AI 中文总结
针对多组数据集测量相同数值变量的情况，提出基于正则化优化的全局对齐主成分分析方法，结合特定组与全局主成分，通过模拟研究和实际数据应用，在捕获组内变化与保持组间可解释性和稳定性间取得良好平衡。
AI 中文摘要
我们针对多组数据集提出了一种新颖的主成分分析（PCA）方法，其中相同的数值变量在不同观测组中进行测量。现有方法要么通过处理全局（合并）数据完全忽略组结构，要么仅专注于局部结构（组内PCA），要么施加共同主成分的严格假设。我们的方法在尊重数据的多组性质的同时，提高了成分的全局可比性。我们通过基于正则化优化的显式对齐机制，将特定组的主成分与全局主成分相结合。我们引入了全局对齐协方差矩阵的概念，在组内协方差矩阵中纳入来自全局主方向的加权贡献。对齐强度由正则化参数控制，可进行调整以实现所需的权衡。通过全面的模拟研究，我们证明了所提出的对齐PCA在捕获组内局部变化与保持组间可解释性和稳定性之间取得了良好的平衡。此外，在应用于2021年加拿大人口普查社会经济数据时，所提出的对齐PCA比合并或区域PCA产生更具可比性和稳定性的特定区域成分。
英文摘要
We propose a novel principal component analysis (PCA) for multi-group datasets, where the same numerical variables are measured across different groups of observations. Existing approaches either ignore group structure entirely by working with global (pooled) data, 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 improving global comparability of components. We combine group-specific principal components with global ones through an explicit alignment mechanism based on regularized optimization. We introduce the notion of globally aligned covariance matrix, incorporating weighted contributions from global principal directions in the group-wise covariance matrix. 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 proposed aligned PCA achieves a favorable compromise between capturing local variation within groups and maintaining interpretability and stability across groups. Furthermore, in an application to the 2021 Canadian Census socioeconomic data, the proposed aligned PCA yields more comparable and stable region-specific components than pooled or region-wise PCA.
