Globally Aligned Principal Component Analysis for Multi-Group Data
H. Fathi, M.A. Cremona and F. Severino
Université Laval, Canada
We propose a novel principal component analysis (PCA) for multi-group datasets, where multiple numerical variables are measured across different groups. Our method combines group-specific principal components with global ones through an explicit alignment mechanism based on regularized optimization. We introduce the notion of a globally aligned covariance matrix that incorporates weighted contributions from global principal directions. In this way, we balance the preservation of within-group variance with global coherence. The alignment strength is controlled by regularization parameters that can be tuned to achieve the desired trade-off. Through a comprehensive simulation study, we demonstrate that the aligned approach achieves a favorable compromise between capturing local variation within groups and maintaining interpretability and stability across groups. The method addresses a fundamental gap in the PCA literature. Existing approaches either ignore group structure entirely, focus exclusively on local structure (group-wise PCA), or impose restrictive assumptions of common principal components. Our approach respects the multi-group nature of data while ensuring global comparability of components. Furthermore, in an application to the 2021 Canadian Census socioeconomic data, the proposed alignment yields more comparable and stable region-specific components than pooled or purely region-wise PCA.
Keywords: Principal Component Analysis, Multi-group data, Dimension reduction. (Use at most 3 keywords)