Time-Varying Covariance Matrices Estimation by Nonlinear Wavelet Thresholding in a Log-Euclidean Riemannian Manifold

Gabriel Bailly and Rainer von Sachs ()
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Gabriel Bailly: Université catholique de Louvain, LIDAM/ISBA, Belgium
Rainer von Sachs: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2024004, LIDAM Discussion Papers ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: 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 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.

Keywords: Nonlinear wavelet thresholding; non-Euclidean geometry; sample covariance matrices; time-varying second-order structure; log-Wishart distribution (search for similar items in EconPapers)
Pages: 45
Date: 2024-02-12
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