Model selection for functional linear regression with hierarchical structure

Abstract

Scalar-on-function regression allows for a scalar response to be dependent on functional predictors; however, not much work has been done when interaction effects between the functional predictors are included. In this paper, we introduce a multiple functional linear regression model with interaction terms. Meanwhile, we enforce the hierarchical structure constraint on the model, that is, interaction terms can be selected into the model only if the associated main effects are in the model. Based on the functional principal component analysis and group smoothly clipped absolute deviation (SCAD) penalty, we propose a new penalized estimation procedure to select the important functional predictors and interactions while automatically obeying the hierarchical structure. With appropriate selection of the tuning parameters, the rates of convergence of the proposed estimators and the consistency of the model selection procedure are established under some regularity conditions. At last, we illustrate the finite sample performance of our proposed methods with some simulation studies and a real data application.