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 ∈ [0, 1]}. By "variable selection" we mean any dimension-reduction method that leads to the replacement of the whole trajectory {X(t), t ∈ [0, 1]}, with a low-dimensional vector (X(t₁),...,X(td)) still keeping a similar classification error. Our proposal for variable selection is based on the idea of selecting the local maxima (t₁,..., td) of the function ${\mathrm{V}}_{\mathrm{x}}^{2}\left(\mathrm{t}\right) = {\mathrm{V}}^{2}\left(\mathrm{X}\left(\mathrm{t}\right), \mathrm{Y}\right)$, where V denotes the "distance covariance" association measure for random variables due to Székely, 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 ${\mathrm{V}}_{\mathrm{x}}^{2}$ is shown. We also show different models for the underlying process X(t) under which the relevant information is concentrated on the maxima of ${\mathrm{V}}_{\mathrm{x}}^{2}$. 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.