Across many scientific disciplines, multiple observations are collected from the same experimental units, and in modern datasets these observations often arise as non-Euclidean random objects. In such settings, the incorporation of random effects is a critical modeling step for efficient estimation and personalized prediction. Although mixed-effects models are well established for scalar outcomes and, more recently, for functional data in Hilbert spaces, general random-effects frameworks for objects in metric spaces remain underdeveloped. In this paper, we propose a nonlinear Fréchet-based algorithm for random-effects modeling of arbitrary random objects defined on a metric space. Using M-estimation theory, we establish conditions under which the proposed metric-space prediction target is consistently estimated under a working random-effects formulation. We then evaluate the empirical performance of the proposed method using both synthetic data and digital health datasets that require practical tools for analyzing random objects in metric spaces, such as multivariate probability distributions and random graphs. We show that, although our method is developed beyond Hilbert spaces, it can outperform existing Hilbert space-based methods.

翻译：在众多科学领域，同一实验单元常被收集多个观测值，而在现代数据集中，这些观测值往往表现为非欧几里得随机对象。在此类设定下，纳入随机效应是实现高效估计与个性化预测的关键建模步骤。尽管混合效应模型在标量结果中已发展成熟，且近年来在希尔伯特空间函数型数据中亦取得进展，但针对度量空间中对象的广义随机效应框架仍不完善。本文提出一种基于非线性Fréchet的算法，用于对定义在度量空间上的任意随机对象进行随机效应建模。借助M估计理论，我们建立了在工作随机效应公式下，所提出的度量空间预测目标可被一致估计的条件。随后，我们通过合成数据及需要实用工具分析度量空间中随机对象的数字健康数据集（如多元概率分布与随机图），评估所提方法的实证表现。结果表明，尽管我们的方法超越希尔伯特空间范畴，但相较于现有基于希尔伯特空间的方法仍能展现更优性能。