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

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.