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.

翻译：在许多纵向研究中，大量变量随时间重复测量且存在显著缺失数据。现有方法（如概率主成分分析，PPCA）难以处理此类不完整的高维纵向数据，因其未能考虑重复测量数据中固有的嵌套变异来源与时间依赖关系。我们提出层级概率主成分分析（HPPCA），一种双层级概率因子模型，可明确分离个体间变异与时变个体内动态。个体内潜在因子通过高斯过程建模。我们开发了EM算法以处理缺失数据与灵活协方差核函数，并通过计算高效的初始化器加速算法收敛。模拟研究表明，即使在高度缺失与模型误设条件下，HPPCA仍能稳健恢复模型参数子空间，且在插补精度上显著优于标准PPCA与多变量函数型PCA。在"研究COVID以促进康复"成人队列的长新冠症状应用分析中，HPPCA有效捕捉了数据的层级结构，其学习特征相较于现有方法显著改善了临床结局预测与屏蔽临床记录的恢复能力。