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Automation and Remote Control

, Volume 68, Issue 11, pp 1917–1930 | Cite as

Minimax a posteriori estimation in the hidden Markov models

  • A. V. Borisov
Estimation in Systems

Abstract

Consideration was given to the minimax estimation in the observation system including a hidden Markov model for continuous and counting observations. The dynamic and observation equations depend on a random finite-dimensional parameter having an unknown distribution with the given support. The conditional expectation of the available observation of some generalized quadratic loss function was used as the risk function. Existence of the saddle point in the formulated minimax problem was proved, and the worst distribution and the minimax estimate as the solution of a simpler dual problem were characterized.

PACS number

05.40.-a 

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References

  1. 1.
    Kats, I.Ya. and Kurzhanskii, A.B., Minimax Multistep Filtration in the Statistically Uncertain Situations, Avtom. Telemekh., 1978, no. 11, pp. 79–87.Google Scholar
  2. 2.
    Martin, C.J. and Mintz, M., Robust Filtering and Prediction for Linear Systems with Uncertain Dynamics: A Game-Theoretic Approach, IEEE Trans. Autom. Control, 1983, vol. 9, pp. 888–896.CrossRefGoogle Scholar
  3. 3.
    Anan’ev, B.I., Minimax Linear Filtration of the Multistep Processs with Uncertain Distribution of Perturbations, Avtom. Telemekh., 1993, no. 10, pp. 131–139.Google Scholar
  4. 4.
    Elliott, R.J., Aggoun, L., and Moore, J.B., Hidden Markov Models: Estimation and Control, Berlin: Springer, 1995.zbMATHGoogle Scholar
  5. 5.
    Borisov, A.V., Preliminary Analysis of the Distribution of States of the Randomly Structured Special-purpose Controlled Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2005, no. 1, pp. 48–62.Google Scholar
  6. 6.
    Liptser, R.Sh. and Shiryaev, A.N., Teoriya martingalov, Moscow: Nauka, 1986. Translated under the title Theory of Martingales, Dordrecht: Kluwer, 1989.zbMATHGoogle Scholar
  7. 7.
    Loève, M., Probability Theory, Princeton, New Jersey: Van Nostrand, 1960. Translated under the title Teoriya veroyatnostei, Moscow: Mir, 1962.zbMATHGoogle Scholar
  8. 8.
    Wong, E. and Hajek, B., Stochastic Processes in Engineering Systems, New York: Springer, 1985.zbMATHGoogle Scholar
  9. 9.
    Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov, Moscow: Nauka, 1974. Translated into English under the title Statistics of Random Processes, Berlin: Springer, 1978.Google Scholar
  10. 10.
    Borisov, A.V., Analysis of the States of the Hidden Markov Processes Generated by the Special Jump Processes, Teor. Veroyatn. Primen., 2006, no. 3, pp. 589–600.Google Scholar
  11. 11.
    Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Parabolic Equations), Moscow: Nauka, 1967.Google Scholar
  12. 12.
    Pardoux, E., Stochastic Partial Differential Equations and Filtering of Diffusion Processes, Stochastics, 1979, vol. 3, pp. 127–167.zbMATHMathSciNetGoogle Scholar
  13. 13.
    Pardoux, E., Filtering of a Diffusion Process with Poisson Type Observation, in Stochastic Control Theory and Stochastic Differential Equations. Lecture Notes in Control and Information Sciences, 1979, vol. 16.Google Scholar
  14. 14.
    Semenikhin, K.V., Minimax Estimation of the Random Elements by the RMS Criterion, Teor. Veroyatn. Primen., 2003, no. 5, pp. 12–25.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • A. V. Borisov
    • 1
  1. 1.Institute of Problems of InformaticsMoscow State Aviation InstituteMoscowRussia

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