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Supervised component generalized linear regression with multiple explanatory blocks: THEME-SCGLR

Bry X., Trottier C., Mortier F., Cornu G., Verron T.. 2016. In : Abdi Hervé (ed.), Esposito Vinzi (ed.), Vincenzo (ed.), Russolillo Giorgio (ed.), Saporta Gilbert (ed.), Trinchera Laura (ed.). The multiple facets of partial least squares and related methods: PLS, Paris, France, 2014. Cham : Springer International Publishing, p. 141-154. (Springer Proceedings in Mathematics and Statistics, 173). International Conference on Partial Least Squares and Related Methods (PLS2014). 8, 2014-05-26/2014-05-28, Paris (France).

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 X1 , ¿ , XR, viewed as explanatory themes. Variables in each X,. are assumed many and redundant. Thus, generalized linear regression demands regularization with respect to each X,.. By contrast, variables in T are assumed selected so as to demand no regularization. Regularization is performed searching each X,. for an appropriate number of orthogonal components that both contribute to model Y and capture relevant structural information in X,.. 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.

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