Bry, Trottier, Mortier, Cornu 2020

Statistical Modelling 20 (1) (2020), 96–119

Component-based regularization of a multivariate GLM with a thematic partitioning of the explanatory variables

Xavier Bry,
Institut Montpelliérain Alexander Grothendieck,
Université Montpellier, CNRS,
Montpellier,
France.

Catherine Trottier,
Institut Montpelliérain Alexander Grothendieck,
Université Montpellier, CNRS,
Montpellier,
France.
e-mail: catherine.trottier@umontpellier.fr
and

Université Paul-Valéry Montpellier,
Montpellier,
France.

Frédéric Mortier,
Forêts et Sociétés,
Université Montpellier, Cirad,
Montpellier,
France.

Guillaume Cornu,
Forêts et Sociétés,
Université Montpellier, Cirad,
Montpellier,
France.

Abstract:

We address component-based regularization of a multivariate generalized linear model (GLM). A vector of random responses Y is assumed to depend, through a GLM, on a set X of explanatory variables, as well as on a set A of additional covariates. X is partitioned into R conceptually homogenous variable groups X₁,…,X_R, viewed as explanatory themes. Variables in each Xr are assumed many and redundant. Thus, generalized linear regression demands dimension reduction and regularization with respect to each X_r. By contrast, variables in A are assumed few and selected so as to demand no regularization. Regularization is performed searching each X_r for an appropriate number of orthogonal components that both contribute to model Y and capture relevant structural information in X_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.

Keywords:

components; multivariate generalized linear model; regularization; SCGLR; dimension reduction.

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