Estimation of Large Covariance and Precision Matrices from Temporally Dependent Observations

TL;DR

This paper develops methods for estimating large covariance and precision matrices from high-dimensional, temporally dependent data, including long-range dependence, with theoretical guarantees and practical tuning procedures.

Contribution

It extends existing covariance and precision matrix estimation techniques to handle long-range temporal dependence in high-dimensional data, providing convergence rates and consistency results.

Findings

Methods applicable to temporally dependent data with long-range memory.

Convergence rates established for covariance and precision matrix estimators.

Proposed cross-validation method performs well in simulations.

Abstract

We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained $\ell_1$ minimization and the $\ell_1$ penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.