Locally Sparse Function on function Regression

TL;DR

This paper introduces a locally sparse functional regression model that allows the regression coefficient to vary across the domain, balancing between concurrent and nonconcurrent models, with efficient algorithms and theoretical guarantees.

Contribution

It proposes a novel locally sparse functional regression approach using overlapping group-Lasso, with computational strategies and theoretical analysis.

Findings

Effective in simulations and real data applications

Balances local and global effects in functional regression

Provides theoretical support for model consistency

Abstract

In functional data analysis, functional linear regression has attracted significant attention recently. Herein, we consider the case where both the response and covariates are functions. There are two available approaches for addressing such a situation: concurrent and nonconcurrent functional models. In the former, the value of the functional response at a given domain point depends only on the value of the functional regressors evaluated at the same domain point, whereas, in the latter, the functional covariates evaluated at each point of their domain have a non-null effect on the response at any point of its domain. To balance these two extremes, we propose a locally sparse functional regression model in which the functional regression coefficient is allowed (but not forced) to be exactly zero for a subset of its domain. This is achieved using a suitable basis representation of the functional regression coefficient and exploiting an overlapping group-Lasso penalty for its estimation. We introduce efficient computational strategies based on majorization-minimization algorithms and discuss appealing theoretical properties regarding the model support and consistency of the proposed estimator. We further illustrate the empirical performance of the method through simulations and two applications related to human mortality and bidding the energy market.