519.218

modeling, methods of statistical signals processing, cyclic random process, periodic random sequence, isomorphism.

The method of statistical processing of cyclic random processes by reducing them to isomorphic periodic random sequences, which significantly simplifies analytical expressions and formulas for calculations, and also reduces the computational complexity of the tasks of statistical processing and computer simulation (generalization) of cyclical signals in intelligent information systems in medicine, technology and economics, which is especially important for their implementation in portable systems with significantly limited computing capacity is developed in the paper. The example of  statistical evaluation of the initial moment function of the first order of cyclic random process of discrete argument using the existing method, and also with the use of the new method of statistical evaluation, based on the procedure of reducing the investigated cyclic random process to isomorphic periodic random sequence, which statistical processing methods are characterized by much less calculations complexity. Isomorphic in terms of order and values, cyclic random processes, in general, differ only in their rhythmic structures (functions of rhythm) and in their totality form an equivalence class. Any class of isomorphic with respect to the order and values of cyclic random processes of a discrete argument, as its subset, contains a subclass of isomorphic with respect to the order and values of periodic random sequences. Based on this fact, the paper developed a method of reducing the statistical processing (estimation, analysis, forecasting) of a cyclic random process of a discrete argument to an isomorphic periodic random sequence. The computational complexity of the known method of statistical estimation of the probabilistic characteristics of a cyclic random process of a discrete argument is investigated and the method of statistical analysis of the probabilistic characteristics of a cyclic random process of a discrete argument developed in this work to the corresponding statistical processing of an isomorphic periodic random sequence is obtained. Examples of statistical estimation of the initial moment function of the first order of a cyclic random process of a discrete argument using the known method are given, as well as with the use of a new method of statistical estimation based on the procedure of reducing the investigated cyclic random process to an isomorphic periodic random sequence, methods of statistical processing of which is characterized by much less computational complexity.

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12. Lupenko S., Lutsyk N., Lapusta Y. Cyclic Linear Random Process As A Mathematical Model Of Cyclic Signals. Acta mechanica et automatica. 2015. № 9 (4). Р. 219–224.
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1. Gardner W. A., Napolitano A., Paura L. Cyclostationarity: Half a century of research. Signal Processing. 2005. № 86 (2006). P. 639–697.
2. Hurd H. L. Periodically Correlated Random Sequences: Spectral Theory and Practice. The University of North Carolina at Chapel Hill Hampton University.
3. Kochel P. Periodically stationary Markovian decision models. Elektron. Informationsverarb. Kybernet. 1980. No. 16. P. 553–567. [Іn German].
4. Nematollahi A. R., Soltani A. R. Discrete time periodically correlated Markov processes. Probability and Mathematical Statistics. 2000. No. 20 (1). P. 127–140.
5. Ghysels E., McCulloch R. E., Tsay R. S. Bayesian Inference for a General Class of Periodic Markov Switching Models. 1993.
6. Ghysels E. On the Periodic Structure of the Business Cycle. Cowles Foundation, Yale Universiti. 1992. No. 1028.
7. Bittanti S., Lorito F., Strada S. Markovian representations of cyclostationary processes, in: L. Gerencser, P. E. Caines (Eds.). Topics in Stochastic Systems: Modelling, Estimation and Adaptive Control. Springer. 1991. Vol. 161. P. 31–46.
8. Lupenko S. A., Osukhivska H. M., Lutsyk N. S., Stadnyk N. B., Zozulia A. M., Shablii N. R. The comparative analysis of mathematical models of cyclic signals structure and processes. Scientific journal of the Ternopil National Technical University. No. 2 (82). 2016. ISSN 1727-7108. Р. 115–127.
9. Lytvynenko I. V. Method of segmentation of determined cyclic signals for the problems related to their processing and modeling. Scientific journal of the Ternopil National Technical University. No. 4 (88).
   2017. ISSN: 2522-4433. Р. 153–169.
10. Lupenko S. A. Theoretical bases of modeling and processing of cyclic signals in information systems: scientific monograph. Lviv: Magnolia Publishing House, 2016. 344 p.
11. Lupenko S., Orobchuk O., Stadnik N., Zozulya A. Modeling and signals processing using cyclic random functions: 13th IEEE International Scientific and Technical Conference on Computer Sciences and Information Technologies (CSIT) (Lviv, September 11–14.). Lviv, 2018. T. 1. Р. 360–363. ISBN 978-1-5386-6463-6. IEEE Catalog Number: CFP18D36-PRT.
12. Lupenko S., Lutsyk N., Lapusta Y. Cyclic Linear Random Process As A Mathematical Model Of Cyclic Signals. Acta mechanica et automatica. 2015. № 9 (4). Р. 219–224.
13. Lytvynenko Y., Lupenko S., Maruschak P. Analysis of multiple cracking of nano-coatings as cyclic random process. Autometry. Novosibirsk: Siberian Branch of the Russian Academy of Sciences, 2013. № 2. P. 68−75.