  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.
