NONPARAMETRIC METHOD OF RECONSTRUCTING PROBABILITY DENSITY ACCORDING TO THE OBSERVATIONS OF A RANDOM VARIABLE
https://doi.org/10.32362/2500-316X-2018-6-3-31-38
Abstract
When investigating the statistical characteristics of a field formed by locally inhomogeneous regions, the problem of reconstructing the probability density function with several vertices on the basis of the results of experimental observations arises. In this case, it is very difficult to apply parametric methods for reconstructing the probability density. Therefore, to restore the probability density, it makes sense to use non-parametric methods of recovery. The Rosenblatt-Parzen method usually used for these purposes has low accuracy and convergence rate. The method proposed in the work of Chentsov N.N. has higher accuracy and convergence rate. However, for multi-vertex distributions its convergence rate is also low. Similar conclusions can be drawn regarding the method proposed in the work of Vapnik V.N. Thus, the problem of developing a technique for reconstructing the multi-vertex probability density on the basis of the results of experimental observations becomes very urgent. The article suggests a nonparametric method of reconstructing probability density according to the observations of a random variable. The method is regular in the sense of Tikhonov regularization and, as the analysis and solution of test problems show, it has sufficiently high accuracy and convergence rate.
About the Authors
A. D. Kryzhanovsky
MIREA - Russian Technological University
Russian Federation
Russian Federation
A. A. Pastushkov
MIREA - Russian Technological University
Russian Federation
Russian Federation
References
1. Kulikov V.B. Reconstruction of multimodal probability densities from experimental data in structures with stochastic properties // Vestnik Nizhegorodskogo universiteta imeni N.I. Lobachevskogo. Matematicheskoye modelirovaniye. Optimal'noye upravleniye (Vestnik of Lobachevsky University of Nizhniy Novgorod. Mathematic Modeling. Optimal Control). 2014. V. 1 (1). P. 248–256. (in Russ.).
2. Vapnik V.N., Stefanyuk A.R. Nonparametric methods for recovering the probability density // Avtomatika i telemehanika (Automation and Remote Control). 1978. № 8. P. 38–52. (in Russ.).
3. Lapko A.V., Lapko V.A. The analysis of efficiency of decomposition methods of measurements interval of the random variable at a probability density estimation // Informatika i sistemy upravleniya (Information Science and Control Systems). 2015. V. 3 (45). P. 84–88. (in Russ.).
4. Vapnik V.N., Glazkova T.G., Kosheev V.A. Algorithms and programs for recovering dependencies / Ed. V.N. Vapnik. Moscow: Nauka Publ., 1984. 815 p. (in Russ.).
5. Chentsov N.N. Estimation of the unknown distribution density by observations // Doklady Akademii Nauk SSSR. 1962. V. 147. № 1. P. 45–48. (in Russ.).
6. Kropotov Yu.A. Methods of estimation models of acoustic signals probability density in telecommunications audio-exchange systems // Sistemy upravleniya, svyazi i bezopasnosti (Systems of Control, Communications and Security). 2017. V. 1. P. 26–39. (in Russ.).
7. Kolmogorov A.N., Fomin S.I. Elements of the theory of functions and functional analysis. M.: Nauka Publ., 1976. 542 p.
8. Kulikov V.B. Reconstruction of multimodal probability densities from experimental data in structures with stochastic properties // Vestnik Nizhegorodskogo universiteta imeni N.I. Lobachevskogo. Matematicheskoye modelirovaniye. Optimal'noye upravleniye (Vestnik of Lobachevsky University of Nizhniy Novgorod. Mathematic Modeling. Optimal Control). 2014. V. 1 (1). P. 248–256. (in Russ.).
9. Il'yin V.A., Sadovnichii V.A., Sendov Bl.X. Mathematical analysis. The initial course. Moscow: Publishing House of Moscow State University, 1985. 662 p.
10. Lapko A.V., Lapko V.A. The analysis of efficiency of decomposition methods of measurements interval of the random variable at a probability density estimation // Informatika i sistemy upravleniya (Information Science and Control Systems). 2015. V. 3 (45). P. 84–88. (in Russ.).
11. Khurgin Ya.I., Yakovlev V.P. Finite functions in physics and engineering. Moscow: Nauka Publ., 1971. 408 p.
12. Trynin A.Yu. Necessary and sufficient conditions for the uniform on a segment sinkapproximations functions of bounded variation // Izvestiya Saratovskogo Universiteta. Novaya Seriya. Matematika. Mekhanika. Informatika (Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics). 2016. V. 16. № 3. P. 288–298. (in Russ.).
13. Chentsov N.N. Estimation of the unknown distribution density by observations // Doklady Akademii Nauk SSSR. 1962. V. 147. № 1. P. 45–48. (in Russ.).
14. Smirnov V.I. Course of Higher Mathematics: in fives vols. V. 5. Moscow: Nauka Publ., 1974. 600 p. (in Russ.).
15. Kolmogorov A.N., Fomin S.I. Elements of the theory of functions and functional analysis. M.: Nauka Publ., 1976. 542 p.
16. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. Moscow: Nauka Publ., 1978. 206 p. (in Russ.).
17. Il'yin V.A., Sadovnichii V.A., Sendov Bl.X. Mathematical analysis. The initial course. Moscow: Publishing House of Moscow State University, 1985. 662 p.
18. Tanana V.P. Methods for solving operator equations. Moscow: Nauka Publ., 1981. 156 p. (in Russ.).
19. Khurgin Ya.I., Yakovlev V.P. Finite functions in physics and engineering. Moscow: Nauka Publ., 1971. 408 p.
20. Pastushkov A.A. Nonparametric method for recovering the probability density from observations of a random variable // Nauchnyi vestnik MIREA (Scientific Bulletin of MIREA). 2009. № 1(6). P. 50–61. (in Russ.).
21. Trynin A.Yu. Necessary and sufficient conditions for the uniform on a segment sinkapproximations functions of bounded variation // Izvestiya Saratovskogo Universiteta. Novaya Seriya. Matematika. Mekhanika. Informatika (Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics). 2016. V. 16. № 3. P. 288–298. (in Russ.).
22. Volper A.I., Khudyaev S.I. Analysis in classes of discontinuous functions and equations of mathematical physics. Moscow: Nauka Publ., 1975 395 p. (in Russ.).
23. Smirnov V.I. Course of Higher Mathematics: in fives vols. V. 5. Moscow: Nauka Publ., 1974. 600 p. (in Russ.).
24. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. Moscow: Nauka Publ., 1978. 206 p. (in Russ.).
25. Tanana V.P. Methods for solving operator equations. Moscow: Nauka Publ., 1981. 156 p. (in Russ.).
26. Pastushkov A.A. Nonparametric method for recovering the probability density from observations of a random variable // Nauchnyi vestnik MIREA (Scientific Bulletin of MIREA). 2009. № 1(6). P. 50–61. (in Russ.).
27. Volper A.I., Khudyaev S.I. Analysis in classes of discontinuous functions and equations of mathematical physics. Moscow: Nauka Publ., 1975 395 p. (in Russ.).
Review
For citations:
Kryzhanovsky A.D., Pastushkov A.A. NONPARAMETRIC METHOD OF RECONSTRUCTING PROBABILITY DENSITY ACCORDING TO THE OBSERVATIONS OF A RANDOM VARIABLE. Russian Technological Journal. 2018;6(3):31-38. (In Russ.) https://doi.org/10.32362/2500-316X-2018-6-3-31-38
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