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ISSN 2415-3400 (Online)
ISSN 1028-821X (Print)

CORRELATION FUNCTIONS FOR LINEAR ADDITIVE MARKOV CHAINS OF HIGHER ORDERS

Vekslerchik, VE, Melnik, SS, Pritula, GM, Usatenko, OV
Organization: 

O. Ya. Usikov Institute for Radiophysics and Electronics of the National Academy of Sciences of Ukraine
12, Proskura st., Kharkov, 61085, Ukraine

E-mail: usatenko@ire.kharkov.ua

https://doi.org/10.15407/rej2019.01.047
Language: russian
Abstract: 

 

Subject and purpose. The task of designing various radio engineering devices, such as filters, delay lines, antennas with a given radiation pattern, requires the development of methods for generating random sequences (the values of the system parameters) with given correlation properties, since the spectral characteristics of the listed and similar to them systems are expressed in terms of the Fourier transforms of correlators. The purpose of this paper is to represent the function of the transition probability of random sequences with long-range correlations in a form convenient for numerical generation of sequences, and to study the statistical properties of the latter.

Method and methodology. An adequate mathematical tool for solving such problems is the higher order Markov chains. The statistical characteristics of these objects are determined by their transition probability function, which in the general case can have a very complex form. In this paper, the transition probability function is assumed to be additive and linear with respect to the values of the random variable. It is assumed that the state space of the sequence belongs to the set of real numbers.

Results. The equations that relate the correlation functions of the sequence to the weight coefficients of the memory function, determined in their turn by the transition probability function, are derived and analytically solved.

Conclusion. It is shown that the correlation functions of the additive Markov chain are completely determined by the variance of the random variable and the weight coefficients of the memory function. The agreement of the obtained analytical results with the results of numerical realization of the additive Markov sequence is demonstrated. Examples of possible correlation scenarios in higher order additive linear chains are given.

Keywords: correlation functions, higher order linear additive Markov chains, Markov sequences, memory function

Manuscript submitted  03.04.2018
PACS: 05.40.-a, 02.50.Ga, 87.10.+e
Radiofiz. elektron. 2019, 24(1): 47-57
Full text  (PDF)

References: 
1. Tichonov, V. I., Harisov, V. N., 2004. Statistical analysis and synthesis of radio engineering devices and systems. Moscow: Radio i svyaz' Publ. (in Russian).
 
2. Amity, N., Galindo, V. W. Ch., 1974. Theory and analysis of phased antenna arrays. Translated from English and ed. by A. F. Chaplin. Moscow: Mir Publ. (in Russian).
 
3. Lukin, K. A., Mogila, A. A., Vyplavin, P. L., 2007. Reception of images using a fixed antenna array, noise signals and aperture synthesizing method. In: V. M. Yako-venko, ed. 2007. Radiofizika i elektronika. Kharkov: IRE NAS of Ukraine Publ. 12(3), pp. 526–531 (in Russian).
 
4. Anderson, D. F., Kurtz, T. G., 2011. Continuous time Markov chain models for chemical reaction networks. In: Koeppl, H., Densmore, D., Setti, G., di Bernardo, M., eds. 2011. Design and analysis of biomolecular circuits: engineering approaches to systems and synthetic biology. New York, NY: Springe, pp. 3–42.
 
5. Atayero, A. A. A., Sheluhin, O., 2013. Integrated Models for Information Communication Systems and Networks: Design and Development. IGI Publishing Hershey, PA.
 
6. Privault, N., 2013. Understanding Markov Chains. Singapore: Springer.
 
7. Tan, W. Y., 2015. Stochastic Models with Applications to Genetics, Cancers, AIDS and Other Biomedical Systems. 2nd ed. World Scientific.
 
8. Andrieu, C., de Freitas, N., Doucet, A., Jordan M. I., 2003. An Introduction to MCMC for Machine Learning. Machine Learning, 50(1–2), pp. 5–43.
 
9. Berchtold, A., Raftery, A. E., 2002. The Mixture Transition Distribution Model for High-Order Markov Chains and Non-Gaussian Time Series. Statistical Science, 17(3), pp. 328–356.
 
10. Ching, W., Ng, M. K., 2006. Markov Chains: Models, Algorithms and Applications. Springer Science&Business Media.
 
11. Kumar, R., Raghu, M., Sarlós, T., Tomkins, A., 2017. Linear Additive Markov Processes. Proc. of the 26th Int. Conf. World Wide Web. Perth, Australia, 03–07 April 2017, pp. 411–419.
 
12. Usatenko, O. V., Yampol'skii, V. A., 2003. Binary N-Step Markov Chains and Long-Range Correlated Systems. Phys. Rev. Lett., 90(11), pp. 110601(4 p.). DOI: https://doi.org/10.1103/PhysRevLett.90.110601
 
13. Usatenko, O. V., Yampol'skii, V. A., Kechedzhy, K. E., Mel'nyk, S. S., 2003. Symbolic stochastic dynamical systems viewed as binary N-step Markov chains. Physical Review E, 68(6), pp. 061107(12 p.). DOI: https://doi.org/10.1103/PhysRevE.68.061107
 
14. Melnyk, S. S., Usatenko, O. V., Yampol'skii, V. A., Golick, V. A., 2005. Competition between two kinds of correlations in literary texts. Phys. Rev. E, 72(2), pp. 026140(7 p.) DOI: https://doi.org/10.1103/PhysRevE. 72.026140
 
15. Melnyk, S. S., Usatenko, O. V., Yampol'skii, V. A., 2006. Memory functions of the additive Markov chains: applications to complex dynamic systems. Physica A, 361(2), pp. 405–415. DOI: https://doi.org/ 10.1016/ j.physa.2005.06.083
 
16. Melnyk, S. S., Usatenko, O. V., Yampol'skii, V. A., Apostolov, S. S., Maiselis, Z. A., 2006. Memory functions and correlations in additive binary Markov chains. J. Phys. A: Math. Gen., 39(46), pp. 14289–14306.
 
17. Usatenko, O. V., Apostolov, S. S., Mayzelis, Z. A., Melnik, S. S., 2010. Random finite-valued dynamical systems: additive Markov chain approach. [pdf] Cambridge: Cambridge Scientific Publ., 2010. 166 p. Available at: http://www.ire.kharkov.ua/~usatenko/papers/!UsatenkoBook-CAMBRIDGE.pdf
 
18. Tikhonov, V. I., Mironov, M. A., 1977. Markov proces-ses. Moscow: Sovetskoe radio Publ. (in Russian).
 
19. Stoica, P., Moses, R. L. 1992. On the unit circle problem: The Schur-Cohn procedure revisited. Signal Process., 26(1), pp. 95–118. DOI: https://doi.org/10.1016/ 0165-1684(92)90057-4
 
20. Bistritz, Y., 1996. Reflections on Schur-Cohn Matrices and Jury-Marden Tables and classification of related unit-circle zero location criteria. Circuits Systems Signal Process., 15(1), pp. 111–136. DOI: https://doi.org/ 10.1007/BF01187696
 
21. Grove, E. A., Ladas, G., 2005. Periodicities in Nonlinear Difference Equations. In: Advances in Discrete Mathematic and Applications. Vol. 4. Chapman & Hall/CRC.