POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS
DOI:
https://doi.org/10.32782/IT/2024-1-4Keywords:
continuous Kolmogorov–Wiener filter, prediction, smoothed heavy-tail data, Chebyshev polynomials of the first kind.Abstract
Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.
References
Kozlovskiy V., Yakymchuk N., Selepyna Y., Moroz S., Tkachuk A. Development of a modified method of network traffic forming. Informatyka, Automatyka, Pomiary W Gospodarce I Ochronie Środowiska. 2023. Vol. 13(1), p. 50–53. doi: 10.35784/iapgos.3452.
Gorev V., Gusev A., Korniienko V. The use of the Kolmogorov–Wiener filter for prediction of heavy-tail stationary processes. CEUR Workshop Proceedings. 2022. Vol. 3156, p. 150–159. Available at: http://ceur-ws.org/Vol-3156/paper9.pdf.
Gorev V., Gusev A., Korniienko V., Shedlovska Y. On the use of the Kolmogorov–Wiener filter for heavytail process prediction. Journal of Cyber Security and Mobility. 2023. Vol. 12(3), p. 315–338. doi: 10.13052/jcsm2245-1439.123.4
Gorev V., Gusev A., Korniienko V., Shedlovska Y. On the continuous Kolmogorov–Wiener filter for prediction of modeled smoothed heavy-tail process. Information Technology: Computer Science, Software Engineering and Cyber Security. 2023. Vol. 1, p. 8–12. doi: 10.32782/IT/2023-1-2.
Polyanin A. D., Manzhirov A. V. Handbook of integral equations. Second edition. New York: Chapman and Hall, 1144 p, 2008. doi: 10.1201/9781420010558
Gorev V., Gusev A., Korniienko V. Aleksieiev M. Kolmogorov–Wiener Filter Weight Function for Stationary Traffic Forecasting: Polynomial and Trigonometric Solutions. Lecture Notes in Networks and Systems. 2021. Vol. 212, p. 111–129. doi: 10.1007/978-3-030-76343-5_7.
Gorev V., Gusev A., Korniienko V. Investigation of the Kolmogorov–Wiener filter for continuous fractal processes on the basis of the Chebyshev polynomials of the first kind. Informatyka, Automatyka, Pomiary W Gospodarce I Ochronie Środowiska. 2020. No. 1, P. 58–61. doi: 10.35784/iapgos.912.
Gorev V., Gusev A., Korniienko V. Approximate solutions for the Kolmogorov-Wiener filter weight function for continuous fractional Gaussian noise. Radio Electronics, Computer Science, Control. 2021. No. 1, p. 29–35. doi: 10.15588/1607-3274-2021-1-3.
Gorev V., Gusev A., Korniienko, V. Kolmogorov–Wiener fіlter for continuous traffic prediction in the GFSD model. Radio Electronics, Computer Science, Control. 2022, No. 3, p. 31–37. doi: 10.15588/1607-3274-2022-3-3.
Gorev V., Gusev A., Korniienko V. On the accuracy of some approximations for the Kolmogorov–Wiener filter weight function for power-law structure function processes. Information Technology: Computer Science, Software Engineering and Cyber Security. 2022. No. 1, p. 9–13. doi: 10.32782/IT/2022-1-2.
