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[[Measuring Precision]]
 
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Revision as of 00:03, 4 June 2015

Dispersion Assumptions

MediaWiki:Sidebar

Figure of Merit

Sighting a Weapon

Mean Radius

Derivation of the Rayleigh Distribution Equation

Measuring Precision

Fliers vs. Outliers

--- Carnac the Magnificent


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



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 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.

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.

In this case the wide shot would be an outlier if it was rejected based on some confidence limit.
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.




Note on spelling: Flier vs flyer has not been well established. We use the former spelling here because flyer seems to be more commonly used to refer to leaflets and architectural features, as opposed to "things that fly".