Circular error probable

"Circular error" redirects here. For the circular error of a pendulum, seependulumandpendulum (mathematics).

Circular error probable(CEP),alsocircular error probabilityorcircle of equal probability,is a measure of a weapon system'sprecisionin themilitary scienceofballistics.It is defined as the radius of a circle, centered on the aimpoint, that is expected to enclose the landing points of 50% of therounds;said otherwise, it is themedianerror radius.That is, if a given munitions design has a CEP of 100 m, when 100 munitions are targeted at the same point, an average of 50 will fall within a circle with a radius of 100 m about that point.

CEP concept and hit probability. 0.2% outside the outmost circle.

There are associated concepts, such as the DRMS (distance root mean square), which is the square root of the average squared distance error, and R95, which is the radius of the circle where 95% of the values would fall in.

The concept of CEP also plays a role when measuring the accuracy of a position obtained by a navigation system, such asGPSor older systems such asLORANandLoran-C.

Concept

Circular bivariate normal distribution

20 hits distribution example

The original concept of CEP was based on acircular bivariate normaldistribution (CBN) with CEP as a parameter of the CBN just as μ and σ are parameters of thenormal distribution.Munitionswith this distribution behavior tend to cluster around themeanimpact point, with most reasonably close, progressively fewer and fewer further away, and very few at long distance. That is, if CEP isnmetres, 50% of shots land withinnmetres of the mean impact, 43.7% betweennand2n,and 6.1% between2nand3nmetres, and the proportion of shots that land farther than three times the CEP from the mean is only 0.2%.

CEP is not a good measure of accuracy when this distribution behavior is not met. Munitions may also have largerstandard deviationof range errors than the standard deviation of azimuth (deflection) errors, resulting in an ellipticalconfidence region.Munition samples may not be exactly on target, that is, the mean vector will not be (0,0). This is referred to asbias.

To incorporate accuracy into the CEP concept in these conditions, CEP can be defined as the square root of themean square error(MSE). The MSE will be the sum of thevarianceof the range error plus the variance of the azimuth error plus thecovarianceof the range error with the azimuth error plus the square of the bias. Thus the MSE results from pooling all these sources of error, geometrically corresponding toradiusof acirclewithin which 50% of rounds will land.

Several methods have been introduced to estimate CEP from shot data. Included in these methods are the plug-in approach of Blischke and Halpin (1966), the Bayesian approach of Spall and Maryak (1992), and the maximum likelihood approach of Winkler and Bickert (2012). The Spall and Maryak approach applies when the shot data represent a mixture of different projectile characteristics (e.g., shots from multiple munitions types or from multiple locations directed at one target).

Conversion

While 50% is a very common definition for CEP, the circle dimension can be defined for percentages.Percentilescan be determined by recognizing that the horizontal position error is defined by a 2D vector which components are two orthogonalGaussianrandom variables(one for each axis), assumeduncorrelated,each having a standard deviation$\sigma$.Thedistance erroris the magnitude of that vector; it is a property of2D Gaussian vectorsthat the magnitude follows theRayleigh distribution,with scale factor$\sigma$.Thedistanceroot mean square(DRMS), is$\sigma _{d}={\sqrt {2}}\sigma$and doubles as a sort of standard deviation, since errors within this value make up 63% of the sample represented by the bivariate circular distribution. In turn, the properties of the Rayleigh distribution are that its percentile at level$F\in [0\%,100\%]$is given by the following formula:

$Q(F,\sigma )=\sigma {\sqrt {-2\ln(1-F/100\%)}}$

or, expressed in terms of the DRMS:

$Q(F,\sigma _{d})=\sigma _{d}{\frac {\sqrt {-2\ln(1-F/100\%)}}{\sqrt {2}}}$

The relation between$Q$and$F$are given by the following table, where the$F$values for DRMS and 2DRMS (twice the distance root mean square) are specific to the Rayleigh distribution and are found numerically, while the CEP, R95 (95% radius) and R99.7 (99.7% radius) values are defined based on the68–95–99.7 rule

Measure of$Q$	Probability$F\,(\%)$
DRMS	63.213...
CEP	50
2DRMS	98.169...
R95	95
R99.7	99.7

We can then derive a conversion table to convert values expressed for one percentile level, to another.Said conversion table, giving the coefficients$\alpha$to convert$X$into$Y=\alpha.X$,is given by:

From$X\downarrow$to$Y\rightarrow$	RMS ($\sigma$)	CEP	DRMS	R95	2DRMS	R99.7
RMS ($\sigma$)	1.00	1.18	1.41	2.45	2.83	3.41
CEP	0.849	1.00	1.20	2.08	2.40	2.90
DRMS	0.707	0.833	1.00	1.73	2.00	2.41
R95	0.409	0.481	0.578	1.00	1.16	1.39
2DRMS	0.354	0.416	0.500	0.865	1.00	1.21
R99.7	0.293	0.345	0.415	0.718	0.830	1.00

For example, a GPS receiver having a 1.25 m DRMS will have a 1.25 m × 1.73 = 2.16 m 95% radius.

See also

• Probable error

Further reading

• Blischke, W. R.; Halpin, A. H. (1966). "Asymptotic Properties of Some Estimators of Quantiles of Circular Error".Journal of the American Statistical Association.61(315): 618–632.doi:10.1080/01621459.1966.10480893.JSTOR2282775.
• Grubbs, F. E. (1964). "Statistical measures of accuracy for riflemen and missile engineers". Ann Arbor, ML: Edwards Brothers.Ballistipedia pdf
• MacKenzie, Donald A.(1990).Inventing Accuracy: A Historical Sociology of Nuclear Missile Guidance.Cambridge, Massachusetts:MIT Press.ISBN978-0-262-13258-9.
• Spall, James C.; Maryak, John L. (1992). "A Feasible Bayesian Estimator of Quantiles for Projectile Accuracy from Non-iid Data".Journal of the American Statistical Association.87(419): 676–681.doi:10.1080/01621459.1992.10475269.JSTOR2290205.
• Winkler, V. and Bickert, B. (2012). "Estimation of the circular error probability for a Doppler-Beam-Sharpening-Radar-Mode," in EUSAR. 9th European Conference on Synthetic Aperture Radar, pp. 368–71, 23/26 April 2012.ieeexplore.ieee.org
• Wollschläger, Daniel (2014), "Analyzing shape, accuracy, and precision of shooting results with shotGroups".Reference manual for shotGroups

External links

• Circular Error ProbableinBallistipedia