Abstract
In this thesis, a response matrix is assumed to depend on a set of explanatory variables, and a set of additional covariates. Explanatory variables are supposed many and redundant, thus demanding dimension reduction and regularization. By contrast, additional covariates contain few selected variables which are forced into the regression model, as they demand no regularization. Originally, the Supervised Component-based Generalized Linear Regression (SCGLR), a Partial Least Squares-type method, and its extension to multiple explanatory variable-blocks, THEME-SCGLR, are designed to extract from the explanatory variables several components jointly supervised by the set of responses. However, this methodology still has some limitations we aim to overcome in this thesis. The first limitation comes from the assumption that all the responses are predicted by the same explanatory space. However, in many practical situations, large sets of responses are not likely to depend exactly on the same explanatory dimensions. As a second limitation, the previous works involving SCGLR assume the responses independent conditional on the explanatory variables. Again, this is not very likely in practice, especially in situations like those in ecology, where a non-negligible part of the explanatory variables could not be measured. To overcome the first limitation, we assume that the responses are partitioned into several unknown groups. We suppose that the responses in each group are predictable from an appropriate number of specific orthogonal supervised components of the explanatory variables. We develop an extension of SCGLR based on a finite mixture model of the responses. The second work relaxes the conditional independence assumption. As in THEME-SCGLR, the response matrix is modeled by a thematic partitioning of the explanatory variables, named ``themes''. Thus, regularization is performed searching each theme for an appropriate number of components that both contribute to predict the response matrix and capture relevant structural information in themes. A set of few latent factors models the ``residual'' covariance matrix of the responses conditional on the components. The approaches presented in this work are tested on many simulation schemes, and then applied on ecology datasets.