Random Matrix Derived Shrinkage of Spectral Precision Matrices

TL;DR

This paper introduces a novel spectral precision matrix shrinkage method based on random matrix theory and Rao-Blackwell estimators, improving estimation accuracy especially for singular spectral matrices in time series analysis.

Contribution

It develops an analytical approach for spectral matrix shrinkage using random matrix theory and Rao-Blackwell estimators, with a new parameter selection method based on predictive risk.

Findings

The new estimators outperform the Ledoit-Wolf inverse in simulations.

The methodology improves spectral precision matrix estimation in EEG data.

Analytical computations replace simulation-based methods.

Abstract

Much research has been carried out on shrinkage methods for real-valued covariance matrices. In spectral analysis of $p$-vector-valued time series there is often a need for good shrinkage methods too, most notably when the complex-valued spectral matrix is singular. The equivalent of the Ledoit-Wolf (LW) covariance matrix estimator for spectral matrices can be improved on using a Rao-Blackwell estimator, and using random matrix theory we derive its form. Such estimators can be used to better estimate inverse spectral (precision) matrices too, and a random matrix method has previously been proposed and implemented via extensive simulations. We describe the method, but carry out computations entirely analytically, and suggest a way of selecting an important parameter using a predictive risk approach. We show that both the Rao-Blackwell estimator and the random matrix estimator of the precision matrix can substantially outperform the inverse of the LW estimator in a time series setting. Our new methodology is applied to EEG-derived time series data where it is seen to work well and deliver substantial improvements for precision matrix estimation.