Figure of Merit

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Figure of Merit The Figure Of Merit (FOM) is the average of the ranges for the width and height of the group.

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 shot dispersion would be a Rayleigh Distribution
    • \(\bar{h} \sim \mathcal{N}(\bar{h},\sigma_h^2), \bar{v} \sim \mathcal{N}(\bar{v},\sigma_v^2)\)
    • Horizontal and vertical dispersion are independent.
    • \(\sigma_h = \sigma_v\) (realistically \(\sigma_h \approx \sigma_v\))
    • \(\rho = 0\)
    • \(PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}\)
Note: It is not necessary to fit \(\sigma\), or to determine the COI, to calculate the FOM.
  • No Fliers
Data transformation Preliminary Cartesian Calculations (Conceptually)
  • Identify two shots that form the horizontal range and measure \(h_{Range}\)
 \(h_{Range} = \max\{\,h_1,\ \dots \ h_n\,\} - \min\{\,h_1,\ \dots\ h_n\}\)
  • Identify two shots that form the vertical range and measure \(v_{Range}\)
 \(v_{Range} = \max\{\,v_1,\ \dots \ v_n\,\} - \min\{\,v_1,\ \dots\ v_n\}\)
Experimental Measure \(FOM = \frac{1}{2}(h_{Range} + v_{Range})\)

Given

Assumptions

Data transformation

Experimental Measure

Outlier Tests

Theoretical Evaluations

Theoretical \(h_{Range}\) or \(v_{Range}\) Distributions

Theoretical \(FOM\) Distribution

yada yada

Theoretical \(FOM\) Distribution
Parameters Needed
\(PDF(r; \sigma)\)
\(CDF(r; \sigma)\)
Mode of PDF
Median of PDF
Mean of PDF
Variance
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

Variance and Its distribution

PDF

CDF

Mode, Median, Mean

Outlier Tests

See Also

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

Diagonal - A different way of combing horizontal and vertical measurement