Intrinsic wavelet regression for surfaces of Hermitian positive definite matrices

TL;DR

This paper introduces intrinsic wavelet denoising techniques for surfaces of Hermitian positive definite matrices, enabling nonparametric estimation of time-varying spectral matrices in multivariate time series analysis.

Contribution

It develops wavelet transforms directly on curved Riemannian manifolds of matrices and analyzes their convergence and practical denoising performance.

Findings

Wavelet coefficient decay rates established for smooth matrix surfaces.

Effective nonlinear thresholding improves spectral matrix estimation.

Application to EEG data demonstrates practical utility.

Abstract

This paper develops intrinsic wavelet denoising methods for surfaces of Hermitian positive definite matrices, with in mind the application to nonparametric estimation of the time-varying spectral matrix of a multivariate locally stationary time series. First, we construct intrinsic average-interpolating wavelet transforms acting directly on surfaces of Hermitian positive definite matrices in a curved Riemannian manifold with respect to an affine-invariant metric. Second, we derive the wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth surfaces of Hermitian positive definite matrices, and investigate practical nonlinear thresholding of wavelet coefficients based on their trace in the context of intrinsic signal plus noise models in the Riemannian manifold. The finite-sample performance of nonlinear tree-structured trace thresholding is assessed by means of simulated data, and the proposed intrinsic wavelet methods are used to estimate the time-varying spectral matrix of a nonstationary multivariate electroencephalography (EEG) time series recorded during an epileptic brain seizure.