Difference between revisions of "Extreme Spread"

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(Variance and Its distribution)
 
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{| class="wikitable"
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| [[File:Bullseye.jpg|50px]] This page is a draft and needs review!
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|}
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= Experimental Summary =
 
= Experimental Summary =
  
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| Assumptions
 
| Assumptions
 
|
 
|
* Ideally the shots would follow a Rayleigh Distribution  
+
* Ideally the shots would have a circular pattern (i.e. the shots would have the Rayleigh Distribution).
** <math>\bar{h} \sim \mathcal{N}(\bar{h},\sigma_h^2), \bar{v} \sim \mathcal{N}(\bar{v},\sigma_v^2)</math>
+
** <math>h \sim \mathcal{N}(\bar{h},\sigma_h^2), v \sim \mathcal{N}(\bar{v},\sigma_v^2)</math>
 
** Horizontal and vertical dispersion are independent.  
 
** Horizontal and vertical dispersion are independent.  
 
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)
 
** <math>\sigma_h = \sigma_v</math> (realistically <math>\sigma_h \approx \sigma_v</math>)
 
** <math>\rho = 0</math>
 
** <math>\rho = 0</math>
** <math>PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}</math>
+
** <math>PDF_{r_i}(r) = \frac{r}{\Re^2}e^{-r^2/2\Re^2}</math>
:: '''Note:''' It is not necessary to fit <math>\sigma</math> to calculate the Figure of Merit.
+
:: '''Note:''' It is not necessary to calculate the COI, nor the constant <math>\Re</math>, to calculate the Extreme Spread.
 
* No Fliers
 
* No Fliers
 
|-
 
|-
 
| Data transformation
 
| Data transformation
| Identify two holes, <math>i, j</math> which are the farthest apart.  
+
| Identify two holes, <math>i, j</math> which are the farthest apart and measure <math>ES</math>.
 +
&nbsp;<math>ES = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}</math>
 
|-
 
|-
 
| Experimental Measure
 
| Experimental Measure
| <math>ES = \sqrt{(x_i - x_j)^2 - (y_i - y_j)^2)}</math>,
+
| <math>ES</math>
 
|}
 
|}
  
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== Assumptions ==
 
== Assumptions ==
  
None are needed to make measurement. However making assumptions about the dispersion will enable theoretical predictions about the measurement.
+
None are needed to make measurement. However some points are worth considering.
  
== Data transformation ==
+
* 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.
  
The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart. 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.
+
:: 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.  
  
== Experimental Measure ==
+
* 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.
  
If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper 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.
+
== Data transformation ==
 
 
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.
 
 
 
== Outlier Tests ==
 
  
= Theoretical <math>ES</math> Distribution =
+
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.
  
Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates. Since the distribution is positively skewed: Mean > Median > Mode.  
+
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.  
  
{| class="wikitable"  
+
:{| class="wikitable"  
|+ Theoretical <math>ES</math> Distribution
+
| [[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.  
|-
 
| Parameters Needed
 
|
 
|-
 
| <math>PDF(r; \sigma)</math>
 
| no direct evaluation, must be simulated via Monte Carlo
 
|-
 
| <math>CDF(r; \sigma)</math>
 
| no direct evaluation, must be simulated via Monte Carlo
 
|-
 
| Mode of PDF)
 
| depends on <math>n</math>, in general Mode increases as number of shots increases.  
 
|-
 
| Median of PDF
 
| depends on <math>n</math>, in general Median increases as number of shots increases.
 
|-
 
| Mean of PDF
 
| depends on <math>n</math>, in general Median increases as number of shots increases
 
|-
 
| Variance
 
| no direct evaluation, must be simulated via Monte Carlo
 
|-
 
| Variance Distribution
 
|
 
|-
 
| (h,v) for all points?
 
| yes for simulation.
 
|-
 
| Symmetric about Mean?
 
| No, skewed to larger values.
 
