# Geodesic Mixed Effects Models for Repeatedly Observed/Longitudinal Random Objects

arXiv: https://arxiv.org/abs/2307.05726 · HTML: https://mlsanity.com/paper/2307.05726/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2023.

## Built on by (most cited first, 4 of 4)

- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [Random-Effects Algorithm for Random Objects in Metric Spaces](https://mlsanity.com/paper/2605.02693/) (2026)

## Abstract (arXiv)

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.