Determining The Number of Principal Components with Schur's Theorem in Principal Component Analysis

Abstract

Principal Component Analysis is a method for reducing the dimensionality of datasets while also limiting information loss. It accomplishes this by producing uncorrelated variables that maximize variance one after the other. The accepted criterion for evaluating a Principal Component’s (PC) performance is λ_j/tr(S) where tr(S) denotes the trace of the covariance matrix S. It is standard procedure to determine how many PCs should be maintained using a predetermined percentage of the total variance. In this study, the diagonal elements of the covariance matrix are used instead of the eigenvalues to determine how many PCs need to be considered to obtain the defined threshold of the total variance. For this, an approach which uses one of the important theorems of majorization theory is proposed. Based on the tests, this approach lowers the computational costs.