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Intrinsic Wavelet Regression for Curves of Hermitian Positive Definite Matrices

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  • Joris Chau
  • Rainer von Sachs

Abstract

Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment. Supplementary materials for this article are available online.

Suggested Citation

  • Joris Chau & Rainer von Sachs, 2021. "Intrinsic Wavelet Regression for Curves of Hermitian Positive Definite Matrices," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(534), pages 819-832, April.
    • Handle: RePEc:taf:jnlasa:v:116:y:2021:i:534:p:819-832
    DOI: 10.1080/01621459.2019.1700129
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    Cited by:

    1. Bailly, Gabriel & von Sachs, Rainer, 2024. "Time-Varying Covariance Matrices Estimation by Nonlinear Wavelet Thresholding in a Log-Euclidean Riemannian Manifold," LIDAM Discussion Papers ISBA 2024004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. 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).
    3. Evangelos E. Ioannidis, 2022. "A new non‐parametric cross‐spectrum estimator," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(5), pages 808-827, September.

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