Orthogonal Mixed-Effects Modeling for High-Dimensional Longitudinal Data: An Unsupervised Learning Approach

The linear mixed-effects model is commonly utilized to interpret longitudinal data, characterizing both the global longitudinal trajectory across all observations and longitudinal trajectories within individuals. However, characterizing these trajectories in high-dimensional longitudinal data presents a challenge. To address this, our study proposes a novel approach, Unsupervised Orthogonal Mixed-Effects Trajectory Modeling (UOMETM), that leverages unsupervised learning to generate latent representations of both global and individual trajectories. We design an autoencoder with a latent space where an orthogonal constraint is imposed to separate the space of global trajectories from individual trajectories. We also devise a cross-reconstruction loss to ensure consistency of global trajectories and enhance the orthogonality between representation spaces. To evaluate UOMETM, we conducted simulation experiments on images to verify that every component functions as intended. Furthermore, we evaluated its performance and robustness using longitudinal brain cortical thickness from two Alzheimer's disease (AD) datasets. Comparative analyses with state-of-the-art methods revealed UOMETM's superiority in identifying global and individual longitudinal patterns, achieving a lower reconstruction error, superior orthogonality, and higher accuracy in AD classification and conversion forecasting. Remarkably, we found that the space of global trajectories did not significantly contribute to AD classification compared to the space of individual trajectories, emphasizing their clear separation. Moreover, our model exhibited satisfactory generalization and robustness across different datasets. The study shows the outstanding performance and potential clinical use of UOMETM in the context of longitudinal data analysis.