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.

翻译：纵向数据的混合效应建模在观测数据为随机对象时面临挑战，此类复杂数据取值于缺乏线性结构的通用度量空间。在此类场景下，经典的加性误差模型与分布假设难以成立。随着技术快速发展，包含协方差矩阵、黎曼流形数据及概率分布等复杂随机对象的纵向数据日益普遍。针对这一挑战，我们提出了一种适用于测地线空间中数据的混合效应回归模型，其中潜在平均响应轨迹为该度量空间中的测地线，而观测值与模型的偏差通过扰动映射或传输进行量化。关键发现是：随机对象场景下的测地线轨迹假设，是标准欧几里得场景中线性假设的自然延伸。此外，通过利用测地路径与全局弗雷歇回归所得路径之间的关联，可从含噪声观测中恢复测地线。基线欧几里得协变量对测地路径的影响通过另一弗雷歇回归步骤建模。我们研究了所提出估计量的渐近收敛性，并通过模拟与真实数据应用进行了验证。