We propose a flexible regression model to study the association between a functional response and multiple functional covariates that are observed on the same domain. Specifically, we relate the mean of the current response to current values of the covariates by a sum of smooth unknown bivariate functions, where each of the functions depends on the current value of the covariate and the time point itself. In this framework, we develop estimation methodology that accommodates realistic scenarios where the covariates are sampled with or without error on a sparse and irregular design, and prediction that accounts for unknown model correlation structure. We also discuss the problem of testing the null hypothesis that the covariate has no association with the response. The proposed methods are evaluated numerically through simulations and two real data applications.

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. Supplementary Material for this article is available online.

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