CEP literature

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The literature on the circular error probable (CEP) is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, error analysis for geopositional asessments, or positional accuracy of navigation and guidance systems like GPS. The statistical foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.

Groups of CEP publications

The following list is by no means intended to be complete. Beware that the quality of the listed publications is not uniformly high. The relevant publications may be roughly categorized into different groups:

List of references

  • Ager, T. P. (2004). An analysis of metric accuracy definitions and methods of computation (Tech. Rep.). Springfield, VA: NIMA InnoVision.
  • Beckmann, P. (1962). Statistical Distribution of the Amplitude and Phase of a Multiply Scattered Field. Journal of Research of the National Bureau of Standards, 66D (3), 231-240. http://nvlpubs.nist.gov/nistpubs/jres/66D/jresv66Dn3p231_A1b.pdf
  • Beckmann, P. (1964). Rayleigh Distribution and Its Generalizations. Radio Science Journal of Research NBS/USNC-URSI, 68D (9), 927-932. http://nvlpubs.nist.gov/nistpubs/jres/68D/jresv68Dn9p927_A1b.pdf
  • Bell, J. W. (1973). A note on CEPs. IEEE Transactions on Aerospace and Electronic Systems, AES-9 (1), 111-112. http://nvlpubs.nist.gov/nistpubs/jres/68D/jresv68Dn9p927_A1b.pdf
  • 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. http://www.jstor.org/stable/2282775
  • Cacoullos, T., & Koutras, M. (1984). Quadratic Forms in Spherical Random Variables: Generalized Noncentral \(\chi^{2}\) Distribution. Naval Research Logistics Quarterly, 31(3), 447-461.
  • Cadwell, J. H. (1964). An Approximation to the Integral of the Circular Gaussian Distribution over an Offset Ellipse. Mathematics of Computation, 18(85), 106-112. http://www.jstor.org/stable/2003411
  • Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181
  • Childs, D. R., Coffey, D. M., & Travis, S. P. (1975). Error statistics for normal random variables. Newport, Rhode Island: U.S. Naval Underwater Systems Center. http://handle.dtic.mil/100.2/ADA011430
  • Chin, G. Y. (1987). Two-Dimensional Measures of Accuracy in Navigational Systems. (Tech. Rep. No. DOT-TSC-RSPA-87-1). Cambridge, MA: U.S. Department of Transportation, Transportation Systems Center. http://ntl.bts.gov/lib/46000/46100/46181/DOT-TSC-RSPA-87-01.pdf
  • Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. http://handle.dtic.mil/100.2/ADA059117
  • Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of \(\chi^{2 }\) random variables. Journal of the Royal Statistical Society, C , 29 , 323–333. http://www.jstor.org/stable/2346911
  • DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf
  • DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368
  • DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368
  • DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026
  • DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.
  • DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054
  • DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.
  • Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.
  • Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498
  • Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: U.S. Air Command and Staff College.
  • Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257
  • Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of \(\chi^{2}\) random variables. Journal of the Royal Statistical Society, C, 33, 332–339. http://www.jstor.org/stable/2347721
  • Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309. http://www.jstor.org/stable/2347778
  • Gilliland, D. 1962. Integral of the Bivariate Normal Distribution over an Offset Circle. Journal of the American Statistical Association, 57, 758-768. http://www.jstor.org/stable/2281806
  • Gilliland, D., Hansen, E. R. 1974. A note on some series representations of the integral of a bivariate normal distribution over an offset circle. Naval Research Logistics Quarterly, 51(1), 207-211.
  • Gillis, J. T. (1991). Computation of the circular error probability integral. IEEE Transactions on Aerospace and Electronic Systems, 27(6), 906-910.
  • Govindarajulu, Z. (1986). A Note on Circular Error Probabilities. Naval Research Logistics Quarterly, 33, 423-429.
  • Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf
  • Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316
  • Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752
  • Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747
  • Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595
  • Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768
  • Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: U.S. National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf
  • Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. http://doi.org/10.1002/j.1538-7305.1947.tb01318.x https://archive.org/details/bstj26-2-318
  • Ignani, B. (2010). Determination of Circular and Spherical Position-Error Bounds in System Performance Analysis. Journal of Guidance, Control, and Dynamics, 33 (4), 1301-1304.
  • Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763
  • Johnson, R., Cottrill, S., & Peebles, P. (1969). A Computation of Radar SEP and CEP. IEEE Transactions on Aerospace and Electronic Systems, AES-5 (2), 353-354.
  • Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450
  • Krempasky, J. J. (2003). CEP equation exact to the fourth order. Navigation: Journal of The Institute of Navigation, 50 (3), 143–149. http://doi.org/10.1002/j.2161-4296.2003.tb00325.x
  • Kuonen, D. (1999). Saddlepoint Approximations for Distributions of Quadratic Forms in Normal Variables. Biometrika, 86 (4), 929-935. http://www.jstor.org/stable/2673596
  • Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53, 853–856.
  • Mathematical Analysis Research Corporation (MARC). (1987). Calculating the CEP (Technical Memorandum 31). Pasadena, CA: Jet Propulsion Laboratory. http://dtic.mil/dtic/tr/fulltext/u2/a196504.pdf
  • Marcum, J. I. (1950). Table of Q-Functions (Tech. Rep. No. RAND-339). Santa Monica, CA: RAND Corporation.
  • McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/
  • Mertikas, S. P. (1985). Error Distributions and Accuracy Measures in Navigation: An Overview (TR 113). Fredericton, NB: Canadian Department of Surveying Engineering. http://www2.unb.ca/gge/Pubs/TR113.pdf
  • Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503
  • Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596
  • Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190
  • Nicholson, D. L., Braddock, D., & McDonald, Inc. (1974). Analytical derivation of an accurate approximation of CEP for elliptical error distributions. IEEE Transactions on Vehicular Technology, 23 (1), 16-19.
  • Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0743066
  • Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828
  • Patnaik, P. B. (1949). The non-central \(\chi^{2}\)- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542
  • Pearson, E. S. (1959). Note on an approximation to the distribution of non-central \(\chi^{2}\). Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533
  • Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.
  • Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105
  • Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.
  • RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf
  • Rogers, S. R. (1993). Comments on "Computation of the Circular Error Probability Integral". IEEE Transactions on Aerospace and Electronic Systems, 29(2), 553-555.
  • Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.
  • Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf
  • Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.
  • Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.
  • Shultz, M. E. (1963). Circular error probability of a quantity affected by a bias (Tech. Rep. ACIC Study Number 6). St. Louis, MO: U.S. Aeronautical Chart & Information Center. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0644106
  • Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751
  • Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html
  • Singh, N. (1970). Spherical probable error (SPE) and its estimation. Metrika, 15 (1), 149–163.
  • Siouris, G. M. (1993). Aerospace Avionics Systems: A Modern Synthesis (Appendix A). New York, NY: Academic Press.
  • Spall, J. C., & Maryak, J. 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. URL http://www.jstor.org/stable/2290205
  • Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828
  • Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828
  • Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR-2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. URL http://handle.dtic.mil/100.2/AD0759989
  • Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Maxwell distribution with emphasis on the SEP (Tech. Rep. No. NWL TR-2954). Dahlgren, VA: U.S. Naval Weapons Laboratory.
  • Thomas, M. A., & Crigler, J. R. (1974). Tolerance Limits for the p-Dimensional Radial Error Distribution. Communications in Statistics, 3(5), 477-483.
  • Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528
  • Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.
  • Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.
  • Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.
  • Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337
  • Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586.