Selecting Hidden Markov Model State Number with Cross-Validated Likelihood

Gilles Celeux (1) , Jean-Baptiste Durand (2)

1 SELECT - Model selection in statistical learning
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems

The problem of estimating the number of hidden states in a hidden Markov model is considered. Emphasis is placed on cross-validated likelihood criteria. Using cross-validation to assess the number of hidden states allows to circumvent the well-documented technical difficulties of the order identification problem in mixture models. Moreover, in a predictive perspective, it does not require that the sampling distribution belongs to one of the models in competition. However, computing cross-validated likelihood for hidden Markov models for which only one training sample is available, involves difficulties since the data are not independent. Two approaches are proposed to compute cross-validated likelihood for a hidden Markov model. The first one consists of using a deterministic half-sampling procedure, and the second one consists of an adaptation of the EM algorithm for hidden Markov models, to take into account randomly missing values induced by cross-validation. Numerical experiments on both simulated and real data sets compare different versions of cross-validated likelihood criterion and penalised likelihood criteria, including BIC and a penalised marginal likelihood criterion. Those numerical experiments highlight a promising behaviour of the deterministic half-sampling criterion.

Mots clés: Hidden Markov Models; Model Selection; Cross-Validation; Missing Values at Random; EM Algorithm This is an electronic version of an article published in Computational Statistics. The original version is available at www.springerlink.com

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Gilles Celeux, Jean-Baptiste Durand. Selecting Hidden Markov Model State Number with Cross-Validated Likelihood. Computational Statistics, 2008, 23 (4), pp.541-564. ⟨10.1007/s00180-007-0097-1⟩. ⟨inria-00193098⟩