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

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  • Chau, Joris
  • von Sachs, Rainer

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.

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  • Chau, Joris & von Sachs, Rainer, 2022. "Time-varying spectral matrix estimation via intrinsic wavelet regression for surfaces of Hermitian positive definite matrices," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:csdana:v:174:y:2022:i:c:s0167947322000573
    DOI: 10.1016/j.csda.2022.107477
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    References listed on IDEAS

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    1. Guo, Wensheng & Dai, Ming & Ombao, Hernando C. & von Sachs, Rainer, 2003. "Smoothing Spline ANOVA for Time-Dependent Spectral Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 643-652, January.
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    4. Fryzlewicz, Piotr, 2007. "Unbalanced Haar technique for nonparametric function estimation," LSE Research Online Documents on Economics 25216, London School of Economics and Political Science, LSE Library.
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    7. Ombao, Hernando & Ringo Ho, Moon-ho, 2006. "Time-dependent frequency domain principal components analysis of multichannel non-stationary signals," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2339-2360, May.
    8. Ombao H. C & Raz J. A & von Sachs R. & Malow B. A, 2001. "Automatic Statistical Analysis of Bivariate Nonstationary Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 543-560, June.
    9. Fryzlewicz, Piotr & Timmermans, Catherine, 2016. "SHAH: SHape-Adaptive Haar wavelets for image processing," LIDAM Reprints ISBA 2016043, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Ombao, Hernando & von Sachs, Rainer & Guo, Wensheng, 2005. "SLEX Analysis of Multivariate Nonstationary Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 519-531, June.
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    Cited by:

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