Editing Circular error probable

{{original-research|date=June 2024}}

{{Short description|Ballistics measure of a weapon system's precision}}
{{Redirect|Circular error|the circular error of a pendulum|pendulum|and|pendulum (mathematics)}}

[[File:Circular error probable - percentage.png|thumb|CEP concept and hit probability. 0.2% outside the outmost circle.]]

'''Circular error probable''' ('''CEP'''),<ref name=tech_paper>Circular Error Probable (CEP), Air Force Operational Test and Evaluation Center Technical Paper 6, Ver 2, July 1987, p. 1</ref> also '''circular error probability'''<ref>{{Cite web | last = Nelson | first = William | year = 1988 | title = Use of Circular Error Probability in Target Detection | url = https://apps.dtic.mil/sti/citations/ADA199190 | archive-url = https://web.archive.org/web/20141028112628/http://www.dtic.mil/dtic/tr/fulltext/u2/a199190.pdf | url-status = live | archive-date = October 28, 2014 | location = Bedford, MA | publisher = The MITRE Corporation; United States Air Force }}</ref> or '''circle of equal probability''',<ref>{{Cite book |author1-link=Robert Ehrlich (physicist) | last = Ehrlich | first = Robert | year = 1985 | title = Waging Nuclear Peace: The Technology and Politics of Nuclear Weapons | location = Albany, NY | publisher = [[State University of New York Press]] | page = [https://books.google.com/books?id=-tEpgCSNV7sC&pg=PA63 63] }}</ref> is a measure of a weapon system's [[Accuracy and precision|precision]] in the [[military science]] of [[ballistics]]. It is defined as the radius of a circle, centered on the aimpoint, that is expected to enclose the landing points of 50% of the [[Round (firearms)|rounds]]; said otherwise, it is the [[median]] error radius, which is a 50% [[confidence interval]].<ref name=tech_paper/><ref>{{Cite book | editor-last = Payne | editor-first = Craig | year = 2006 | title = Principles of Naval Weapon Systems | location = Annapolis, MD | publisher = [[Naval Institute Press]] | page = [https://books.google.com/books?id=F3q59-hcGDoC&pg=PA342&dq=%22precisely+50%22 342] }}</ref> That is, if a given munitions design has a CEP of 100&nbsp;m, when 100 munitions are targeted at the same point, an average of 50 will fall within a circle with a radius of 100&nbsp;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 the [[standard 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 as [[GPS]] or older systems such as [[LORAN]] and [[Loran-C]].

==Concept==
[[File:Multivariate Gaussian.png|thumb|Circular bivariate normal distribution]]
[[File:Circular error probable - example.png|thumb|20 hits distribution example]]

The original concept of CEP was based on a [[multivariate normal distribution#Bivariate case|circular bivariate normal]] distribution (CBN) with CEP as a parameter of the CBN just as μ and σ are parameters of the [[normal distribution]]. [[Munition]]s with this distribution behavior tend to cluster around the [[mean]] 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. Munitions may also have larger [[standard deviation]] of range errors than the standard deviation of azimuth (deflection) errors, resulting in an elliptical [[confidence region]]. Munition samples may not be exactly on target, that is, the mean vector will not be (0,0). This is referred to as [[Unbiased estimator|bias]].

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]] of the range error plus the variance of the azimuth error plus the [[covariance]] 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]] of a [[circle]] 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]]s can be determined by recognizing that the horizontal position error is defined by a 2D vector which components are two orthogonal [[normal distribution|Gaussian]] [[random variable]]s (one for each axis), assumed [[uncorrelated]], each having a standard deviation <math>\sigma</math>. The ''distance error'' is the magnitude of that vector; it is a property of [[multivariate normal distribution|2D Gaussian vectors]] that the magnitude follows the [[Rayleigh distribution]], with scale factor <math>\sigma</math>. The ''distance [[root mean square]]'' (DRMS), is <math>\sigma_d=\sqrt{2}\sigma</math> 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 <math>F \in [0\%, 100\%]</math> is given by the following formula:

