Dynamic principal component analysis from a global perspective

Abstract

Principal component analysis has been one of prominent techniques in multi-dimensional statistics, and the core is to estimate the eigenvalues in descending ordering with their corresponding eigenvectors. However, when data are collected longitudinally in many scientific applications, the eigenvalues become dynamic over time, and the ordering of them may not be well-defined, for instance, does not coincide with the pointwise ordering. To deal with this issue, we propose a new framework, namely the dynamic principal component analysis. This addresses the identifiability of principal components from a global perspective, and transforms the problem into a regression model for data situated on the orthogonal matrix group space. The one-step unrolling method is exploited to solve the regression problem with a suitably constructed regular base curve. The minimax rate of the proposed estimators is established through theoretical analysis of the one-step unrolling and its connection to smoothing spline in the context of manifold-valued data.