More symmetric about mean as the number of shots increases.  
 
 
|}
 
|}
  
 +
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.
  
== Parameters Needed ==
+
== Experimental Measure ==
 
 
== PDF ==
 
 
 
== CDF ==
 
  
== Mode, Median, Mean, Variance, %RSD of Mean ==
+
No calculation needs to be done to get the measurement. The single physical measurement is the data sought for the target.
 
 
{| class="wikitable"
 
|+ Table columns for "ES Values from Monte Carlo Simulation" Table
 
|-
 
| number of shots
 
|
 
|-
 
| Mode
 
|
 
|-
 
| Median
 
|
 
|-
 
| Mean
 
|
 
|-
 
| "Normality Error"
 
| As sort of a crude indication of normality let's use the value:
 
 
 
"Normality Error" = <math> \frac{\frac{CDF(25) - CDF(75)}{2} - Mean}{Mean} {\dot 100}</math>
 
 
 
So we measure half the distance between the 25<sup>th</sup> percentile and the 75<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.
 
* + value means positively skewed,
 
* - value means negatively skewed.  
 
|-
 
| Variance
 
|
 
|-
 
| %RSD
 
|
 
|}
 
 
 
 
 
{| class="wikitable"
 
|+ Theoretical <math>ES Values from Monte Carlo Simulation</math> Distribution
 
|-
 
! number of shots
 
! Mode
 
! Median
 
! Mean
 
! "Normality Error"
 
! Variance
 
! %RSD
 
|-
 
| 2
 
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| 4
 
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| 5
 
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| 6
 
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== Sample Variance and Its distribution ==
 
 
 
== Outlier Tests ==
 
  
 
= See Also =
 
= See Also =
  
[[Dispersion Assumptions]] - A discussion of the different cases for shot dispersion
+
[[Projectile Dispersion Classifications]] - A discussion of the different cases for projectile dispersion
 
 
Other measurements practical for range use are:
 
 
 
* [[Covering Circle Radius]] - about same precision as Extreme Spread if Rayleigh distributed
 
* [[Diagonal]] - somewhat better precision than Extreme Spread if Rayleigh distributed
 
* [[Figure of Merit]] - somewhat better precision than Extreme Spread if Rayleigh distributed
 

Latest revision as of 18:42, 9 January 2024

Bullseye.jpg This page is a draft and needs review!

Experimental Summary

Given
  • set of n shots {\( (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) \)}

All of the (h,v) positions do not need to be known so a ragged hole will suffice.

Assumptions
  • Ideally the shots would have a circular pattern (i.e. the shots would have the Rayleigh Distribution).
    • \(h \sim \mathcal{N}(\bar{h},\sigma_h^2), v \sim \mathcal{N}(\bar{v},\sigma_v^2)\)
    • Horizontal and vertical dispersion are independent.
    • \(\sigma_h = \sigma_v\) (realistically \(\sigma_h \approx \sigma_v\))
    • \(\rho = 0\)
    • \(PDF_{r_i}(r) = \frac{r}{\Re^2}e^{-r^2/2\Re^2}\)
Note: It is not necessary to calculate the COI, nor the constant \(\Re\), to calculate the Extreme Spread.
  • No Fliers
Data transformation Identify two holes, \(i, j\) which are the farthest apart and measure \(ES\).

 \(ES = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}\)

Experimental Measure \(ES\)

Given

The requirements for this test are very basic. Just a target with \(n\) 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.

Assumptions

None are needed to make measurement. However some points are worth considering.

  • 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.
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.
  • 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.

Data transformation

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.

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.

Bullseye.jpg A cheap ($10-$20) vernier caliper works fine. There is no need for a $2,000 one that measures to 1/10,000th 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.

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.

Experimental Measure

No calculation needs to be done to get the measurement. The single physical measurement is the data sought for the target.

See Also

Projectile Dispersion Classifications - A discussion of the different cases for projectile dispersion