Time-varying spectral matrix estimation via intrinsic wavelet regression for surfaces of Hermitian positive definite matrices

Joris Chau and Rainer von Sachs
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Rainer von Sachs: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2022016, LIDAM Reprints ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: Intrinsic wavelet transforms and denoising methods are introduced for the purpose of time-varying Fourier spectral matrix estimation. A non-degenerate time-varying spectral matrix constitutes a surface of Hermitian positive definite matrices across time and frequency and any spectral matrix estimator ideally adheres to these geometric constraints. Spectral matrix estimation of a locally stationary time series by means of linear or nonlinear wavelet shrinkage naturally respects positive definiteness at each time-frequency point, without any postprocessing. Moreover, the spectral matrix estimator enjoys equivariance in the sense that it does not nontrivially depend on the chosen basis or coordinate system of the multivariate time series. The algorithmic construction is based on a second-generation average-interpolating wavelet transform in the space of Hermitian positive definite matrices equipped with an affine-invariant metric. The wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth surfaces of Hermitian positive definite matrices are derived. Furthermore, practical nonlinear thresholding based on the trace of the matrix-valued wavelet coefficients is investigated. Finally, the time-varying spectral matrix of a nonstationary multivariate electroencephalography (EEG) time series recorded during an epileptic brain seizure is estimated.

Keywords: Multivariate nonstationary time series; Time-varying spectral matrix estimation; Hermitian positive definite matrices; Surface wavelet transform; Riemannian manifold; Affine-invariant metric (search for similar items in EconPapers)
Date: 2021-03-25
Note: In: Computational Statistics & Data Analysis, 2022, 107477
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Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvar:2022016

DOI: 10.1016/j.csda.2022.107477

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