Component-based regularization of a multivariate GLM with a thematic partitioning of the explanatory variables. (February 2020) - Bry, Xavier; Trottier, Catherine; Mortier, Frédéric; Cornu, Guillaume
We address component-based regularization of a multivariate generalized linear model (GLM). A vector of random responsesY is assumed to depend, through a GLM, on a setX of explanatory variables, as well as on a setA of additional covariates.X is partitioned intoR conceptually homogenous variable groupsX 1, …, X R, viewed as explanatory themes. Variables in eachX r are assumed many and redundant. Thus, generalized linear regression demands dimension reduction and regularization with respect to eachX r . By contrast, variables inA are assumed few and selected so as to demand no regularization. Regularization is performed searching eachX r for an appropriate number of orthogonal components that both contribute to modelY and capture relevant structural information inX r . To estimate a single-theme model, we first propose an enhanced version of Supervised Component Generalized Linear Regression (SCGLR), based on a flexible measure of structural relevance of components, and able to deal with mixed-type explanatory variables. Then, to estimate the multiple-theme model, we develop an algorithm encapsulating this enhanced SCGLR: THEME-SCGLR. The method is tested on simulated data and then applied to rainforest data in order to model the abundance of tree species.