Geodesic Mixed Effects Models for Repeatedly Observed/Longitudinal Random Objects

TL;DR

This paper introduces a novel mixed-effects regression model for longitudinal data with complex random objects in metric spaces, extending classical models to non-linear geodesic trajectories and providing methods for estimation and inference.

Contribution

It develops a geodesic mixed-effects model for random objects in metric spaces, connecting geodesic paths with Fréchet regression, and studies the asymptotic properties of the estimators.

Findings

Geodesic trajectories generalize linear models for complex data.

Estimation methods recover geodesics from noisy observations.

Model performs well in simulations and real data applications.

Abstract

Mixed effect modeling for longitudinal data is challenging when the observed data are random objects, which are complex data taking values in a general metric space without linear structure. In such settings the classical additive error model and distributional assumptions are unattainable. Due to the rapid advancement of technology, longitudinal data containing complex random objects, such as covariance matrices, data on Riemannian manifolds, and probability distributions are becoming more common. Addressing this challenge, we develop a mixed-effects regression for data in geodesic spaces, where the underlying mean response trajectories are geodesics in the metric space and the deviations of the observations from the model are quantified by perturbation maps or transports. A key finding is that the geodesic trajectories assumption for the case of random objects is a natural extension of the linearity assumption in the standard Euclidean scenario. Further, geodesics can be recovered from noisy observations by exploiting a connection between the geodesic path and the path obtained by global Fr\'echet regression for random objects. The effect of baseline Euclidean covariates on the geodesic paths is modeled by another Fr\'echet regression step. We study the asymptotic convergence of the proposed estimates and provide illustrations through simulations and real-data applications.