Intrinsic Wavelet Regression for Curves of Hermitian Positive Definite Matrices

Abstract

Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment. Supplementary materials for this article are available online.

Keywords:

• Fourier spectral matrix
• Hermitian positive definite matrices
• Intrinsic wavelet transform
• Multivariate time series
• Riemannian manifold
• Wavelet thresholding

Supplementary Materials

Additional background material and detailed proofs of the presented results are available in the online supplementary materials, and the accompanying R-package pdSpecEst is publicly available on CRAN (Chau 2017).

Acknowledgments

We thank the UC Irvine Space-Time Modeling Group and Dr. Emad Eskandar (Massachusetts General Hospital) for the local field potential data to illustrate the methodology and the anonymous referees for their suggestions that helped improving the presentation of this work.

Additional information

Funding

The authors gratefully acknowledge financial support from the following agencies and projects: the Belgian Fund for Scientific Research FRIA/FRS-FNRS (J. Chau), the contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the “Communauté française de Belgique” (R. von Sachs), IAP research network P7/06 of the Belgian government (R. von Sachs).