Thomas Verron

Supervised component generalized linear regression using a PLS-extension of the Fisher scoring algorithm

In the current estimation of a GLM model, the correlation structure of regressors is not used as the basis on which to lean strong predictive dimensions. Looking for linear combinations of regressors that merely maximize the likelihood of the GLM has two major consequences: (1) collinearity of regressors is a factor of estimation instability, and (2) as predictive dimensions may lean on noise, both predictive and explanatory powers of the model are jeopardized. For a single dependent variable, attempts have been made to adapt PLS regression, which solves this problem in the classical Linear Model, to GLM estimation. In this paper, we first discuss the methods thus developed, and then propose a technique, Supervised Component Generalized Linear Regression (SCGLR), that combines PLS regression with GLM estimation in the multivariate context. SCGLR is tested on both simulated and real data.

THEME-SEER: a multidimensional exploratory technique to analyze a structural model using an extended covariance criterion

In this work, we present a new approach to path modeling based on an extended multiple covariance criterion: system extended multiple covariance (SEMC). SEMC is suitable to measure the quality of any structural equations system. We show why SEMC may be preferred to criteria based on usual covariance of components and also to criteria based on residual sums of squares. We give a pursuit algorithm ensuring that SEMC increases and converges. When one wishes to extract more than one component per variable group, a problem arises of component hierarchy. To solve it, we define a local nesting principle of component models that makes the role of each component statistically clear. We then embed the pursuit algorithm in a more general algorithm that extracts sequences of locally nested models. We finally provide a component backward selection strategy. The technique is applied to cigarette data to model the generation of chemical compounds in smoke through tobacco combustion. Copyright

Exploring a physico-chemical multi-array explanatory model with a new multiple covariance-based technique: structural equation exploratory regression.

In this work, we consider chemical and physical variable groups describing a common set of observations (cigarettes). One of the groups, minor smoke compounds (minSC), is assumed to depend on the others (minSC predictors). PLS regression (PLSR) of m inSC on the set of all predictors appears not to lead to a satisfactory analytic model, because it does not take into account the experts knowledge. PLS path modeling (PLSPM) does not use the multidimensional structure of predictor groups. Indeed, the expert needs to separate the influence of several pre-designed predictor groups on minSC, in order to see what dimensions this influence involves. To meet these needs, we consider a multi-group component-regression model, and propose a method to extract from each group several strong uncorrelated components that fit the model. Estimation is based on a global multiple covariance criterion, used in combination with an appropriate nesting approach. Compared to PLSR and PLSPM, the structural equation exploratory regression (SEER) we propose fully uses predictor group complementarity, both conceptually and statistically, to predict the dependent group.

Multidimensional Exploratory Analysis of a Structural Model Using a Class of Generalized Covariance Criteria

Our aim is to explore a structural model: several variable groups describing the same observations are assumed to be structured around latent dimensions that are linked through a linear model that may have several equations. This type of model is commonly dealt with by methods assuming that the latent dimension in each group is unique. However, conceptual models generally link concepts which are multidimensional. We propose a general class of criteria suitable to measure the quality of a Structural Equation Model (SEM). This class contains the covariance criteria used in PLS Regression and the Multiple Covariance criterion of the SEER method. It also contains quartimax-related criteria. All criteria in the class must be maximized under a unit norm constraint. We give an equivalent unconstrained maximization program, and algorithms to solve it. This maximization is used within a general algorithm named THEME (Thematic Equation Model Exploration), which allows to search the structures of groups for all dimensions useful to the model. THEME extracts locally nested structural component models.

Supervised Component Generalized Linear Regression with Multiple Explanatory Blocks: THEME-SCGLR

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, …, 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.