Positive-Definite Multivariate Spectral Estimation: A Geometric Wavelet Approach
(2017) , 47 pages
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- Abstract
- In nonparametric estimation of the autocovariance matrices or the spectral density matrix of a second-order stationary multivariate time series, it is important to preserve positive-definiteness of the estimator. This in order to ensure interpretability of the estimator as a covariance or spectral matrix, but also to avoid computational issues in e.g. simulation or bootstrapping. To this purpose, we consider multivariate spectral estimation on the Riemannian manifold of Hermitian and positive-definite matrices { based on a geometric wavelet approach. Nonlinear wavelet curve denoising on the Riemannian manifold allows one to capture not only local smoothness behavior in the spectral matrix across frequency, but also varying degrees of smoothness across components of the spectral matrix. Moreover, and in contrast to existing approaches, the wavelet-based spectral estimator enjoys the important property that it is equivariant to permutations of the components of the time series. In addition to spectral estimation, we propose computationally fast clustering of spectral matrices based on their wavelet domain representations, exploiting the fact that smooth curves on the Riemannian manifold are summarized by few high-energy wavelet coefficients. The spectral estimation and clustering methods are applied to analyze a brain signal time series dataset recorded over the course of an associative learning experiment.
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Citations
Chau, V. V., & von Sachs, R. (2017). Positive-Definite Multivariate Spectral Estimation: A Geometric Wavelet Approach (ISBA Discussion Paper 2017/02). https://hdl.handle.net/2078.5/178935