Circular error probable
Circular error probable (CEP), 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, 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
.Concept
The original concept of CEP was based on a
CEP is not a good measure of accuracy when this distribution behavior is not met.
To incorporate accuracy into the CEP concept in these conditions, CEP can be defined as the square root of the
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.
or, expressed in terms of the DRMS:
The relation between and are given by the following table, where the 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
Measure of | Probability |
---|---|
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 to convert into , is given by:
From to | RMS () | CEP | DRMS | R95 | 2DRMS | R99.7 |
---|---|---|---|---|---|---|
RMS () | 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.
Warning: often, sensor datasheets or other publications state "RMS" values which in general, but not always,[7] stand for "DRMS" values. Also, be wary of habits coming from properties of a 1D normal distribution, such as the 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
References
- ^ a b Circular 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.
- State University of New York Press. p. 63.
- 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]
- ^ 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 × standard deviation", which is true only if we are speaking about the standard deviation of the underlying 1D variable, defined as above.
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. JSTOR 2282775.
- Grubbs, F. E. (1964). "Statistical measures of accuracy for riflemen and missile engineers". Ann Arbor, ML: Edwards Brothers. Ballistipedia pdf
- ISBN 978-0-262-13258-9.
- 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. JSTOR 2290205.
- 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