Gohberg-Semencul Estimation of Toeplitz Structured Covariance Matrices and Their Inverses

TL;DR

This paper introduces novel likelihood-based estimators for Toeplitz structured covariance matrices and their inverses, ensuring positive definiteness and combining advantages of existing methods, validated through extensive simulations.

Contribution

It develops positive definiteness enforcing constraints for Gohberg-Semencul parameterization and proposes hyperparameter tuning and a closed-form estimator for improved covariance estimation.

Findings

Estimators perform well in simulations for Toeplitz structured CMs.

The proposed methods ensure positive definiteness and improve estimation accuracy.

Computationally efficient closed-form estimator validated across multiple scenarios.

Abstract

When only few data samples are accessible, utilizing structural prior knowledge is essential for estimating covariance matrices and their inverses. One prominent example is knowing the covariance matrix to be Toeplitz structured, which occurs when dealing with wide sense stationary (WSS) processes. This work introduces a novel class of positive definiteness ensuring likelihood-based estimators for Toeplitz structured covariance matrices (CMs) and their inverses. In order to accomplish this, we derive positive definiteness enforcing constraint sets for the Gohberg-Semencul (GS) parameterization of inverse symmetric Toeplitz matrices. Motivated by the relationship between the GS parameterization and autoregressive (AR) processes, we propose hyperparameter tuning techniques, which enable our estimators to combine advantages from state-of-the-art likelihood and non-parametric estimators. Moreover, we present a computationally cheap closed-form estimator, which is derived by maximizing an approximate likelihood. Due to the ensured positive definiteness, our estimators perform well for both the estimation of the CM and the inverse covariance matrix (ICM). Extensive simulation results validate the proposed estimators' efficacy for several standard Toeplitz structured CMs commonly employed in a wide range of applications.