# Talk:Measuring Precision

Ok, I'm happy with title on this and general layout. Still some clean up to do in the various measures. My notion is just to present the conceptual idea for the measures on this page. Each dispersion measure title would be linked to a wiki page describing that measure in more detail.

Herb (talk) 22:11, 3 June 2015 (EDT)

3.4 Elliptical Error Probable (EEP)

Need to do so more literature searching on this. I suspect that this includes Hoyt case where rho <> 0. I'm thinking of constraining it to ellipses oriented along horizontal and vertical axis. You could then convert to circle with a simple rescaling.

Herb (talk) 22:11, 3 June 2015 (EDT)

- EEP is fully analogous to CEP: For a given ellipse and a fixed bivariate normal shot distribution, you can either ask "what is the probability of a hit whithin my ellipse? Or you can ask "without changing the shape of the ellipse (its aspect ratio), how much do I have to blow it up (scale both axes uniformly) such that it is expected to cover, say, 50% of the hits?" In the latter case, the length of the major and minor axis is calculated by applying the scaling factor to the lengths of the ellipse that was originally specified. A special case results if the ellipse has the same center, shape and orientation as the ellipse that characterizes the bivariate shot distribution.

- EEP thus has nothing to do with the Hoyt distribution. The Hoyt distribution is the distribution of radial error with respect to the true center of a bivariate normal shot distribution that is correlated and has unequal variances. This is fully analogous to the Rayleigh distribution that is also the distribution of radial error, but in the more restricted case of equal variances and 0 correlation. Just like the Rayleigh distribution, the Hoyt distribution can be used to calculate CEP, but under more liberal assumptions.

- Sorry that I'm expressed myself so poorly. I'll refer to an orthogonal ellipse as an ellipse oriented along horizontal or vertical axis. By a nonorthogonal ellipse I mean one for which the major axis is at an angle to both the horizontal or vertical axis.
- For an orthogonal ellipse then \(\sigma_h \neq \sigma_v\). For such ellipses you could turn them into a circle with a simple rescaling. Easy enough to do on paper, but a bit laborious. The scaling factor is from ratio of variances.
- For an nonorthogonal ellipse then still \(\sigma_h \neq \sigma_v\) but "conversion" to a circle is a lot messier. You really need a computer program to do it. The issue isn't really the need for a computer but the idea that the computer is fitting multiple parameters and making multiple clip level decisions. A lot more mysticism as to what Oz is doing behind the curtain.
- So the issue that I'm wonder about is if it is worthwhile to set up the orthogonal ellipses as a special case since you're just "fitting" the one scaling constant around COI. For instance muzzle velocity variations causing vertical stringing on the target. You'd probably use a computer to do it anyway, it is just that the orthogonal constraint greatly reduces the amount of mysticism of what the computer is doing.

- thanks! I now understand that an ellipse along axes doesn't imply correlation.

3.8 Hoyt Distribution Parameters (Bivariate Normal Distribution Parameters)

Think changing name for this to "Hoyt Error Probable" to fit in with other such measures.

Herb (talk) 22:11, 3 June 2015 (EDT)

I think the two sections

- 3.10 Radial Standard Deviation of the Rayleigh Distribution
- 3.11 Rayleigh Distribution Shape Parameter

Should probably be merged. My notion is to swizzle Radial Standard Deviation so that it fits Rayleigh Distribution better. ie sqrt(2) factor.

Herb (talk) 22:11, 3 June 2015 (EDT)

- Rayleigh Distribution Shape Parameter just doesn't sound like a measure. I am just seduced by the sexy name. Inclusion of "standard deviation" is nice. "Radial" meaning a circle is good too. Mulling this over for the 43rd time in my mind, how about "Rayleigh Radial Scale Factor"?? "Radial" is really the keyword. Fits in with wikipedia use too. What do you think?

- You're right, bad terminology is hard to overcome. Electric circuits still are calculated as if a positive charge is moving instead of the electron. So a somewhat fresh start might be good.

- Just to be clear, at the top of this thread I was thinking of redefining RSD as \(r = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (r_i - \bar{r})^2}\),
- where \(\bar{r}\) is the mean radius. Thus Mean Radius and Radial Standard deviation would relate back to the Rayleigh Distribution. The point is that if \(\sigma_h \neq \sigma_v\) then you shouldn't be assuming that the Rayleigh Distribution models the shots. "Radius" and "Radial" inherently make you think of a circular pattern. Maybe "Mean Radius Variance"? That would really do what I want and make the two tie together. To let you in on a secret, I think the mean radius would be a better predictor of variance than the actual variance measurement (with a big assumption that the pattern is truly round. If pattern isn't round then the MR estimator of RSD wouldn't be robust.)

Herb (talk) 13:27, 4 June 2015 (EDT)

- Just to be clear, at the top of this thread I was thinking of redefining RSD as \(r = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (r_i - \bar{r})^2}\),

3.12 String Method

Need some literature reference for this. No point in just pulling crap measures out of thin air. Found a couple of forum references. I remember it being something like mean radius and looked it up. Sexy in that it is a nice example of Rice Distribution.

Herb (talk) 23:50, 3 June 2015 (EDT)

*This is just*Mean Radius * n*, so I would note it under the Mean Radius section as evidence that MR is nothing new. David (talk) 12:18, 4 June 2015 (EDT)*

4 Dispersion Measures about COI

... being more repetitive: The figure caption *"A circular dispersion is the Rayleigh distribution."* is a bit misleading: The distribution of shots (in the sense of (h,v) coordinates) is assumed to be circular normal. Then the distances to the true COI follows a Rayleigh distribution.

4.4 Elliptical Error Probable (EEP)

Also here: *"Elliptical Error Probable assumes that the shots follow the Hoyt distribution"* - if the shots (in the sense of (h,v) coordinates) are assumed to follow a bivariate normal distribution, the their distances to the true COI follows a Hoyt distribution.

*"The EEP is the only measurement considered which is appropriate for a non-circular distribution.*" - I wouldn't say that. CEP for elliptical shot distributions is fine if you're interested in the probability of hitting a disc around the COI. If you want to characterize dispersion of the shots ((h,v) coordinates), then CEP does of course not provide as much information about an elliptical shot group as the EEP.

*"The EEP(50) measurement were based on the median values then it would be a robust estimator."* - I find this problematic since the empirical median is a bad estimator for the true median. That's why there are things like the Hodges-Lehmann pseudo-median or the Hogg (1967) estimator. There are robust estimators for the covariance matrix like the MCD estimator.

5.1 Dispersion Measures From POA

Figure caption *"Rice Distribution - Shots dispersed about COI followed the Rayleigh distribution, but distance for each shot measured to the offset POA."* - see above: The distances of the shots to the true COI follow a Rayleigh distribution (not the shots themselves), and the distances of the shots to the origin follow a Rice distribution.