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Circular error probable

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Ballistics measure of a weapon system's precision

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

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

Circular error probable (CEP),^[1] alsocircular error probability^[2] orcircle of equal probability,^[3] is a measure of a weapon system'sprecision in themilitary science ofballistics. 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 themedian error radius, which is a 50%confidence interval.^[1]^[4] 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.

There are associated concepts, such as the DRMS (distance root mean square), which is the square root of the average squared distance error, a form of thestandard deviation. Another is the R95, which is the radius of the circle where 95% of the values would fall, a 95%confidence interval.

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

Concept

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The original concept of CEP was based on acircular bivariate normal distribution (CBN) with CEP as a parameter of the CBN just as μ and σ are parameters of thenormal distribution.Munitions with this distribution behavior tend to cluster around themean impact point, with most reasonably close, progressively fewer and fewer further away, and very few at long distance. That is, if CEP isn metres, 50% of shots land withinn metres of the mean impact, 43.7% betweenn and2n, and 6.1% between2n and3n metres, 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 deviation of 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 thevariance of the range error plus the variance of the azimuth error plus thecovariance of 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 toradius of acircle within 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

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While 50% is a very common definition for CEP, the circle dimension can be defined for percentages.Percentiles can be determined by recognizing that the horizontal position error is defined by a 2D vector which components are two orthogonalGaussian random variables (one for each axis), assumeduncorrelated, each having a standard deviation $\sigma$ . Thedistance error is the magnitude of that vector; it is a property of2D Gaussian vectors that 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 $Q {\displaystyle Q}$ and $F {\displaystyle F}$ are given by the following table, where the $F {\displaystyle F}$ 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 $Q {\displaystyle Q}$	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.^[5]^[6] Said conversion table, giving the coefficients $\alpha$ to convert $X {\displaystyle X}$ 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.

References

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^^a ^bCircular Error Probable (CEP), Air Force Operational Test and Evaluation Center Technical Paper 6, Ver 2, July 1987, p. 1
^Nelson, William (1988)."Use of Circular Error Probability in Target Detection". Bedford, MA: The MITRE Corporation; United States Air Force.Archived(PDF) from the original on October 28, 2014.
^Ehrlich, Robert (1985).Waging Nuclear Peace: The Technology and Politics of Nuclear Weapons. Albany, NY:State University of New York Press. p. 63.
^Payne, Craig, ed. (2006).Principles of Naval Weapon Systems. Annapolis, MD:Naval Institute Press. p. 342.
^Frank van Diggelen, "GPS Accuracy: Lies, Damn Lies, and Statistics",GPS World, Vol 9 No. 1, January 1998
^Frank van Diggelen, "GNSS Accuracy – Lies, Damn Lies and Statistics",GPS World, Vol 18 No. 1, January 2007. Sequel to previous article with similar title[1][2]