Modélisation des bi-grappes et sélection des variables pour des données de grande
dimension : application aux données d’expression génétique

Clustering is a classical method to analyse gene expression data. When applied to the rows (e.g. genes), each column belongs to all  clusters. However, it is often observed that the genes of a subset of genes are co-regulated and co-expressed in a subset of conditions, but behave almost independently under other conditions. For these reasons, biclustering techniques have been proposed to look for sub-matrices of a data matrix. Biclustering is a simultaneous clustering  of rows and columns of a data matrix. Most of the biclustering algorithms proposed in the literature have no statistical foundation. It is interesting to pay attention to the underlying models of these algorithms and develop statistical models to obtain  significant biclusters. In this thesis, we review some biclustering algorithms that seem to be most popular. We group these algorithms in accordance  to the type of homogeneity in the bicluster and the type of overlapping that may be encountered. We shed light on statistical models that can justify these algorithms. It turns out that some techniques can be justified in a Bayesian framework. We develop an extension of the biclustering plaid model  in a Bayesian framework and we propose a measure of  complexity for biclustering. The deviance information criterion (DIC) is used to select the number of biclusters. Studies on gene expression data and simulated data give satisfactory results.  To our knowledge, the biclustering algorithms assume that genes and experimental conditions are independent entities. These algorithms do not incorporate prior biological information that could be available  on genes and conditions. We introduce a new Bayesian plaid model for gene expression data which integrates biological knowledge and takes into account the pairwise interactions  between genes and between conditions via a Gibbs field. Dependence between these entities is made from relational graphs, one for genes and another for conditions. The graph of the genes and conditions is constructed by the k-nearest neighbors and allows to define  a priori distribution of labels as auto-logistic models. The similarities of genes are calculated using  gene ontology (GO). To estimate the parameters, we adopt a hybrid procedure that mixes MCMC with a variant of the Wang-Landau algorithm. Experiments on simulated and real data show the performance of our approach.  It should be noted that there may be several variables of noise in microarray data. These variables may mask the true structure of the clustering. Inspired by the plaid model, we propose a model  that simultaneously finds the true clustering structure and identifies discriminating variables. We propose a new model to solve the problem. It assumes that an observation can be explained by more than one cluster. This problem is addressed by using a binary latent vector, so the estimation is obtained via the Monte Carlo EM algorithm. Importance Sampling is used to reduce the computational cost of the Monte Carlo sampling at each step of the EM algorithm.  Numerical examples demonstrate the usefulness of these methods in terms of variable selection and clustering.

Keyphrases: Clustering, Gene Ontology, gene expression, Deviance information criterion, Wang-Landau algorithm, auto-logistic models, Variable selection, plaid model, Monte Carlo EM algorithm, Importance Sampling