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 crossvalidation 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 halfsampling 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 crossvalidation. 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.

The hidden Markov tree models were introduced by Crouse, Nowak and Baraniuk in 1998 for modeling nonindependent, non-Gaussian wavelet transform coefficients. In their article, they developed the equivalent of the forward-backward algorithm for hidden Markov tree models and termed it the "upward-downward algorithm". This algorithm is subject to the same numerical limitations as the forward-backward algorithm for hidden Markov chains. In this paper, adapting the ideas of Devijver from 1985, we propose a new "upward-downward" algorithm, which is a true smoothing algorithm and is immune to numerical underflow. Furthermore, we propose a Viterbi-like algorithm for global restoration of the hidden state tree. The contribution of those algorithms as diagnosis tools is illustrated through the modeling of statistical dependencies between wavelet coefficients with a special emphasis on local regularity changes.

Because irregular bearing generates major agronomic issues in fruit-tree species, particularly in apple, the selection of regular cultivars is desirable. Here, we aimed to define methods and descriptors allowing a diagnostic for bearing behaviour during the first years of tree maturity, when tree production is increasing. Flowering occurrences were collected at whole-tree and (annual) shoot scales on a segregating apple population. At both scales, the number of inflorescences over the years was modelled. Two descriptors were derived from model residuals: a new biennial bearing index, based on deviation around yield trend over years and an autoregressive coefficient, which represents dependency between consecutive yields. At the shoot scale, entropy was also considered to represent the within-tree flowering synchronicity. Clusters of genotypes with similar bearing behaviours were built. Both descriptors at the whole-tree and shoot scales were consistent for most genotypes and were used to discriminate regular from biennial and irregular genotypes. Quantitative trait loci were detected for the new biennial bearing index at both scales. Combining descriptors at a local scale with entropy showed that regular bearing at the tree scale may result from different strategies of synchronization in flowering at the local scale. The proposed methods and indices open an avenue to quantify bearing behaviour during the first years of tree maturity and to capture genetic variations. Their extension to other progenies and species, possible variants of descriptors, and their use in breeding programmes considering a limited number of years or fruit yields are discussed.