Variable selection in functional data classification: a maxima-hunting proposal

Abstract

Variable selection is considered in the setting of supervised binary classification with functional data $\{X(t),t\in [0,1]\}$. By "variable selection" we mean any dimension-reduction method which leads to replace the whole trajectory $\{X(t),t\in [0,1]\}$, with a low-dimensional vector $(X(t_1),\dots ,X(t_k))$ still keeping a similar classification error. Our proposal for variable selection is based on the idea of selecting the local maxima $(t_1,\dots ,t_k)$ of the function ${\mathcal{V}}_X^2(t)={\mathcal{V}}^2(X(t),Y)$, where $\mathcal{V}$ 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. This paper includes (a) Some theoretical motivation: a result of consistent estimation on the maxima of ${\mathcal{V}}_X^2$ is shown. We also show different theoretical models for the underlying process $X(t)$ under which the relevant information in concentrated in the maxima of ${\mathcal{V}}_X^2$. (b) An extensive empirical study, including about 400 simulated models and real data examples, aimed at comparing our variable selection method with other standard proposals for dimension reduction.

References in corpus (8)

• Brownian distance covariance
• Methodology and theory for partial least squares applied to functional data
• Nonparametric sparsity and regularization
• Factor models and variable selection in high-dimensional regression analysis
• An RKHS formulation of the inverse regression dimension-reduction problem
• Fractals with point impact in functional linear regression
• Inverse regression for longitudinal data
• Fast DD-classification of functional data

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