Intrinsic wavelet regression for curves of Hermitian positive definite matrices

In multivariate time series analysis, non-degenerate autocovariance and spectral density matrices are necessarily Hermitian and positive definite and it is important to preserve these properties in any estimation procedure. Our main contribution is the development of intrinsic wavelet transforms and nonparametric wavelet regression for curves in the non-Euclidean space of Hermitian positive definite matrices. The primary focus is on the construction of intrinsic average-interpolation wavelet transforms in the space equipped with a natural invariant Riemannian metric. In addition, we derive the wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth curves of Hermitian positive definite matrices. The intrinsic wavelet transforms are computationally fast and nonlinear wavelet shrinkage or thresholding captures localized features, such as cups or kinks, in the matrix-valued curves. In the context of nonparametric spectral estimation, the intrinsic linear or nonlinear wavelet spectral estimator satisfies the important property that it is equivariant under a change of basis of the time series, in contrast to most existing approaches. The finite-sample performance of the intrinsic wavelet spectral estimator based on nonlinear tree-structured trace thresholding is benchmarked against several state-of-the-art nonparametric curve regression procedures in the Riemannian manifold by means of simulated time series data.