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In the military science of ballistics, circular error probable (CEP) (also circular error probability or circle of equal probability) is a measure of a weapon system's precision. It is defined as the radius of a circle, centered on the mean, whose perimeter is expected to include the landing points of 50% of the rounds; said otherwise, it is the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
error radius. That is, if a given munitions design has a CEP of 100 m, when 100 munitions are targeted at the same point, 50 will fall within a circle with a radius of 100 m around their average impact point. (The distance between the target point and the average impact point is referred to as
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
.) 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 as GPS or older systems such as
LORAN LORAN, short for long range navigation, was a hyperbolic radio navigation system developed in the United States during World War II. It was similar to the UK's Gee system but operated at lower frequencies in order to provide an improved range ...
and Loran-C.


Concept

The original concept of CEP was based on a circular bivariate normal distribution (CBN) with CEP as a parameter of the CBN just as μ and σ are parameters of the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
. Munitions with this distribution behavior tend to cluster around the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ar ...
impact point, with most reasonably close, progressively fewer and fewer further away, and very few at long distance. That is, if CEP is ''n'' metres, 50% of shots land within ''n'' metres of the mean impact, 43.7% between ''n'' and ''2n'', and 6.1% between ''2n'' and ''3n'' 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. Precision-guided munitions generally have more "close misses" and so are not normally distributed. Munitions may also have larger standard deviation of range errors than the standard deviation of azimuth (deflection) errors, resulting in an elliptical
confidence region In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an ''n''-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, a ...
. Munition samples may not be exactly on target, that is, the mean vector will not be (0,0). This is referred to as
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
. To incorporate accuracy into the CEP concept in these conditions, CEP can be defined as the square root of the mean square error (MSE). The MSE will be the sum of the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of the range error plus the variance of the azimuth error plus the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
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 to
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
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

While 50% is a very common definition for CEP, the circle dimension can be defined for percentages.
Percentile In statistics, a ''k''-th percentile (percentile score or centile) is a score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percentage fall ...
s can be determined by recognizing that the horizontal position error is defined by a 2D vector which components are two orthogonal
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
random variables (one for each axis), assumed
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, ther ...
, each having a standard deviation \sigma. The ''distance error'' is the magnitude of that vector; it is a property of 2D Gaussian vectors that the magnitude follows the Rayleigh distribution, with a standard deviation \sigma_d=\sqrt\sigma, called the ''distance root mean square'' (DRMS). In turn, the properties of the Rayleigh distribution are that its percentile at level F\in \%,100\%/math> is given by the following formula: :Q(F,\sigma)=\sigma \sqrt or, expressed in terms of the DRMS: :Q(F,\sigma_d)=\sigma_d \frac 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 the
68–95–99.7 rule In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, t ...
We can then derive a conversion table to convert values expressed for one percentile level, to another.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 titl

/ref> Said conversion table, giving the coefficients \alpha to convert X into Y=\alpha.X, is given by: For example, a GPS receiver having a 1.25 m DRMS will have a 1.25 m\times1.73 = 2.16 m 95% radius. Warning: often, sensor datasheets or other publications state "RMS" values which in general, ''but not always'',For instance, the International Hydrographic Organization, in the IHO standard for hydrographic survey S-44 (fifth edition) defines "the 95% confidence level for 2D quantities (e.g. position) is defined as 2.45 x standard deviation", which is true only if we are speaking about the standard deviation of the underlying 1D variable, defined as \sigma above. stand for "DRMS" values. Also, be wary of habits coming from properties of a 1D
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
, such as the 68–95–99.7 rule, 68-95-99.7 rule, in essence trying to say that "R95 = 2DRMS". As shown above, these properties simply ''do not'' translate to the distance errors. Finally, mind that these values are obtained for a theoretical distribution; while generally being true for real data, these may be affected by other effects, which the model does not represent.


See also

*
Probable error In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) ''The Oxford Dictiona ...


References


Further reading

* * * Grubbs, F. E. (1964). "Statistical measures of accuracy for riflemen and missile engineers". Ann Arbor, ML: Edwards Brothers
Ballistipedia pdf
* * Daniel Wollschläger (2014), "Analyzing shape, accuracy, and precision of shooting results with shotGroups"
Reference manual for shotGroups
* 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
{{Refend


External links


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