Additive Function-on-Function Regression

TL;DR

This paper introduces a new additive function-on-function regression method that models responses based on entire covariate trajectories, providing efficient estimation, prediction, and inference tools for complex functional data.

Contribution

It develops a novel spline and eigenbasis-based estimation approach for additive function-on-function regression, including prediction and inference procedures for realistic data scenarios.

Findings

Effective in finite samples through simulations

Handles correlated errors and irregular data

Demonstrated on real datasets

Abstract

We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications.