Theory of Probability and Mathematical Statistics
Construction of the Karhunen‒Loéve model for the input Gaussian process applied to the linear system, taking into account the output
Yu. V. Kozachenko, I. V. Rozora
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Abstract: In the article, we study the simulation of the input signal, which is applied to a linear system with known impulse response function. System response is an output process. With the help of the Karhunen–Loeve expansion, a model is constructed which approximates the input process taking into account the output with predefined accuracy and reliability in the Banach space C([0, T ]). Also, a partial case is investigated, in which the input process is the Brownian motion.
Keywords: Simulation, Gaussian Process, Karhunen‒Loéve expansion, accuracy and reliability.
Bibliography: 1. D. Brillinger, Time Series: Data Analysis and Theory, San Francisco: Holden-Day, 1981.
2. V. Buldygin, Yu. Kozachenko, Metric characterization of random variables and random processes , Amer. Math. Soc., Providence, RI, 2000.
3. I. Gikhman, A. Skorokhod, M. Yadrenko, Probability Theory and Mathematical Statistics, "Vyscha Shkola", Kyiv, 1988. (in Russian)
4. S. Iermakov, G. Mikhaylov, Statistical Modelling , "Nauka",Moskow, 1982. (in Russian)
5. Yu. Kozachenko, A. Olenko, A liasing-truncation Errors in Sampling Approximations of Sub-Gaussian Signals, IEEE Trannsactions on Information Theory, 62 (2016), No 10, 5831-5838.
6. Yu. Kozachenko, A. Olenko, Whittaker-Kotel'nikov-Shannon approximation of φ-sub-Gaussian random processes, Journal of Mathematical Analysis and Applications. 443 (2016), No 2, 926-946.
7. Yu. Kozachenko, A. Olenko and O. Polosmak, Uniform convergence of compactly supported wavelet expansions of Gaussian random processes, Communications in Statistics ‒ Theory and Methods, 43 (2014), No 10-12, 2549-2562.
8. Yu. Kozachenko, A. Pashko, I. Rozora, Simulation of Stochastic Processes and fields , Zadruga, Kyiv, 2007. (in Ukrainian)
9. Yu. Kozachenko, O. Pogorilyak, I. Rozora and A. Tegza, Simulation of Stochastic Processes with Given Accuracy and Reliability, ISTE Press - Elsevier, 2016.
10. Yu. Kozachenko, I. Rozora, Simulation of Gaussian stochastic processes , Random Oper. and Stochastic Equ., 11 (2003), No 3, 275-296.
11. Yu. Kozachenko, I. Rozora, Accuracy and Reliability of models of stochastic processes of the space Sub
φ(Ω), Theor. Probability and Math. Statist., 71 (2005), 105-117.
12. Yu. Kozachenko, I. Rozora, On cross-correlogram estimators of impulse response function, Theor. Probability and Math. Statist., 93 (2016), 79-91.
13. Yu. Kozachenko, I. Rozora, A Criterion For Testing Hypothesis About Impulse Response Function, Statistics, optimization & information computing, 4 (2016), No 3, 214-232.
14. Yu. Kozachenko, I. Rozora and Ye. Turchyn On an expansion of random processes in series, Random Operators and Stochastic Equ., 15 (2007), 15-33.
15. Yu. Kozachenko, I. Rozora and Ye. Turchyn Properties of Some Random Series , Communications in Statistics ‒ Theory and Methods, 40 (2011), No 19-20, 3672-3683.
16. Yu. Kozachenko Yu., T. Sottinen, O. Vasylyk, Simulation of weakly self-similar stationary increment Sub
φ(Ω)-processes: a series expansion approach, Methodology and computing in applied probability, 7 (2005), 379-400.
17. P. Kramer, O. Kurbanmuradov, K. Sabelfeld, Comparative Analysis of Multiscale Gaussian Random Field Simulation Algorithms, Journal of Computational Physics, 226 September (2007), 897-924.
18. G. Mikhaylov, A. Voitishek, Numerical statistical modelling , "Academiya", Moscow, 2006. (in Russian)
19. A. Pashko A., I. Rozora, Accuracy of simulation for the network traffic in the form of fractional Brownian Motion , 14-th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering, TCSET, Proceedings, April (2018), 840-845.
20. S. Prigarin, The methods of numerical simulation for stochastic processes and fields, IVMiMG, Novosibirsk, 2005. (in Russian)
21. I.Rozora, Simulation of Gaussian stochastic process with respect to derivative , Prykladna statystyka, actuarna i finansova matematyka, 1-2 (2008), 139-147. (in Ukrainian)
22. I. Rozora, Simulation accuracy of strictly φ -Sub-Gaussian stochastic processes in the space L
2[0,T], Obchuslyuvalna ta prykladna matematyka, 2 (2009), No 98, 68-76. (in Ukrainian)
23. I. Rozora, Statistical hypothesis testing for the shape of impulse response function , Communications in Statistics ‒ Theory and Methods, 47 (2018), No 6, 1459-1474.
24. I. Rozora, M. Lyzhechko, On the modeling of linear system input stochastic processes with given accuracy and reliability , Monte Carlo Methods Appl., 24 (2018), No 2, 129-137.
25. K. Sabelfeld, Monte Carlo methods in boundary-value problems, "Nauka", Novosibirsk, 1989. (in Russian)
26. H. Trikomi, Integral equations, In.Lit., Moscow, 1960. (in Russian)