Varying-coefficient functional linear regression

TL;DR

This paper introduces a varying-coefficient functional linear regression model that incorporates scalar covariates as additional arguments, enhancing flexibility and performance in longitudinal data analysis.

Contribution

It extends functional linear regression to include scalar predictors as varying coefficients, combining smoothing and principal component regularization.

Findings

The method outperforms traditional functional linear regression on longitudinal data.

The approach demonstrates consistency and favorable asymptotic properties.

Practical implementation shows improved modeling flexibility.

Abstract

Functional linear regression analysis aims to model regression relations which include a functional predictor. The analog of the regression parameter vector or matrix in conventional multivariate or multiple-response linear regression models is a regression parameter function in one or two arguments. If, in addition, one has scalar predictors, as is often the case in applications to longitudinal studies, the question arises how to incorporate these into a functional regression model. We study a varying-coefficient approach where the scalar covariates are modeled as additional arguments of the regression parameter function. This extension of the functional linear regression model is analogous to the extension of conventional linear regression models to varying-coefficient models and shares its advantages, such as increased flexibility; however, the details of this extension are more challenging in the functional case. Our methodology combines smoothing methods with regularization by truncation at a finite number of functional principal components. A practical version is developed and is shown to perform better than functional linear regression for longitudinal data. We investigate the asymptotic properties of varying-coefficient functional linear regression and establish consistency properties.