Functional Mixtures-of-Experts

We consider the statistical analysis of heterogeneous data for clustering and prediction purposes, in situations where the observations include functions, typically time series. We extend the modeling with Mixtures-of-Experts (ME), as a framework of choice in modeling heterogeneity in data for prediction and clustering with vectorial observations, to this functional data analysis context. We first present a new family of functional ME (FME) models, in which the predictors are potentially noisy observations, from entire functions, and the data generating process of the pair predictor and the real response, is governed by a hidden discrete variable representing an unknown partition, leading to complex situations to which the standard ME framework is not adapted. Second, we provide sparse and interpretable functional representations of the FME models, thanks to Lasso-like regularizations, notably on the derivatives of the underlying functional parameters of the model, projected onto a set of continuous basis functions. We develop dedicated expectation--maximization algorithms for Lasso-like regularized maximum-likelihood parameter estimation strategies, to encourage sparse and interpretable solutions. The proposed FME models and the developed EM-Lasso algorithms are studied in simulated scenarios and in applications to two real data sets, and the obtained results demonstrate their performance in accurately capturing complex nonlinear relationships between the response and the functional predictor, and in clustering.