Which principal components are most sensitive in the change detection problem?

Abstract

Principal component analysis (PCA) is often used in anomaly detection and statistical process control tasks. For bivariate normal data, we prove that the minor projection (the least varying projection) of the PCA‐rotated data is the most sensitive to distributional changes, where sensitivity is defined as the Hellinger distance between the projections' marginal distributions before and after a change. In particular, this is almost always the case if only one parameter of the bivariate normal distribution changes, that is, the change is sparse. Simulations indicate that the minor projections are the most sensitive for a large range of changes and pre‐change settings in higher dimensions as well, including changes that are very sparse. This motivates using only a few of the minor projections for detecting sparse distributional changes in high‐dimensional data.