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1 IMT - Institut de Mathématiques de Toulouse UMR5219 2 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : Given a pair of random vectors $X,Y$, we consider the problem of approximating $Y$ by $\bcX=\{\bc 1X,\dots,\bc MX\}$ where $\bc$ is a measurable set-valued function.We give meaning to the approximation by using the principles of vector quantization which leads to the definition of a multifunction regression problem.The formulated problem amounts at quantizing the conditional distributions of $Y$ given $X$.We propose a nonparametric estimate of the solutions of the multifunction regression problem by combining the method of $M$-means clustering with the nonparametric smoothing technique of $k$-nearest neighbors.We provide an asymptotic analysis of the estimate and we derive a convergence rate for the excess risk of the estimate.The proposed methodology is illustrated on simulated examples and on a speed-flow traffic data set emanating from the context of road traffic forecasting.

Keywords : Regression analysis vector quantization nonparametric statistics clustering k-means Set-valued function Multifunction

Author: Jean-Michel Loubes - Bruno Pelletier -



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