Summary: We address component-based regularization of a multivariate Generalized Linear Model (GLM). A set of random responses \(Y\) is assumed to depend, through a GLM, on a set \(X\) of explanatory variables, as well as on a set \(T\) of additional covariates. \(X\) is partitioned into \(R\) conceptually homogeneous blocks \(X_{1},\dots, X_R\), viewed as explanatory themes. Variables in each \(X_r\) are assumed many and redundant. Thus, generalized linear regression demands regularization with respect to each \(X_r\). By contrast, variables in \(T\) are assumed 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 \(X_r\). We propose a very general criterion to measure structural relevance (SR) of a component in a block, and show how to take SR into account within a Fisher-scoring-type algorithm in order to estimate the model. We show how to deal with mixed-type explanatory variables. The method, named THEME-SCGLR, is tested on simulated data, and then applied to rainforest data in order to model the abundance of tree-species.
For the entire collection see [Zbl 1356.62003].