Generalized linear model based on latent factors and supervised components

Abstract

In a context of component-based multivariate modeling we propose to model the residual dependence of the responses. Each response of a response vector is assumed to depend, through a Generalized Linear Model, on a set of explanatory variables. The vast majority of explanatory variables are partitioned into conceptually homogeneous variable groups, viewed as explanatory themes. Variables in themes are supposed many and some of them are highly correlated or even collinear. Thus, generalized linear regression demands dimension reduction and regularization with respect to each theme. Besides them, we consider a small set of “additional” covariates not conceptually linked to the themes, and demanding no regularization. Supervised Component Generalized Linear Regression proposed to both regularize and reduce the dimension of the explanatory space by searching each theme for an appropriate number of orthogonal components, which both contribute to predict the responses and capture relevant structural information in themes. In this paper, we introduce random latent variables (a.k.a. factors) so as to model the covariance matrix of the linear predictors of the responses conditional on the components. To estimate the model, we present an algorithm combining supervised component-based model estimation with factor model estimation. This methodology is tested on simulated data and then applied to an agricultural ecology dataset.