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doi: 10.17586/2226-1494-2023-23-3-646-651

Estimation of the moments of a quantized random variable

M. I. Lomakin, A. V. Dokukin


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Article in Russian

For citation:
Lomakin M.I., Dokukin A.V. Estimation of the moments of a quantized random variable. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2023, vol. 23, no. 3, pp. 646–651 (in Russian). doi: 10.17586/2226-1494-2023-23-3-646-651


Abstract
A significant part of the research on the problems of quantization of random variables is devoted to practical aspects of optimal quantization in the sense of filling in information. For these purposes, certain quantitative characteristics of quantized random variables are used, such as: mathematical expectation, variance and mean square deviation. At the same time, to determine the quantitative characteristics of quantized random variables, as a rule, well-known parametric distributions are used: uniform, exponential, normal and others. In real situations, it is not possible to identify the initial parametric distribution based on the available statistical information. In this paper, a nonparametric model is proposed for determining such numerical characteristics of a quantized random variable as the highest initial moments. The mathematical formalization of the problem of estimating the higher initial moments of a quantized random variable in the conditions of incomplete data represented by small samples of a quantized random variable is performed in the form of an optimization model of a certain integral of a piecewise continuous function satisfying certain conditions. The final estimates of the highest initial moments of the quantized random variable are found as extreme (lower and upper) estimates of a certain integral on a set of distribution functions with given moments equal to the sample moments of the quantized random variable. A model of the higher initial moments of a quantized random variable is presented in the form of a definite integral of a piecewise continuous function; in the general case, the problem of finding extreme (lower and upper) estimates of the higher initial moments of a quantized random variable on a set of distribution functions with given moments is solved. Examples of finding higher initial moments and optimal quantization of a random variable are given. The obtained results can be used by specialists in evaluating and optimizing the quantization of various information presented by random signals.

Keywords: random variable, sample, quantum, probability, moments, distribution function

References
  1. Rozhkov N.F. Signal Conversion Methods and Noise-Immunity Coding. Omsk, Omsk State Technical University Publ., 2003, 107 p. (in Russian)
  2. Orlov Iu.N. Basics of Quantization of Degenerate Dynamical Systems. Moscow, MIPT Publ., 2004, 236 p. (in Russian)
  3. Hering J.G. From slide rule to big data: how data science is changing water science and engineering. Journal of Environmental Engineering, 2019, vol. 145, no. 8, pp. 02519001. https://doi.org/10.1061/(asce)ee.1943-7870.0001578
  4. An T. Science opportunities and challenges associated with SKA big data. Science China Physics, Mechanics and Astronomy, 2019, vol. 62, no. 8, pp. 989531.  https://doi.org/10.1007/s11433-018-9360-х
  5. Savanov V.L. Effect of quantization on random values moments calculation accuracy. Automation and Remote Control, 1972, vol. 3, no. 10, pp. 1626–1632.
  6. Grizutenko S.S., Stepanova E.A. An optimum quantification of a random value. Tehnika radiosvjazi, 2011, no. 16, pp. 55–59. (in Russian)
  7. Zachateyskiy E.D., Lavruhin A.A. Analysis and simulation of an optimization algorithm for signals quantization. Journal of Transsib Railway Studies, 2013, no. 2(14), pp. 84–90. (in Russian)
  8. Averina T.A. Modified algorithm of statistical modeling of systems with a random quantization period. Vestnik Saratovskogo gosudarstvennogo tehnicheskogo universiteta, 2011, vol. 4, no. 4(62), pp. 212–218. (in Russian)
  9. Andronov A.M., Bokoev T.I. Optimal quantization of information in the sense of filling. Izvestija AN SSSR. Ser. Tehnicheskaja kibernetika, 1979, no. 3, pp. 154–158. (in Russian)
  10. Smagin V.A., Paramonov I.Yu. Model of optimal probabilistic quantization of information in the space with guaranteed restriction of volume quantum influence zone. Journal of Instrument Engineering,2016, vol. 59, no. 1, pp. 32–37. (in Russian). https://doi.org/10.17586/0021-3454-2016-59-1-32-37
  11. Smagin V.А., Bubnov V.P. Mathematical model of determinated and random processes in the form of consistent hyperfractal distribution. Transport automation research, 2019, vol. 5, no. 2, pp. 148–159. (in Russian). https://doi.org/10.20295/2412-9186-2019-5-2-145-159
  12. Smagin V.A., Bubnov V.P. Optimum in sense of filling quantization of the information in the presence of errors in quanta. Information and Space, 2021, no. 1, pp. 40–48. (in Russian)
  13. Smagin V.A., Bubnov V.P., Sultonov S.K. Mathematical models for calculating the quantitative characteristics of the optimal quantization of information. Modern Transportation Systems and Technologies, 2021, vol. 7, no. 1, pp. 46–58. (in Russian). https://doi.org/10.17816/transsyst20217146-58
  14. Orlov A.I. Asymptotics of quantization and the number of gradations choice in sociological questionnaires. Mathematical Methods and Models in Sociology, Moscow, Institute of Philosophy of the USSR Academy of Science, 1977, pp. 42–55. (in Russian)
  15. Borovkov A.A. Probability Theory. Moscow, Nauka' Publ., 1986, 432 p. (in Russian)
  16. Levin B.R., Shvartc V. Probabilistic Models and Methods in Communication and Control Systems. Moscow, Radio i svjaz' Publ., 1985, 312 p. (in Russian)
  17. Krein M.G., Nudelman A.A. The Markov Moment Problem and Extremum Problems: Ideas and Problems of P.L. Chebyshev and A.A. Markov and their Further Development. Moscow, Nauka Publ., 1973, 551 p. (in Russian)
  18. Kashtanov V.A. On minimax strategies under restrictions on distribution moments. Basic Questions of the Reliability Theory and Practice, Moscow, Sovetskoe radio Publ., 1980, pp. 143–154. (in Russian)
  19. Lomakin M., Buryi A., Dokukin A., Strekha А., Niyazova J., Balvanovich A. Estimation of quality indicators based on sequential measurements analysis. International Journal for Quality Research, 2020, vol. 40, no. 1, pp. 147–162. https://doi.org/10.24874/ijqr14.01-10
  20. Lomakin M.I., Sukhov A.V., Dokukin A.V., Niyazova Y.M. Assessment of reliability indicators of space vehicles under conditions of incomplete data. Cosmic Research, 2021, vol. 59, no. 3, pp. 199–203. https://doi.org/10.1134/S0010952521030072
  21. Buryi A.S., Lomakin M.I., Sukhov A.V. Quality assessment of “stress-strength” models in the conditions of big data.International Journal of Innovative Technology and Exploring Engineering, 2020, vol. 9, no. 3, pp. 3276–3281. https://doi.org/10.35940/ijitee.c8982.019320
  22. Lomakin M.I., Niyazova YU.M., Dokukin A.V., Zlydnev M.I., Garin A.V. Evaluation of the quality of business processes of an enterprise in conditions of incomplete data. Svarochnoe proizvodstvo, 2022, no. 4, pp. 52–58. (in Russian). https://doi.org/10.34641/SP.2022.1049.4.030


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