EDP Sciences Journals List
Home Document
Subscriber Authentication Point
Issue ESAIM: PS
Volume 13, 2009
Page(s) 38 - 50
DOI 10.1051/ps:2007048
Published online 21 February 2009

ESAIM: PS, January 2009, Vol. 13, p. 38-50
DOI: 10.1051/ps:2007048

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

Antoine Chambaz1 and Catherine Matias2

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


Received October 17, 2006. Revised July 6, 2007. Published online 21 February 2009

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).


Mathematics Subject Classification. 62B10, 62B15, 62M07

Key words: Markov regime, order estimation, hidden states, conditional memory, hidden Markov model


© EDP Sciences, SMAI 2009


What is OpenURL?

The OpenURL standard is a protocol for transmission of metadata describing the resource that you wish to access. An OpenURL link contains article metadata and directs it to the OpenURL server of your choice. The OpenURL server can provide access to the resource and also offer complementary services (specific search engine, export of references...). The OpenURL link can be generated by different means.
  • If your librarian has set up your subscription with an OpenURL resolver, OpenURL links appear automatically on the abstract pages.
  • You can define your own OpenURL resolver with your EDPS Account. In this case your choice will be given priority over that of your library.
  • You can use an add-on for your browser (Firefox or I.E.) to display OpenURL links on a page (see http://www.openly.com/openurlref/). You should disable this module if you wish to use the OpenURL server that you or your library have defined.