VARIABLE SELECTION IN FUNCTIONAL DATA CLASSIFICATION: A MAXIMA-HUNTING PROPOSAL

Variable selection is considered in the setting of supervised binary classification with functional data {X(t), t is an element of [0, 1]}. By "variable selection" we mean any dimension-reduction method that leads to the replacement of the whole trajectory {X(t), t is an element of [0, 1]}, with a low-dimensional vector (X (t(1)),, X(t(d))) still keeping a similar classification error. Our proposal for variable selection is based on the idea of selecting the local maxima (t(1),...,t(d)) of the function V-X(2)(t) = V-2(X(t), Y), where V denotes the "distance covariance" association measure for random variables due to Szekely, Rizzo, and Bakirov (2007). This method provides a simple natural way to deal with the relevance vs. redundancy trade-off which typically appears in variable selection. A result of consistent estimation for the maxima of V-X(2) is shown. We also show different models for the underlying process X(t) under which the relevant information is concentrated on the maxima of V-X(2). An extensive empirical study is presented, including about 400 simulated models and data examples aimed at comparing our variable selection method with other standard proposals for dimension reduction.