h</math> and <math>\mu_v</math> and these estimates are at best only good to a scant few significant figures. Thus the difference between ''approximately equal to'' and ''strictly equal to'' is really under experimental control. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but practically there are limits. | ||
| + | |||
| + | ==Case 1, Equal variances and uncorrelated == | ||
| + | Given: <br /> | ||
| + | #<math>\sigma_h \approx \sigma_v</math><br /> | ||
| + | #<math>\rho \approx 0</math><br /> | ||
| + | # The radial distance for each shot, ''i'', is <math>r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}</math> | ||
| + | then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:<br /> | ||
| + | <math>f(r) = \frac{r}{\sigma_{RSD}^2} e^{-r^2/(2\sigma_{RSD}^2)}, \quad r \geq 0,</math> and <math>\sigma_{RSD}</math> is the distribution shape factor known as the Radial Standard Deviation.<br /> | ||
| + | |||
| + | This is really the best case for shot dispersion. Shot groups would be round. The following analysis models are appropriate: | ||
| + | * Circular Error Probable (CEP) | ||
| + | * Covering Circle Radius (CCR) | ||
| + | * Group Size (GS) | ||
| + | * Figure of Merit (FOM) | ||
| + | * Mean Radius (MR) | ||
| + | * Rayleigh Distribution | ||
| + | |||
| + | Notes: | ||
| + | # In this case that the FOM, and Group Size are different measurements. | ||
| + | # The group size would only depend on the two shots most distant in separation. 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. | ||
| + | # For the measures for the CCR, the GS and then FOM measurements a target would a ragged hole would be acceptable, but for the rest the {''h,v''} position of each shot must be known. | ||
| + | |||
| + | ==Case 2, Equal variances and correlated == | ||
| + | Given:<br /> | ||
| + | #<math>\sigma_h \approx \sigma_v</math><br /> | ||
| + | #<math>\rho \neq 0</math><br /> | ||
| + | # The {''h,v''} position of each shot must be known. | ||
| + | Then the following analysis models are appropriate: | ||
| + | * Elliptic Error Probable | ||
| + | |||
| + | ==Case 3, Unequal variances and uncorrelated == | ||
| + | Given:<br /> | ||
| + | # <math>\sigma_h \neq \sigma_v</math><br /> | ||
| + | # <math>\rho = 0</math><br /> | ||
| + | # The {''h,v''} position of each shot must be known. | ||
| + | Then the following analysis models are appropriate: | ||
| + | * Diagonal | ||
| + | * Individual Horizontal and Vertical variances | ||
| + | |||
| + | In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively. The mathematical formula for the dispersion distribution would be:<br /> | ||
| + | <math> | ||
| + | f(h,v) = | ||
| + | \frac{1}{2 \pi s_h s_v} | ||
| + | \exp\left( | ||
| + | -\frac{1}{2}\left[ | ||
| + | \frac{h^2}{s_h^2} + | ||
| + | \frac{v^2}{s_v^2} | ||
| + | \right] | ||
| + | \right) | ||
| + | </math> | ||
| + | |||
| + | ==Case 4, Unequal variances and correlated == | ||
| + | Given:<br /> | ||
| + | #<math>\sigma_h \neq \sigma_v</math><br /> | ||
| + | #<math>\rho \neq 0</math><br /> | ||
| + | # The {''h,v''} position of each shot must be known. | ||
| + | |||
| + | This is really the most complex case for shot dispersion. Shot groups would be elliptical or egg-shaped. The mathematical analysis would require the full version of the bivariate Gaussian distribution. The mathematical formula for the dispersion distribution would be:<br /> | ||
| + | <math> | ||
| + | f(h,v) = | ||
| + | \frac{1}{2 \pi s_h s_v \sqrt{1-\rho^2}} | ||
| + | \exp\left( | ||
| + | -\frac{1}{2(1-\rho^2)}\left[ | ||
| + | \frac{h^2}{s_h^2} + | ||
| + | \frac{v^2}{s_v^2} - | ||
| + | \frac{2\rho h v}{s_h s_v} | ||
| + | \right] | ||
| + | \right) | ||
| + | </math> | ||
Revision as of 23:46, 23 May 2015
Herb's talk page
Contents
Modeling Dispersion
Before considering the mathematical models that will be used for the actual actual calculations, let's consider the intrinsic character of how shots are dispersed. The Normal distribution is the broadly assumed probability model used for a single random variable and it is characterized by its mean \((\bar{x}\) and standard deviation \((\sigma)\). Since we are interested in shot dispersion on a two-dimensional target we will assume that the two dimensional analog of the Normal distribution, the Bivariate Normal Distribution, describes, at least approximately, the dispersion of a gunshots about their center point, \((\bar{h}\) and \(\bar{v}\). The bivariate normal distribution also has separate parameters for the standard deviation in each dimension, \(\sigma_h and \sigma_v\), as well as a correlation parameter ρ. The full bivariate normal distribution is thus:
\(
f(h,v) =
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}}
\exp\left(
-\frac{1}{2(1-\rho^2)}\left[
\frac{(h-\bar{h})^2}{\sigma_h^2} +
\frac{(v-\bar{v})^2}{\sigma_v^2} -
\frac{2\rho(h-\bar{h})(v-\bar{v})}{\sigma_h \sigma_v}
\right]
\right)
\)
For the overall equation note that the following restrictions apply:
\(-1 ≤ \rho ≤ 1\)
\( \sigma_h>0 \) and \( \sigma_v>0 \)
Since we are primarily interested in the dispersion component, the overall assumption is that weapon is properly sighted so that the center of impact is the same as the point of aim. In practice this can be achieved with a simple translation of the horizontal and vertical coordinates from absolute values to values relative to the average point of impact. Therefore the terms controlled by sighting in the gun \((\bar{h} and \bar{v}\) drop out in the simplification of the dispersion equation.
