Generalized Multivariate Functional Additive Mixed Models for Location, Scale, and Shape

TL;DR

This paper introduces a comprehensive Bayesian regression framework for modeling multilevel, multivariate functional data of mixed types, capturing complex dependencies and covariate effects with a flexible, modular approach.

Contribution

It develops a novel generalized multivariate functional additive mixed model that handles mixed data types and complex dependencies using shared latent Gaussian processes and functional principal components.

Findings

Validated through simulation studies confirming model accuracy.

Successfully applied to traffic data in Berlin, demonstrating practical utility.

Flexible in incorporating various covariate effects and data types.

Abstract

We propose a flexible regression framework to model the conditional distribution of multilevel generalized multivariate functional data of potentially mixed type, e.g. binary and continuous data. We make pointwise parametric distributional assumptions for each dimension of the multivariate functional data and model each distributional parameter as an additive function of covariates. The dependency between the different outcomes and, for multilevel functional data, also between different functions within a level is modelled by shared latent multivariate Gaussian processes. For a parsimonious representation of the latent processes, (generalized) multivariate functional principal components are estimated from the data and used as an empirical basis for these latent processes in the regression framework. Our modular two-step approach is very general and can easily incorporate new developments in the estimation of functional principal components for all types of (generalized) functional data. Flexible additive covariate effects for scalar or even functional covariates are available and are estimated in a Bayesian framework. We provide an easy-to-use implementation in the accompanying R package 'gmfamm' on CRAN and conduct a simulation study to confirm the validity of our regression framework and estimation strategy. The proposed multivariate functional model is applied to four dimensional traffic data in Berlin, which consists of the hourly numbers and mean speed of cars and trucks at different locations.