Functional regression with multivariate responses

TL;DR

This paper introduces a novel functional regression method for multivariate responses that leverages correlations among responses and predictors, improving estimation and prediction accuracy, especially with many predictors, and is implemented in R.

Contribution

It develops a new approach for multivariate functional regression that optimally decomposes coefficient functions and handles high-dimensional predictors with a smooth-sparse penalty.

Findings

Method improves prediction accuracy over separate response modeling.

Proposed penalty effectively performs curve selection in high-dimensional settings.

Asymptotic theory supports the method's consistency as sample size and predictors grow.

Abstract

We consider the functional regression model with multivariate response and functional predictors. Compared to fitting each individual response variable separately, taking advantage of the correlation between the response variables can improve the estimation and prediction accuracy. Using information in both functional predictors and multivariate response, we identify the optimal decomposition of the coefficient functions for prediction in population level. Then we propose methods to estimate this decomposition and fit the regression model for the situations of a small and a large number $p$ of functional predictors separately. For a large $p$, we propose a simultaneous smooth-sparse penalty which can both make curve selection and improve estimation and prediction accuracy. We provide the asymptotic results when both the sample size and the number of functional predictors go to infinity. Our method can be applied to models with thousands of functional predictors and has been implemented in the R package FRegSigCom.