An RKHS model for variable selection in functional regression

TL;DR

This paper introduces an RKHS-based approach for variable selection in functional regression, replacing entire trajectories with selected impact points to improve model interpretability and performance.

Contribution

It proposes a novel RKHS framework for selecting impact points in functional regression, ensuring consistent estimation and demonstrating effectiveness through simulations and real data.

Findings

RKHS approach effectively identifies impact points

Method outperforms traditional L2-based models in simulations

Real data examples confirm practical utility

Abstract

A mathematical model for variable selection in functional regression models with scalar response is proposed. By "variable selection" we mean a procedure to replace the whole trajectories of the functional explanatory variables with their values at a finite number of carefully selected instants (or "impact points"). The basic idea of our approach is to use the Reproducing Kernel Hilbert Space (RKHS) associated with the underlying process, instead of the more usual L2[0,1] space, in the definition of the linear model. This turns out to be especially suitable for variable selection purposes, since the finite-dimensional linear model based on the selected "impact points" can be seen as a particular case of the RKHS-based linear functional model. In this framework, we address the consistent estimation of the optimal design of impact points and we check, via simulations and real data examples, the performance of the proposed method.