Scaling in the eigenvalue fluctuations of correlation matrices

Abstract

The spectra of empirical correlation matrices, constructed from multivariate data, are widely used in many areas of sciences, engineering, and social sciences as a tool to understand the information contained in typically large data sets. In the past two decades, random-matrix-theory-based tools such as the nearest-neighbor eigenvalue spacing and eigenvector distributions have been employed to extract the significant modes of variability present in such empirical correlations. In this work we present an alternative analysis in terms of the recently introduced spacing ratios, which does not require the cumbersome unfolding process. It is shown that the higher-order spacing ratio distributions for the Wishart ensemble of random matrices, characterized by the Dyson index β , are related to the first-order spacing ratio distribution with a modified value of codimension β′. This scaling is demonstrated for the Wishart ensemble and also for the spectra of empirical correlation matrices drawn from the observed stock market and atmospheric pressure data. Using a combination of analytical and numerics, such scalings in spacing distributions are also discussed.