Smooth Lasso Estimator for the Function-on-Function Linear Regression Model

TL;DR

This paper introduces the S-LASSO estimator for function-on-function linear regression, enhancing interpretability and smoothness of coefficient functions through a combined penalization approach, with proven consistency and practical advantages demonstrated via simulations and real data.

Contribution

The paper proposes a novel S-LASSO estimator that combines sparsity and smoothness penalties for better interpretability and estimation in function-on-function regression models.

Findings

S-LASSO outperforms existing estimators in estimation and prediction.

The estimator is proven to be estimation and sign consistent.

Practical applications demonstrate its effectiveness on real datasets.

Abstract

A new estimator, named S-LASSO, is proposed for the coefficient function of the Function-on-Function linear regression model. The S-LASSO estimator is shown to be able to increase the interpretability of the model, by better locating regions where the coefficient function is zero, and to smoothly estimate non-zero values of the coefficient function. The sparsity of the estimator is ensured by a \textit{functional LASSO penalty}, which pointwise shrinks toward zero the coefficient function, while the smoothness is provided by two roughness penalties that penalize the curvature of the final estimator. The resulting estimator is proved to be estimation and pointwise sign consistent. Via an extensive Monte Carlo simulation study, the estimation and predictive performance of the S-LASSO estimator are shown to be better than (or at worst comparable with) competing estimators already presented in the literature before. Practical advantages of the S-LASSO estimator are illustrated through the analysis of the \textit{Canadian weather}, \textit{Swedish mortality} and \textit{ship CO\textsubscript{2} emission data}. The S-LASSO method is implemented in the \textsf{R} package \textsf{slasso}, openly available online on CRAN.