Cleaning large-dimensional covariance matrices for correlated samples.

Abstract

We elucidate the problem of estimating large-dimensional covariance matrices in the presence of correlations between samples. To this end, we generalize the Marčenko-Pastur equation and the Ledoit-Péché shrinkage estimator using methods of random matrix theory and free probability. We develop an efficient algorithm that implements the corresponding analytic formulas based on the Ledoit-Wolf kernel estimation technique. We also provide an associated open-source Python library, called shrinkage, with a user-friendly API to assist in practical tasks of estimation of large covariance matrices. We present an example of its usage for synthetic data generated according to exponentially decaying autocorrelations.