Selecting hidden Markov model state number with cross-validated likelihood

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.     

Variable selection in model-based clustering and discriminant analysis with a regularization approach

Advances in Data Analysis and Classification - Several methods for variable selection have been proposed in model-based clustering and classification. These make use of backward or forward...     

Accounting for inspection errors and change in maintenance behaviour

We propose a way to account for inspection errors in a particularframework. We consider a situation where the lifetime of a systemdepends essentially of a particular part. A deterioration ofthis part is regarded as an unacceptable state for the safetyof the system and a major renewal is deemed necessary. Thusthe statistical analysis of the deterioration time distributionof this part is of primary interest for the preventive maintenanceof the system. In this context, we faced the following problem.In the early life of the system, unwarranted renewals of thepart are decided upon, caused by overly cautious behaviour.Such unnecessary renewals make the statistical analysis of deteriorationtime data difficult and can induce and underestimation of themean life of the part. To overcome this difficulty, we proposeto regard the problem as an incomplete data model. We presentits estimation under the maximum likelihood methodology. Numericalexperiments show that this approach eliminates the pessimisticbias in the estimation of the mean life of the part. We alsopresent a Bayesian analysis of the problem which can be usefulin a small sample setting.     

Variable selection in model-based discriminant analysis

A general methodology for selecting predictors for Gaussian generative classification models is presented. The problem is regarded as a model selection problem. Three different roles for each possible predictor are considered: a variable can be a relevant classification predictor or not, and the irrelevant classification variables can be linearly dependent on a part of the relevant predictors or independent variables. This variable selection model was inspired by a previous work on variable selection in model-based clustering. A BIC-like model selection criterion is proposed. It is optimized through two embedded forward stepwise variable selection algorithms for classification and linear regression. The model identifiability and the consistency of the variable selection criterion are proved. Numerical experiments on simulated and real data sets illustrate the interest of this variable selection methodology. In particular, it is shown that this well ground variable selection model can be of great interest to improve the classification performance of the quadratic discriminant analysis in a high dimension context.