Covariance Operator Estimation via Adaptive Thresholding

TL;DR

This paper proposes an adaptive thresholding method for estimating sparse covariance operators in nonstationary processes, improving accuracy over traditional methods by accounting for local variance and correlation structures.

Contribution

It introduces a novel adaptive thresholding estimator for covariance operators that leverages variance estimates, with theoretical error bounds and demonstrated advantages in nonstationary contexts.

Findings

Adaptive thresholding outperforms universal thresholding in simulations.

Theoretical bounds on estimation error are derived.

Method effectively handles nonstationary processes with varying variance.

Abstract

This paper studies sparse covariance operator estimation for nonstationary processes with sharply varying marginal variance and small correlation lengthscale. We introduce a covariance operator estimator that adaptively thresholds the sample covariance function using an estimate of the variance component. Building on recent results from empirical process theory, we derive an operator norm bound on the estimation error in terms of the sparsity level of the covariance and the expected supremum of a normalized process. Our theory and numerical simulations demonstrate the advantage of adaptive threshold estimators over universal threshold and sample covariance estimators in nonstationary settings.