Function-on-Function Mixture of Experts Regression Models

International audienceThe relationship between the target variable and the predictors that one tries to estimate through a regression model is generally assumed to be identical for all the subjects. However, for unknown reasons or because of unobserved explanatory variables, this relationship may be heterogeneous. We introduce a method to relax this assumption with a regression structure by a group of individuals based on the framework of mixture models. In its original formulation (dedicated to unsupervised learning or explanatory modelling), the group membership probability of an individual is independent of its covariates. Knowing to which regression group a new individual belongs quickly proved difficult in the context of predictive modelling. To address this issue, the mixture model was modified to make this probability depend on the covariates: this is the mixture of experts model. The mixture model and its extension are well known, and implementation tools have been developed in the classical case, i.e. when the target and explanatory variables are both scalars. In the case where we are in the presence of functional observations for both variables, it would be relevant to develop these mixture models as well. This problem has already been tackled but, to the best of our knowledge, only when the predictors are functional. We plan to develop here the mixture of experts model in the case where both predictors and target variable are functional