Abstract
Some problems arising in solving various applied problems of economy, recognition, sociology, biology, and modeling of catastrophes are considered. Such problems can be solved using methods of the theory of Markov random processes with local interaction. General characteristics of such processes and a number of concrete applied problems that can be modelled with their help are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. B. Averintsev, “Description of Markov random fields using Gibbs conditional probabilities,” Probability Theory and Its Application, 17, 21–35 (1972).
M. B. Averintsev, “Gibbs description of random fields whose conditional probabilities may vanish,” Probl. Peredachi Inf., No. 4, 86–89 (1975).
T. Liggett, Markov Processes with Local Interaction [Russian translation], Mir, Moscow (1989).
R. K. Chornei, H. Daduna, and P. S. Knopov, Control of Spatially Structured Random Processes and Random Fields with Applications, Springer, New York (2006).
N. B. Vasilyev and O. K. Kozlov, “Reversible Markov chains with local interactions,” in: Multicomponent Random Systems, Nauka, Moscow (1978), pp. 83–100.
O. N. Stavskaya, “Gibbs invariant measures for Markov chains on finite lattices with local interaction,” Mat. Sb., 92, No. 3, 402–419 (1971).
P. A. David and D. Foray, “Percolation Structures, Markov random fields, and the economics of EDI Standard Diffusion,” in: Global Telecommunications Strategies and Technological Changes, North-Holland, Amsterdam (1993).
Ya. G. Sinai, Theory of Phase Transitions [in Russian], Nauka, Moscow (1980).
D. Ruel, Statistical Mechanics [Russian translation], Mir, Moscow (1971).
P. S. Knopov, “Markov fields and their applications in economics,” J. of Math. Sci., 97, No. 2, 3923–3931 (1999).
M. I. Schlesinger and V. Hlavac, Ten Lectures on Statistical and Structural Pattern Recognition (Computational Imaging and Vision), Kluwer, Dordrecht-Boston-London (2002).
G. L. Gimelfarb and A. V. Zalesnyi, “Bayesian segmentation of images by using Markov fields models defined by Gibbs probability distributions,” in: Proc. 3th All-Union. Conf. on Computer-Aided Image Processing Systems, Leningrad (1989), pp. 22–23.
A. V. Zalesnyi, Algorithms of Digital Processing of Images Described by Markov Random Fields, Ph.D. Thesis, V. M. Glushkov Institute of Cybernetics, Academy of Sciences of Ukraine, Kyiv (1991).
A. Golodnikov, P. Knopov, P. Pardalos, and S. Uryasev, “Optimization in the space of distribution functions and applications in the Bayes analysis,” in: Probabilistic Constrained Optimization: Methodology and Applications, Kluwer, Dordrecht–Boston–London (2000), pp. 102–131.
A. N. Golodnikov, P. S. Knopov, and V. A. Pepelyaev, “Some approaches to pattern recognition problems,” Cybernetics and Systems Analysis, 43, No. 6, 810–821 (2007).
R. P. Kinderman and J. L. Snell, “On the relation between Markov random fields and social networks,” J. of Math. Soc., 7, 1–13 (1980).
Yu. Ermoliev, A. Gaivoronski, and M. Makowski, “Robust design of networks under risks,” in: Coping with Uncertainty: Robust Solutions, Springer, Heidelberg (2010), pp. 101–137.
Yu. M. Ermol’yev, T. Yu. Ermol’yeva, G. McDonald, and V. I. Norkin, “Problems of insurance of catastrophic risks,” Cybernetics and Systems Analysis, 37, No. 2, 220–234 (2001).
R. K. Chornei, H. Daduna, and P. S. Knopov, “Controlled Markov fields with finite state space on graphs,” Stochastic Models, 21, 847–874 (2005).
I. M. Gelfand and M. L. Tsetlin, “Mathematical modeling of mechanisms of the central nervous system,” in: Models of the Structurally Functional Organization of Some Biological Systems, Nauka, Moscow (1966), pp. 9–26.
M. L. Tsetlin, Investigations on Automata Theory and Modeling of Biological Systems [in Russian], Nauka, Moscow (1969).
A. M. Leontovich, I. I. Pyatetskii–Shapiro, and O. N. Stavskaya, “Some mathematical problems connected with morphogenesis,” Automat. Telemekh., No. 4, 94–107 (1970).
M. Apter, Cybernetics and Development [Russian translation], Mir, Moscow (1970).
A. G. Gurvich, The Theory of the Biological Field [in Russian], Sov. Nauka, Moscow (1944).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 15–32, May–June 2011.
Rights and permissions
About this article
Cite this article
Knopov, P.S., Samosonok, A.S. On Markov stochastic processes with local interaction for solving some applied problems. Cybern Syst Anal 47, 346–359 (2011). https://doi.org/10.1007/s10559-011-9317-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-011-9317-3