Representation of the stochastic matrix sets with given properties based on autonomous automatic models
V.M. Zakharov* , S.V. Shalagin** , B.F. Eminov***
A.N. Tupolev Kazan National Research Technical University, Kazan, 420111 Russia
E-mail: *gilvv@mail.ru,**sshalagin@mail.ru, ***bulfami@mail.ru
Received May 28, 2019
DOI: 10.26907/2541-7746.2019.3.456-467
For citation: Zakharov V.M., Shalagin S.V., Eminov B.F. Representation of the stochastic matrix sets with given properties based on autonomous automatic models. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 3, pp. 456–467. doi: 10.26907/2541-7746.2019.3.456-467. (In Russian)
Abstract
This paper considers the methods of construction (presentation) of the sets of ergodic stochastic matrices using automaton models and determination of the power estimates of the generated sets. The research aimed at developing algorithms for constructing the sets of ergodic stochastic matrices with rational elements with given structures and limit vector based on the automaton probabilistic and deterministic models represented by autonomous automata, as well as at estimating the power of the obtained sets of stochastic matrices depending on the dimensions of the given automaton models. The constructed sets of ergodic stochastic matrices are focused on solving the problem of classification of automaton probabilistic models according to certain criteria (parameters) of similarities or differences between the structures of ergodic matrices by the methods of applied multidimensional mathematical statistics. The developed algorithms enable to form a variety of sets of stochastic matrices by changing the transition functions in the considered automata, output functions in autonomous deterministic automaton, and random entry of probabilistic automaton. It was shown that the transition function of autonomous probabilistic automaton allows makes it possible to develop, based on the proposed functioning algorithm, a set of ergodic stochastic matrices with different probabilistic automaton powers and structures based on the permutations of a set of states with repetitions and changes in the probability distribution of the input random variable. It was demonstrated that the sets of ergodic stochastic matrices with different power characteristics and a limit vector based on rearrangements in a set of output letters with repetitions can be formed by changing the functions of the the autonomous deterministic automaton with the help of the developed algorithm. The estimates of the powers of the sets of ergodic stochastic matrices with rational elements represented by autonomous probabilistic and deterministic automata for given restrictions were provided. These estimates reflect the dependencies of the values of powers of the generated sets of stochastic matrices on the dimension characteristics of the automaton models. The proposed construction algorithms of the sets of ergodic stochastic matrices are complementary with respect to the solved problems.
Keywords: set of stochastic matrices, structure, limit vector, autonomous automatic models, set power estimates
Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 18-01-00120).
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