On the number of principal components in high dimensions

TL;DR

This paper introduces a new method for determining the number of principal components to retain in high-dimensional PCA by sequentially testing skewness of residual scores, demonstrating consistency and effectiveness.

Contribution

It proposes a novel skewness-based sequential testing procedure for estimating the number of principal components in high-dimensional settings.

Findings

Estimator is consistent in high dimensions.

Performs well in simulation studies.

Provides reasonable estimates on real data.

Abstract

We consider the problem of how many components to retain in the application of principal component analysis when the dimension is much higher than the number of observations. To estimate the number of components, we propose to sequentially test skewness of the squared lengths of residual scores that are obtained by removing leading principal components. The residual lengths are asymptotically left-skewed if all principal components with diverging variances are removed, and right-skewed if not. The proposed estimator is shown to be consistent, performs well in high-dimensional simulation studies, and provides reasonable estimates in a number of real data examples.