Circular Error Probable

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Circular Error Probable (CEP)

The Circular Error Probable \(CEP(p)\) for \(p \in [0,1)\) is the estimated radius of the smallest circle that is expected to cover proportion \(p\) of the shot group.

Systematic Accuracy Bias

Some approaches to estimating CEP conflate the question of precision with the question of accuracy, or "sighting in."

The simpler case only tries to estimate precision, and computes CEP about the sample center.

The general case allows that the point-of-aim is offset from the true center point-of-impact. In the literature this is referred to as systematic accuracy bias. Including systematic accuracy bias sets the center of the circle to the point of aim, which means the sample center will probably be offset from that and CEP will be correspondingly larger.


Several different methods for estimating \(CEP(p)\) have been proposed which are based on different assumptions about the underlying distribution of coordinates (see the CEP literature overview for references and the shotGroups package for a free open source implementation):

  • The general correlated normal estimator (DiDonato & Jarnagin, 1961a; Evans, 1985) is based on the assumption of bivariate normality of the shot coordinates. It allows the x- and y-coordinates to be correlated and have different variances. This estimate can be based on the closed-form solution for the distribution of radial error in the bivariate normal distribution re-written in polar coordinates (radius and angle; Hoyt, 1947; Paris, 2009).
    • If systematic accuracy bias is taken into account, numerical integration of the multivariate normal distribution around an offset circle is required. The calculation of the correlated normal estimator is difficult and requires numerical approaches only available in specialized software.
  • The Grubbs-Pearson estimator (Grubbs, 1964) shares its assumptions with the general correlated normal estimator. It is based on the Pearson three-moment central \(\chi^{2}\)-approximation (Imhof, 1961; Pearson, 1959) of the cumulative distribution function of radial error in bivariate normal variables. This approach has the advantage that its calculation is much easier than the exact distribution and does not require special software. For \(p \geq 0.25\), the approximation to the true cumulative distribution function is very close but can diverge from it for \(p < 0.25\) with some distribution shapes.
  • The Grubbs-Patnaik estimator (Grubbs, 1964) differs from the Grubbs-Pearson estimator insofar as it is based on the Patnaik two-moment central \(\chi^{2}\)-approximation (Patnaik, 1949) of the true cumulative distribution function of radial error. For \(p < 0.5\) with some distribution shapes, the approximation can diverge significantly from the true cumulative distribution function.
  • The Grubbs-Liu estimate was not proposed by Grubbs but can be constructed following the same principle as his original estimators. It differs from them insofar as it is based on the recent Liu, Tang, and Zhang (2009) four-moment non-central \(\chi^{2}\)-approximation of the true cumulative distribution function of radial error.
  • The Ethridge (1983) estimator is not based on the assumption of bivariate normality of \((x,y)\)-coordinates but uses a robust unbiased estimator for the median radius (Hogg, 1967). This estimator "assumes that the square root of the radial miss distances follows the logarithmic generalized exponential power distribution." (Williams, 1997). It is only available for \(p = 0.5\).
  • The modified RAND R-234 estimator is an early example of CEP and is based on lookup tables that have later been fitted with a regression model to accommodate systematic accuracy bias (Pesapane & Irvine, 1977). It assumes a mostly circular distribution of \((x,y)\)-coordinates. In its original form it was only available for \(p = 0.5\), but McMillan & McMillan (2008) proposed an extension to levels \(p = 0.9\) and \(p = 0.95\) based on numerical simulations.

Comparing CEP estimators

If the true variances of x- and y-coordinates as well as their covariance is known then the closed-form general correlated normal estimator is ideal.

When we are confident in asserting a bivariate normal model for shot dispersion the Grubbs estimators are excellent approximations for reasonable values of p and ellipticity.

  • To date most comparison studies have only used the Grubbs-Patnaik estimator.
  • The Grubbs-Pearson estimator has the theoretical advantage over the Grubbs-Patnaik estimator that the approximating distribution matches the true distribution not only in mean and variance but also in skewness. Both the Grubbs-Pearson and Grubbs-Patnaik estimators are easy to calculate with standard software as long as the central \(\chi^{2}\)-distribution is available (as it is, for example, in spreadsheets).
  • If systematic accuracy bias is taken into account, the Grubbs-Liu estimator has the theoretical advantage over the Grubbs-Pearson estimator that the approximating distribution matches the true distribution not only in mean, variance, and skewness but also in kurtosis. If systematic accuracy bias is ignored, the Grubbs-Liu estimator is equivalent to the Grubbs-Pearson estimator. Its calculation is less complicated than the exact correlated normal estimator but requires the non-central \(\chi^{2}\)-distribution. This distribution might not be available in general tools like spreadsheets, but it is implemented in all statistics packages.

One shortcoming of the Grubbs estimators is that it is not possible to incorporate the confidence intervals of the variance estimates into the CEP estimate. This is particularly relevant to small samples where the variance estimates themselves are subject to considerable error.

In the special case where we assume uncorrelated bivariate normal data with equal variances the Rayleigh estimator does provide true confidence intervals, and it is easy to calculate using spreadsheets.

The Ethridge estimator stands out because it does not require bivariate normality of the \((x,y)\)-coordinates. It generalizes to three-dimensional data and can accommodate systematic accuracy bias, but it is limited to the 50% CEP.

Small Samples

For small samples we are more sensitive to which estimator is least bias and most efficient. This question has been studied, e.g., by Williams (1997).

A related question is which estimator is most robust to a very small number of outliers (or Fliers) that may result from clear operator error. See the literature overview for more comparison studies.