Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices

Abstract

Nowadays, analytical instruments that produce a data matrix for one chemical sample enjoy a widespread popularity. However, for a successful analysis of these data an accurate estimate of the pseudorank of the matrix is often a crucial prerequisite. A large number of methods for estimating the pseudorank are based on the eigenvalues obtained from principal component analysis (PCA). In this paper methods are discussed that exploit the essential similarity between the residuals of PCA of the test data matrix and the elements of a random matrix. In the literature of PCA these methods are commonly denoted as parallel analysis. Attention is paid to several aspects that have to be considered when applying such methods. For some of these aspects asymptotic results can be found in the statistical literature. In this study Monte Carlo simulations are used to investigate the practical implications of these theoretical results. It is shown that for sufficiently large matrices the distribution of the measurement error does not significantly influence the results. Down to a very small signal-to-noise ratio the ratio of the number of rows and the number of columns constitutes the major influence on the expected value of the eigenvalues associated with the residuals. The consequences are illustrated for two functions of the eigenvalues, i.e. the logarithm of the eigenvalues and Malinowski's reduced eigenvalues. Both methods are graphical and have been applied in the past with considerable success for a variety of data. Malinowski's reduced eigenvalues are of special interest since they have been used to construct an F-test. Finally, a modification is proposed for pseudorank estimation methods that are based on the principle of parallel analysis.