Nonlinear wavelet threshold estimation of time-varying covariance matrices in a log-Euclidean manifold

We tackle the problem of estimating time-varying covariance matrices (TVCM; i.e., covariance matrices with entries being time-dependent curves) whose elements show inhomogeneous smoothness over time (e.g., pronounced local peaks). To address this challenge, wavelet denoising estimators are particularly appropriate. Specifically, we model TVCM using a signal-noise model within the Riemannian manifold of symmetric positive definite matrices (endowed with the log-Euclidean metric) and use the intrinsic wavelet transform, designed for curves in Riemannian manifolds. Within this non-Euclidean framework, the proposed estimators preserve positive definiteness. Although linear wavelet estimators for smooth TVCM achieve good results in various scenarios, they are less suitable if the underlying curve features singularities. Consequently, our estimator is designed around a nonlinear thresholding scheme, tailored to the characteristics of the noise in covariance matrix regression models. The effectiveness of this novel nonlinear scheme, equipped with a variety of new intrinsic thresholding rules, is assessed by deriving mean-squared error consistency and by numerical simulations, and its practical application is demonstrated on TVCM of electroencephalography (EEG) data showing abrupt transients over time.