Difference between revisions of "Mean Radius"

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(See Also)
(Experimental Summary)
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|-
 
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| Data transformation
 
| Data transformation
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| Measure positions <math>(h_i, v_i)</math> for each shot, <math>i</math>.
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|-
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| Experimental Measure
 
| Preliminary Cartesian Calculations
 
| Preliminary Cartesian Calculations
 
* <math>\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2 </math>
 
* <math>\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2 </math>
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: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
 
: Thus the Cartesian Coordinate (<math>\bar{h}, \bar{v}</math>) is translated to Polar Coordinate <math>(r = 0, \theta = 0)</math><br />
 
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).
 
* <math>\theta</math> Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).
 +
 +
<math>\overline{r_n}</math> - the average radius of ''n'' shots
 +
 +
<math>\overline{r_n} = \sum_{i=1}^n r_i / n</math><br />
 
|-
 
|-
| Experimental Measure
+
| Outlier Tests
| <math>\bar{r_n}</math> - the average radius of ''n'' shots
+
|
<math>\bar{r_n} = \sum_{i=1}^n r_i / n</math><br />
 
 
|}
 
|}
  

Revision as of 11:55, 14 June 2015

Mean Radius

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

Experimental Summary

yada yada

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
  • The dispersion of shot \(i\) follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, \(\theta\) will be entirely random and independent of radius.
    • \(h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)\)
    • Horizontal and vertical dispersion are independent.
    • \(\sigma_h = \sigma_v\) (realistically \(\sigma_h \approx \sigma_v\))
    • \(\rho = 0\)
    • \(PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}\)
Note: It is not necessary to fit \(\sigma\) to calculate the Mean Radius.
  • No Fliers
Data transformation Measure positions \((h_i, v_i)\) for each shot, \(i\).
Experimental Measure Preliminary Cartesian Calculations
  • \(\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2 \)
  • \(\bar{v} = \frac{1}{n-1} \sum_{i=1}^n v_i^2 \)

Shot impact positions converted from Cartesian Coordinates

  • \(r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}\)
Thus the Cartesian Coordinate (\(\bar{h}, \bar{v}\)) is translated to Polar Coordinate \((r = 0, \theta = 0)\)
  • \(\theta\) Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).

\(\overline{r_n}\) - the average radius of n shots

\(\overline{r_n} = \sum_{i=1}^n r_i / n\)

Outlier Tests

Given

Assumptions

Data transformation

Experimental Measure

Outlier Tests

Theoretical \(r(1)\) Distribution

Distribution for a single shot as a function of r.


Theoretical \(r(1)\) Distribution
Parameters Needed \(\sigma\) - Rayleigh shape parameter fit to experimental shot distribution
\(PDF_{r(1)}(r; \sigma)\) \(\frac {r}{\sigma^2} \exp\Big \{-\frac {r^2}{2\sigma^2} \Big\}\)
\(CDF_{r(1)}(r; \sigma)\) \( 1 - \exp\Big \{-\frac {r^2}{2\sigma^2} \Big\}\)
Mode of \(PDF_{r(1)\) \(\sigma\)
Median of \(PDF_{r(1)\) \(\sigma\sqrt{\ln{4}}\)
Mean of \(PDF_{r(1)\) \(\sigma\sqrt{\frac{\pi}{2}}\)
Variance of \(PDF_{r(1)\) \(\frac{(4-\pi)}{2}\sigma^2\)
Variance Distribution
(h,v) for all points? Yes
Symmetric about Mean? No, skewed to larger values.

More symmetric as number of shots increases.

Parameters Needed

yada yada

Variance and Its distribution

yada yada

PDF

yada yada

CDF

Mode, Median, Mean

Outlier Tests

Theoretical \(\overline{r(n)}\) Distribution

Given:

  • \(n\) shots were taken on a target
  • The average mean radius, \(\overline{r(n)}\), was calculated
  • The Rayleigh shape parameter \(\sigma\) for an individual shot is known.

then using \(r\) as a variable, the properties of the distribution of the average mean radius for \(n\) shots is investigated in this section.

Theoretical \(\bar{r_n}\) Distribution
Parameters Needed \(n\) - n of shots in sample

\(\sigma\) - Rayleigh shape parameter from individual shot distribution

\(PDF(\bar{r_n}; n, \sigma)\) \(\frac{\Gamma(n,2\sigma^2)}{n}\)

where \(\Gamma(n,2\sigma^2)\) is the Gamma Distribution

\(CDF(r; n, \sigma)\)
Mode of PDF) \(\bar{r_n}\)
Median of PDF no closed form, but \(\approx 1.177\bar{r_n}\)
Mean of PDF \(\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}\)
Variance
Variance Distribution
(h,v) for all points? Yes
Symmetric about Measure? No, skewed to larger values.

More symmetric as number of shots increases.

NSPG Invariant Yes
Robust No

Parameters Needed

yada yada

Variance and Its distribution

yada yada

PDF

yada yada

CDF

yada yada

Mode, Median, Mean

yada yada

Outlier Tests

yada yada

Studentized Mean Radius

need table for this...

Outlier Tests

See Also

Projectile Dispersion Classifications - Discussion of other models for shot dispersion