# Intrinsic wavelet regression for curves of Hermitian positive definite matrices

## TL;DR

This paper develops intrinsic wavelet transforms for curves of Hermitian positive definite matrices, enabling improved spectral estimation of multivariate time series with guarantees of positive definiteness and invariance properties.

## Contribution

It introduces intrinsic wavelet methods in the Riemannian space of positive definite matrices, with convergence analysis and practical applications to spectral estimation.

## Key findings

- Intrinsic wavelet thresholding guarantees positive definite estimates.
- Method is equivariant under change of basis.
- Outperforms benchmark estimators in simulations.

## 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.