Determining the number of components in principal components analysis: A comparison of statistical, crossvalidation and approximated methods

Principal component analysis is one of the most commonly used multivariate tools to describe and summarize data. Determining the optimal number of components in a principal component model is a fundamental problem in many fields of application. In this paper we compare the performance of several methods developed for this task in different areas of research. We consider statistical methods based on results from random matrix theory (Tracy-Widom and Kritchman-Nadler testing procedures), cross-validation methods (namely the well characterized element wise k-fold algorithm, ekf, and its corrected version cekf) and methods based on numerical approximation (SACV and GCV). The performance of these methods is assessed on both simulated and real life data sets. In both cases, differential behavior of the considered methods is observed, for which we propose theoretical explanations