Looking at the overall equation two different mutually exclusive simplifications can be made:
- Either \(\sigma_h = \sigma_v\) (equal variances) or \(\sigma_h \neq \sigma_v\) (unequal variances).
- Obviously if we could measure both \(\sigma_h\) and \(\sigma_v\) 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.
- Either \(\rho = 0\) (uncorrelated) or \(\rho \neq 0\) (correlated).
The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation. Before discussing the individual cases consider the following. To the extent that either \(\sigma_h \neq \sigma_v\) or \(\rho \neq 0\) then elliptical shot groups will result instead of circular shot groups. If the shot groups are not round then we have two options. We can either use a mathematical model for analysis that allows for stringing, or we can scale the raw data to remove the differences.
There isn't any theoretical ballistic requirement that requires correlation between the horizontal and vertical dispersion of gunshots. Therefore, most statistical measures implicitly assume \(\rho = 0\). In general if \(\rho \neq 0\), then there would be an elliptical shaped group with the major axis oriented at some angle to the horizontal or vertical axis.
We do know that targets can often exhibit vertical or horizontal stringing as evidenced by an elliptical shaped group along the vertical or horizontal axis respectively. Obviously in such cases \(\sigma_h \neq \sigma_v\).
- (1) The primary source of horizontal stringing is crosswind.
- If we measure the wind while shooting we can bound and remove a “wind correction” term from that axis. E.g., "Suppose the orthogonal component of wind is ranging at random from 0-10mph during the shooting. Given lag-time t this will expand the no-wind horizontal dispersion at the target by \(\sigma_w\)."[1] Since variances are additive we could adjust \(\sigma_h\) via the equation \({\sigma'}_h^2 = \sigma_h^2 - \sigma<sub>wind</sub>^2\).
- (2) The primary source of vertical stringing is muzzle velocity.
- We can actually measure with a chronograph and then correct for that source of variation. E.g., If standard deviation of muzzle velocity is \(\s_{mv}\) then, given the bullet's ballistic model for the given target distance, the vertical spread attributable to that is some \(\s_v\). Here too we can remove this known source of dispersion from our samples via the equation \(s_v^2 = \s_v^2 - f(\sigma_{mv}^2\). This adjustment is shown in several of the examples:
Four Special Cases for Dispersion
Neglecting flyers, and assuming perfect aim, the overall assumptions about shot dispersion result in four cases for statistical analysis.
Note that in the cases below the assumptions use approximately equal to \((\approx)\)) rather than strictly equal to (=). 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. There is absolutely no method by which the true population values for \(\sigma_h\) and \(\sigma_v\) can be determined. We can only get estimates for the factors \(\mu_h\) and \(\mu_v\) and these estimates are at best only good to a scant few significant figures. Thus the difference between approximately equal to and strictly equal to is really under experimental control. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but practically there are limits.
Given:
- \(\sigma_h \approx \sigma_v\)
- \(\rho \approx 0\)
- The radial distance for each shot, i, is \(r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}\)
then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:
\(f(r) = \frac{r}{\sigma_{RSD}^2} e^{-r^2/(2\sigma_{RSD}^2)}, \quad r \geq 0,\) and \(\sigma_{RSD}\) is the distribution shape factor known as the Radial Standard Deviation.
This is really the best case for shot dispersion. Shot groups would be round. The following analysis models are appropriate:
- Circular Error Probable (CEP)
- Covering Circle Radius (CCR)
- Group Size (GS)
- Figure of Merit (FOM)
- Mean Radius (MR)
- Rayleigh Distribution
Notes:
- In this case that the FOM, and Group Size are different measurements.
- The group size would only depend on the two shots most distant in separation. 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.
- For the measures for the CCR, the GS and then FOM measurements a target would a ragged hole would be acceptable, but for the rest the {h,v} position of each shot must be known.
Given:
- \(\sigma_h \approx \sigma_v\)
- \(\rho \neq 0\)
- The {h,v} position of each shot must be known.
Then the following analysis models are appropriate:
- Elliptic Error Probable
Given:
- \(\sigma_h \neq \sigma_v\)
- \(\rho = 0\)
- The {h,v} position of each shot must be known.
Then the following analysis models are appropriate:
- Diagonal
- Individual Horizontal and Vertical variances
In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively. The mathematical formula for the dispersion distribution would be:
\(
f(h,v) =
\frac{1}{2 \pi s_h s_v}
\exp\left(
-\frac{1}{2}\left[
\frac{h^2}{s_h^2} +
\frac{v^2}{s_v^2}
\right]
\right)
\)
Given:
- \(\sigma_h \neq \sigma_v\)
- \(\rho \neq 0\)
- The {h,v} position of each shot must be known.
This is really the most complex case for shot dispersion. Shot groups would be elliptical or egg-shaped. The mathematical analysis would require the full version of the bivariate Gaussian distribution. The mathematical formula for the dispersion distribution would be:
\( f(h,v) = \frac{1}{2 \pi s_h s_v \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{h^2}{s_h^2} + \frac{v^2}{s_v^2} - \frac{2\rho h v}{s_h s_v} \right] \right) \)
- ↑ Wind deflection is a function of the ballistic curve and distance, but can be expressed as a simple product of the cross-wind velocity and lag time. For more information on the "lag rule" see Bryan Litz, Applied Ballistics for Long Range Shooting, 2nd Edition (2011) A4; or Robert McCoy, Modern Exterior Ballistics, 2nd Edition (2012) 7.27.
