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Finite-dimensional models for hidden Markov chains

Part of: Markov processes Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Lakhdar Aggoun  and
Robert J. Elliott
Lakhdar Aggoun*
Affiliation:
University of Alberta
Robert J. Elliott*
Affiliation:
University of Alberta
*
* Postal address: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
* Postal address: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
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Abstract

A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the number of jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm to improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.

Keywords

RECURSIVE FILTERUNNORMALIZED DENSITYREFERENCE PROBABILITYPARAMETER ESTIMATIONEM ALGORITHM

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 93E11: Filtering 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 146 - 160
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research partially supported by NSERC Grant A7964.

References

[1] Baum, L. E., Petrie, T., Soules, G. and Weiss, N. (1970) A maximization technique occuring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math, Statist 41, 164–171.Google Scholar
[2] Boel, R., Varayia, P. and Wong, E. (1975) Martingales on jump processes I. Representation results. SIAM J. Control 13, 999–1021.CrossRefGoogle Scholar
[3] Boel, R., Varayia, P. and Wong, E. (1975) Martingales on jump processes II. Applications. SIAM J. Control 13, 1022–1061.Google Scholar
[4] Brémaud, P. M. (1981) Point Processes and Queues. Springer-Verlag, Berlin.Google Scholar
[5] Davis, M. H. A., Segall, A. and Kailath, T. (1975) Nonlinear filtering with counting observations. IEEE Trans. Inf. Theory 21, 143–149.Google Scholar
[6] Dembo, A. and Zeitouni, O. (1986) Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm. Stoch. Proc. Appl. 23, 91–113.Google Scholar
[7] Elliott, R. J. (1982) Stochastic Calculus and Applications. Applications of Mathematics 18. Springer-Verlag, Berlin.Google Scholar
[8] Elliott, R. J. (1993) New finite dimensional filters and smoothers for noisily observed Markov chains. IEEE Trans. Inf. Theory 39, 265–271.Google Scholar
[9] Elliott, R. J. (1992) A partially observed control problem for Markov chains. Appl. Math. Optim. 25, 151–169.Google Scholar
[10] Elliott, R. J. (1990) Martingales associated with finite Markov chains. In Seminar on Stochastic Processes , ed. Cinlar, E., Williams, R. and Fitzsimmons, P., 1, 61–172, Birkhauser, Boston.Google Scholar
[11] Elliott, R. J. (1985) A special semimartingale derivation of smoothing and prediction equations. Syst. Control Lett. 6, 287–289.Google Scholar
[12] Rabiner, L. R. (1989) A tutorial on hidden models and selected applications in speech recognition. Proc. IEEE 77, 257–286.Google Scholar
[13] Segall, A. and Kailath, T. (1975) Radon–Nikodym derivatives with respect to measures induced by discontinuous independent-increment processes. Ann. Prob. 3, 449–464.Google Scholar
[14] Van Schuppen, J. (1975) Filtering prediction and smoothing for counting processes observations, a martingale approach. SIAM J. Appl. Math. 32, 552–570.Google Scholar
[15] Wong, E. and Hajek, B. (1985) Stochastic Processes in Engineering Systems. Springer-Verlag, Berlin.Google Scholar