Functional Additive Mixed Models

TL;DR

This paper introduces a comprehensive additive mixed modeling framework for correlated functional data, capable of handling complex random effects, various covariate effects, and different observation densities, with practical software implementation.

Contribution

It presents a novel, flexible framework for modeling correlated functional responses with multiple random effects and covariate effects, implemented in accessible R software.

Findings

Reliable recovery of effects demonstrated in simulations

Handles small sample sizes effectively

Scales well to large datasets

Abstract

We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both spline-based and functional principal component-based terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easy-to-use open source software in the pffr() function for the R-package refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well and also scales to larger data sets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.