# Hierarchical Probabilistic Principal Component Analysis of Longitudinal Data

arXiv: https://arxiv.org/abs/2604.22015 · HTML: https://mlsanity.com/paper/2604.22015/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 23 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [Generalized probabilistic principal component analysis of correlated data](https://mlsanity.com/paper/1808.10868/) (2018)
- [FDApy: a Python package for functional data](https://mlsanity.com/paper/2101.11003/) (2021)
- [On the use of the Gram matrix for multivariate functional principal components analysis](https://mlsanity.com/paper/2306.12949/) (2023)
- [A dynamic probabilistic principal components model for the analysis of longitudinal metabolomic data](https://mlsanity.com/paper/1312.2393/) (2013)
- [Principal Component Analysis of Two-dimensional Functional Data with Serial Correlation](https://mlsanity.com/paper/2303.04408/) (2023)

## Abstract (arXiv)

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.