Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-17T15:23:41.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Number of hidden states and memory: a joint order estimation problem for Markov chains with Markov regime

Published online by Cambridge University Press:  21 February 2009

Antoine Chambaz  and
Catherine Matias
Antoine Chambaz
Affiliation:
Laboratoire MAP5, UMR CNRS 8145, Université René Descartes, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; Antoine.Chambaz@univ-paris5.fr
Catherine Matias
Affiliation:
Laboratoire Statistique et Génome, UMR CNRS 8071, Tour Évry 2, 523 pl. des Terrasses de l'Agora, 91000 Évry, France; matias@genopole.cnrs.fr
Get access

Abstract

This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order (k, m) of an observed MCMR process, relying on information theoretic arguments. The novelty of our work relies in the joint estimation of two structural parameters. Furthermore, the different models in competition are not nested. In an asymptotic framework, we prove that a penalized maximum likelihood estimator is strongly consistent without prior bounds on k and m. We complement our theoretical work with a simulation study of its behaviour. We also study numerically the behaviour of the BIC criterion. A theoretical proof of its consistency seems to us presently out of reach for MCMR, as such a result does not yet exist in the simpler case where m = 0 (that is for hidden Markov models).

Keywords

Markov regimeorder estimationhidden statesconditional memoryhidden Markov model
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 38 - 50
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blackwell, D. and Koopmans, L., On the identifiability problem for functions of finite Markov chains. Ann. Math. Stat. 28 (1957) 1011–1015. CrossRef
S. Boucheron and E. Gassiat, Order estimation and model selection, in Inference in hidden Markov models, Olivier Cappé, Eric Moulines, and Tobias Rydén (Eds.), Springer Series in Statistics. New York, NY: Springer (2005).
Boys, R.J. and Henderson, D.A., Bayesian, A approach to DNA sequence segmentation. Biometrics 60 (2004) 573–588. CrossRef
O. Cappé, E. Moulines and T. Rydén (Eds.), Inference in hidden Markov models. Springer Series in Statistics (2005).
Csiszár, I. and Talata, Z., Context tree estimation for not necessarily finite memory processes, via BIC and MDL. IEEE Trans. Info. Theory 52 (2006) 1007–1016. CrossRef
Davisson, L.D., McEliece, R.J., Pursley, M.B. and Wallace, M.S., Efficient universal noiseless source codes. IEEE Trans. Inf. Theory 27 (1981) 269–279. CrossRef
Dempster, A.P., Laird, N.M. and Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. Ser. B 39 (1977) 1–38. With discussion.
Y. Ephraim and N. Merhav, Hidden Markov processes. IEEE Trans. Inform. Theory, special issue in memory of Aaron D. Wyner 48 (2002) 1518–1569.
L. Finesso, Consistent estimation of the order for Markov and hidden Markov chains. Ph.D. Thesis, University of Maryland, ISR, USA (1991).
Fuh, C-D., Efficient likelihood estimation in state space models. Ann. Stat. 34 (2006) 2026–2068. CrossRef
Gassiat, E. and Boucheron, S., Optimal error exponents in hidden Markov model order estimation. IEEE Trans. Info. Theory 48 (2003) 964–980. CrossRef
Hannan, E.J., The estimation of the order of an ARMA process. Ann. Stat. 8 (1980) 1071–1081. CrossRef
Ito, H., Amari, S.I. and Kobayashi, K., Identifiability of hidden Markov information sources and their minimum degrees of freedom. IEEE Trans. Inf. Theory 38 (1992) 324–333. CrossRef
Kieffer, J.C., Strongly consistent code-based identification and order estimation for constrained finite-state model classes. IEEE Trans. Inf. Theory 39 (1993) 893–902. CrossRef
Leroux, B.G., Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 (1992) 127–143. CrossRef
Liu, C.C. and Narayan, P., Order estimation and sequential universal data compression of a hidden Markov source by the method of mixtures. IEEE Trans. Inf. Theory 40 (1994) 1167–1180.
MacKay, R.J., Estimating the order of a hidden markov model. Canadian J. Stat. 30 (2002) 573–589. CrossRef
Nicolas, P., Bize, L., Muri, F., Hoebeke, M., Rodolphe, F., Ehrlich, S.D., Prum, B. and Bessières, P., Mining bacillus subtilis chromosome heterogeneities using hidden Markov models. Nucleic Acids Res. 30 (2002) 1418–1426. CrossRef
P. Nicolas, A.S. Tocquet and F. Muri-Majoube, SHOW User Manual. URL: http://www-mig.jouy.inra.fr/ssb/SHOW/show_doc.pdf (2004). Software available at URL: http://www-mig.jouy.inra.fr/ssb/SHOW/.
Pötscher, B.M., Estimation of autoregressive moving-average order given an infinite number of models and approximation of spectral densities. J. Time Ser. Anal. 11 (1990) 165–179. CrossRef
C.P. Robert, T. Rydén and D.M. Titterington, Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method. J. R. Stat. Soc., Ser. B, Stat. Methodol. 62 (2000) 57–75.
Rydén, T., Estimating the order of hidden Markov models. Statistics 26 (1995) 345–354. CrossRef
Shtar'kov, Y.M., Universal sequential coding of single messages. Probl. Inf. Trans. 23 (1988) 175–186.