Hierarchical Probabilistic Principal Component Analysis of Longitudinal Data

TL;DR

This paper introduces HPPCA, a hierarchical probabilistic PCA model designed for longitudinal data with missing values, capturing nested variance and temporal dynamics to improve data imputation and outcome prediction.

Contribution

HPPCA explicitly models between-subject and within-subject variation using Gaussian processes, advancing analysis of incomplete longitudinal datasets.

Findings

HPPCA outperforms standard PPCA and multivariate functional PCA in imputation accuracy.

HPPCA effectively captures hierarchical structure in longitudinal data.

Application to COVID-19 symptoms improved clinical outcome prediction.

Abstract

In many longitudinal studies, a large number of variables are measured repeatedly over time, with substantial missing data. Existing methods, such as probabilistic principal component analysis (PPCA), are ill-equipped to handle such incomplete, high-dimensional longitudinal data, as they fail to account for the nested sources of variation and temporal dependency inherent in repeated measures. We introduce hierarchical probabilistic principal component analysis (HPPCA), a two-level probabilistic factor model that explicitly separates between-subject variance from time-varying within-subject dynamics. The within-subject latent factors are modeled by a Gaussian process. We develop an EM algorithm to handle missing data and flexible covariance kernels, accelerated by computationally efficient initializers. Simulation studies demonstrated that HPPCA robustly recovers model parameters subspaces and substantially outperforms both standard PPCA and multivariate functional PCA in imputation accuracy, even under heavy missingness and model misspecification. An application to the long COVID symptoms in the Researching COVID to Enhance Recovery adult cohort revealed that HPPCA effectively captured the data's hierarchical structure and its learned features significantly improved the prediction of clinical outcomes and the recovery of masked clinical records compared to exisiting methods.