Multivariate Functional Additive Mixed Models

Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary like precipitation, temperature, and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by e.g.\ repeated measurements or crossed study designs. Modeling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modeling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.