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Intrinsic wavelet regression for curves and surfaces of Hermitian positive definite matrices

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STSW02 - Statistics of geometric features and new data types

Co-author: Joris Chau (ISBA, UC Louvain)

In multivariate time series analysis, non-degenerate autocovariance and spectral density matrices are necessarily Hermitian and positive definite. An estimation methodology which preserves these properties is developed based on intrinsic wavelet transforms being applied to nonparametric wavelet regression for curves in the non-Euclidean space of Hermitian positive definite matrices. Via intrinsic average-interpolation in a Riemannian manifold equipped with a natural invariant Riemannian metric, we derive the wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth curves. Applying this more specifically to nonparametric spectral density estimation, an important property of the intrinsic linear or nonlinear wavelet spectral estimator under the invariant Riemannian metric is that it is independent of the choice of coordinate system of the time series, in contrast to most existing approaches. As a generalisation of this one-dimensional denoising of matrix-valued curves in the Riemannian manifold we also present higher-dimensional intrinsic wavelet transforms, applied in particular to time-varying spectral estimation of non-stationary multivariate time series, i.e. surfaces of Hermitian positive definite matrices.

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