:<math>Q(F, \sigma) = \sigma \sqrt{-2\ln(1 - F/100\%)}</math>

or, expressed in terms of the DRMS:

:<math>Q(F, \sigma_d) = \sigma_d \frac{\sqrt{-2\ln(1 - F/100\%)}}{\sqrt{2}}</math>

The relation between <math>Q</math> and <math>F</math> are given by the following table, where the <math>F</math> 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]]

{|class="wikitable"
|-
! Measure of <math>Q</math>
! Probability <math>F \, (\%)</math>
|-
| 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.<ref name=gps>Frank van Diggelen, "[http://gpsworld.com/gps-accuracy-lies-damn-lies-and-statistics/ GPS Accuracy: Lies, Damn Lies, and Statistics]", ''GPS World'', Vol 9 No. 1, January 1998</ref><ref name="gnss">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 [http://www.gpsworld.com/gpsgnss-accuracy-lies-damn-lies-and-statistics-1134] [http://www.frankvandiggelen.com/wp-content/uploads/2009/03/2007-gps-world-accuracy-article-0107-van-diggelen-1.pdf]</ref> Said conversion table, giving the coefficients <math>\alpha</math> to convert <math>X</math> into <math>Y=\alpha.X</math>, is given by:

{|class="wikitable"
|-
! From <math>X \downarrow</math> to <math>Y \rightarrow</math>
! RMS (<math>\sigma</math>)
! CEP
! DRMS
! R95
! 2DRMS
! R99.7
|-
! RMS (<math>\sigma</math>)
| 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&nbsp;m DRMS will have a 1.25&nbsp;m × 1.73 = 2.16&nbsp;m 95% radius.

==See also==
* [[Probable error]]

==References==
{{reflist}}

==Further reading==
{{Refbegin}}
* {{cite journal|jstor=2282775|title=Asymptotic Properties of Some Estimators of Quantiles of Circular Error|last1=Blischke|first1=W. R.|last2=Halpin|first2=A. H.|journal=Journal of the American Statistical Association|volume=61|issue=315|pages=618–632|year=1966|doi=10.1080/01621459.1966.10480893}}
* Grubbs, F. E. (1964). "Statistical measures of accuracy for riflemen and missile engineers". Ann Arbor, ML: Edwards Brothers. [http://ballistipedia.com/images/3/33/Statistical_Measures_for_Riflemen_and_Missile_Engineers_-_Grubbs_1964.pdf Ballistipedia pdf]
* {{Cite book | last = MacKenzie | first = Donald A. | author-link = Donald Angus MacKenzie | year = 1990 | title = Inventing Accuracy: A Historical Sociology of Nuclear Missile Guidance | location = Cambridge, Massachusetts | publisher = [[MIT Press]] | isbn = 978-0-262-13258-9 | url-access = registration | url = https://archive.org/details/inventingaccurac00dona }}
* {{cite journal|jstor=2290205|title=A Feasible Bayesian Estimator of Quantiles for Projectile Accuracy from Non-iid Data|last1=Spall|first1=James C.|last2=Maryak|first2=John L.|journal=Journal of the American Statistical Association|volume=87|issue=419|pages=676–681|year=1992|doi=10.1080/01621459.1992.10475269}}
* 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.&nbsp;368–71, 23/26 April 2012. [https://ieeexplore.ieee.org/document/6217081 ieeexplore.ieee.org]
* Wollschläger, Daniel (2014), "Analyzing shape, accuracy, and precision of shooting results with shotGroups". [http://cran.fhcrc.org/web/packages/shotGroups/vignettes/shotGroups.pdf  Reference manual for shotGroups]
{{Refend}}

==External links==
* [http://ballistipedia.com/index.php?title=Circular_Error_Probable Circular Error Probable] in [http://ballistipedia.com Ballistipedia]

[[Category:Applied probability]]
[[Category:Military terminology]]
[[Category:Aerial bombs]]
[[Category:Artillery operation]]
[[Category:Ballistics]]
[[Category:Weapon guidance techniques]]
[[Category:Accuracy and precision]]
[[Category:Statistical distance]]
[[Category:Combat modeling]]