Sample Mean Versus Sample Fréchet Mean for Combining Complex Wishart Matrices: A Statistical Study
| Autor: | L. Zhuang, Andrew T. Walden |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Wishart distribution
Technology extrinsic mean POSITIVE-DEFINITE MATRICES multi-variable power spectra Complex Wishart matrix 02 engineering and technology 01 natural sciences partial coherence metrics Combinatorics SPECTRAL DENSITY 010104 statistics & probability Estimation of covariance matrices Engineering MD Multidisciplinary 0202 electrical engineering electronic engineering information engineering Applied mathematics BRAIN CONNECTIVITY COVARIANCE 0101 mathematics Electrical and Electronic Engineering DISTANCES risk Mathematics Riemannian distance Science & Technology Riemannian manifold Frechet mean Covariance matrix intrinsic mean Truncated mean Estimator Engineering Electrical & Electronic 020206 networking & telecommunications PRECISION MATRIX Covariance Fréchet mean RIEMANNIAN METRICS Signal Processing convex loss functions Networking & Telecommunications Random matrix |
| Zdroj: | IEEE Transactions on Signal Processing. 65:4551-4561 |
| ISSN: | 1941-0476 1053-587X |
| DOI: | 10.1109/tsp.2017.2713763 |
| Popis: | The space of covariance matrices is a non-Euclidean space. The matrices form a manifold which if equipped with a Riemannian metric becomes a Riemannian manifold, and recently this idea has been used for comparison and clustering of complex valued spectral matrices, which at a given frequency are typically modelled as complex Wishart-distributed random matrices. Identically distributed sample complex Wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator. However, using the Riemannian geometry their so-called sample Fr´echet mean can also be found. We derive the expected value of the determinant of the sample Fr´echet mean and the expected value of the sample Fr´echet mean itself. The population Fr´echet mean is shown to be a scaled version of the true covariance matrix. The risk under convex loss functions for the standard sample mean is never larger than for the Fr´echet mean. In simulations the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix, namely partial coherence. |
| Databáze: | OpenAIRE |
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