Subspace estimation and prediction methods for hidden Markov models

Subspace estimation and prediction methods for hidden Markov models

Sofia Andersson and Tobias Rydén

Source: Ann. Statist. Volume 37, Number 6B (2009), 4131-4152.

Abstract

Hidden Markov models (HMMs) are probabilistic functions of finite Markov chains, or, put in other words, state space models with finite state space. In this paper, we examine subspace estimation methods for HMMs whose output lies a finite set as well. In particular, we study the geometric structure arising from the nonminimality of the linear state space representation of HMMs, and consistency of a subspace algorithm arising from a certain factorization of the singular value decomposition of the estimated linear prediction matrix. For this algorithm, we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation, and that the m-step linear predictor computed from the estimated system matrices is consistent, i.e., converges to the true optimal linear m-step predictor.

Primary Subjects: 62M09

Secondary Subjects: 62M10, 62M20, 93B15, 93B30

Keywords: Hidden Markov model; linear innovation representation; prediction error representation; subspace estimation; consistency

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Permanent link to this document: http://projecteuclid.org/euclid.aos/1256303539
Digital Object Identifier: doi:10.1214/09-AOS711

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References

[1] Anderson, B. D. O. (1999). The realization problem for hidden Markov models. Math. Control Signals Systems 12 80–120.

Mathematical Reviews (MathSciNet): MR1685090

Zentralblatt MATH: 0949.93015

Digital Object Identifier: doi:10.1007/PL00009846

[2] Andersson, S. and Rydén, T. (2009). Subspace estimation and prediction methods for hidden Markov models: Algorithms and consistency. Technical Report 2009:7, Centre for Mathematical Sciences, Lund University. Available at http://arxiv.org/abs/0907.4418v1.

[3] Andersson, S., Rydén, T. and Johansson, R. (2003). Linear optimal prediction and innovations representations of hidden Markov models. Stochastic Process. Appl. 108 131–149.

Mathematical Reviews (MathSciNet): MR2008604

Zentralblatt MATH: 1075.62626

Digital Object Identifier: doi:10.1016/S0304-4149(03)00086-3

[4] Bauer, D. (1998). Some asymptotic theory for the estimation of linear systems using maximum likelihood methods of subspace algorithms. Ph.D. thesis, Technische Univ. Wien, Vienna.

[5] Baum, L. E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 1554–1563.

Mathematical Reviews (MathSciNet): MR202264

Digital Object Identifier: doi:10.1214/aoms/1177699147

Project Euclid: euclid.aoms/1177699147

[6] Baum, L. E., Petrie, T., Soules, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41 164–171.

Mathematical Reviews (MathSciNet): MR287613

Digital Object Identifier: doi:10.1214/aoms/1177697196

Project Euclid: euclid.aoms/1177697196

[7] Bickel, P. J., Ritov, Y. and Rydén, T. (1998). Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 1614–1635.

Mathematical Reviews (MathSciNet): MR1647705

Zentralblatt MATH: 0932.62097

Digital Object Identifier: doi:10.1214/aos/1024691255

Project Euclid: euclid.aos/1024691255

[8] Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.

Mathematical Reviews (MathSciNet): MR2159833

[9] Chu, M. T. and Guo, Q. (1998). A numerical method for the inverse stochastic spectrum problem. SIAM J. Matrix Anal. Appl. 19 1027–1039.

Mathematical Reviews (MathSciNet): MR1636512

Zentralblatt MATH: 0917.65037

Digital Object Identifier: doi:10.1137/S0895479896292418

[10] Deistler, M., Peternell, K. and Scherrer, W. (1995). Consistency and relative efficiency of subspace methods. Automatica 31 1865–1875.

Mathematical Reviews (MathSciNet): MR1364690

Digital Object Identifier: doi:10.1016/0005-1098(95)00089-6

[11] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer, New York.

Mathematical Reviews (MathSciNet): MR1312160

[12] Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models: Estimation and Control. Springer, New York.

