# Generalized probabilistic principal component analysis of correlated data

arXiv: https://arxiv.org/abs/1808.10868 · HTML: https://mlsanity.com/paper/1808.10868/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Aug 2018.

## Built on by (most cited first, 6 of 6)

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- [Spatiotemporal Besov Priors for Bayesian Inverse Problems](https://mlsanity.com/paper/2306.16378/) (2023)
- [Dynamic Bayesian Learning for Spatiotemporal Mechanistic Models](https://mlsanity.com/paper/2208.06528/) (2022)
- [Fast data inversion for high-dimensional Ornstein-Uhlenbeck processes from noisy measurements](https://mlsanity.com/paper/2501.01324/) (2025)
- [Exact and general decoupled solutions of the LMC Multitask Gaussian Process model](https://mlsanity.com/paper/2310.12032/) (2023)
- [Hierarchical Probabilistic Principal Component Analysis of Longitudinal Data](https://mlsanity.com/paper/2604.22015/) (2026)

## Abstract (arXiv)

Principal component analysis (PCA) is a well-established tool in machine learning and data processing. The principal axes in PCA were shown to be equivalent to the maximum marginal likelihood estimator of the factor loading matrix in a latent factor model for the observed data, assuming that the latent factors are independently distributed as standard normal distributions. However, the independence assumption may be unrealistic for many scenarios such as modeling multiple time series, spatial processes, and functional data, where the outcomes are correlated. In this paper, we introduce the generalized probabilistic principal component analysis (GPPCA) to study the latent factor model for multiple correlated outcomes, where each factor is modeled by a Gaussian process. Our method generalizes the previous probabilistic formulation of PCA (PPCA) by providing the closed-form maximum marginal likelihood estimator of the factor loadings and other parameters. Based on the explicit expression of the precision matrix in the marginal likelihood that we derived, the number of the computational operations is linear to the number of output variables. Furthermore, we also provide the closed-form expression of the marginal likelihood when other covariates are included in the mean structure. We highlight the advantage of GPPCA in terms of the practical relevance, estimation accuracy and computational convenience. Numerical studies of simulated and real data confirm the excellent finite-sample performance of the proposed approach.