Mathematical Reviews (MathSciNet): MR1323178

[13] Golub, G. H. and Loan, C. F. V. (1996). Matrix Computations, 3rd ed. Johns Hopkins Univ. Press, Baltimore.

Mathematical Reviews (MathSciNet): MR1417720

[14] Greub, W. (1975). Linear Algebra, 4th ed. Springer, New York.

Mathematical Reviews (MathSciNet): MR369382

[15] Hannan, E. J. and Deistler, M. (1988). The Statistical Theory of Linear Systems. Wiley, New York.

Mathematical Reviews (MathSciNet): MR940698

[16] Hjalmarsson, H. and Ninness, B. (1998). Fast, non-iterative estimation of hidden Markov models. In Proc. IEEE Conf. Acoustics, Speech and Signal Process (ICASSP’98) 2253–2256. Seattle.

[17] Hong-Zhi, A., Zhao-Guo, C. and Hannan, E. J. (1982). Autocorrelation, autoregression and autoregressive approximation. Ann. Statist. 10 926–936. (Correction note (1983) 11 1018.)

Mathematical Reviews (MathSciNet): MR707955

Digital Object Identifier: doi:10.1214/aos/1176346271

[18] Katayama, T. (2005). Subspace Methods for System Identification. Springer, New York.

Mathematical Reviews (MathSciNet): MR2256895

Zentralblatt MATH: 1118.93002

[19] Koski, T. (2001). Hidden Markov Models for Bioinformatics. Kluwer, Dordrecht.

Mathematical Reviews (MathSciNet): MR1888250

Zentralblatt MATH: 0983.92001

[20] Kotsalis, G., Megretski, A. and Dahleh, M. A. (2008). Balanced truncation for a class of stochastic jump linear systems and model reduction for hidden Markov models. IEEE Trans. Autom. Control 53 2543–2557.

Mathematical Reviews (MathSciNet): MR2474825

Digital Object Identifier: doi:10.1109/TAC.2008.2006931

[21] Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127–143.

Mathematical Reviews (MathSciNet): MR1145463

Zentralblatt MATH: 0738.62081

Digital Object Identifier: doi:10.1016/0304-4149(92)90141-C

[22] MacDonald, I. L. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman and Hall, London.

Mathematical Reviews (MathSciNet): MR1692202

Zentralblatt MATH: 0868.60036

[23] Oodaira, H. and Yoshihara, K.-I. (1971). The law of iterated logarithm for stationary processes satisfying mixing conditions. Kōdai Math. Sem. Rep. 23 311–334.

Mathematical Reviews (MathSciNet): MR307311

Digital Object Identifier: doi:10.2996/kmj/1138846370

Project Euclid: euclid.kmj/1138846370

[24] Orsi, R. (2006). Numerical methods for solving inverse eigenvalue problems for nonnegative matrices. SIAM J. Matrix Anal. Appl. 28 190–212.

Mathematical Reviews (MathSciNet): MR2218949

Zentralblatt MATH: 1113.65034

Digital Object Identifier: doi:10.1137/050634529

[25] Ortega, J. M. (1987). Matrix Theory. A Second Course. Plenum Press, New York.

Mathematical Reviews (MathSciNet): MR878977

Zentralblatt MATH: 0654.15001

[26] Overschee, P. V. and Moor, B. D. (1996). Subspace Identification for Linear Systems. Kluwer, Dordrecht.

[27] Peternell, K. (1995). Identification of linear dynamic systems by subspace and realization-based algorithms. Ph.D. thesis, Technische Univ. Wien, Vienna.

[28] Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77 257–284.

[29] Rugh, W. J. (1993). Linear System Theory. Prentice-Hall, Englewood Cliffs, NJ.

Mathematical Reviews (MathSciNet): MR1211190

Zentralblatt MATH: 0892.93002

[30] Vidyasagar, M. (2005). The relaization problem for hidden Markov models: The complete realization problem. In Proc. IEEE Conf. Decision Control 6632–6637. Seville.

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