Sample mean versus sample Frechet mean for combining complex Wishart matrices: a statistical study
File(s)
Accepted version
Author(s)
Zhuang, L
Walden, AT
Type
Journal Article
Abstract
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.
Date Issued
2017-06-08
Date Acceptance
2017-05-29
Citation
IEEE Transactions on Signal Processing, 2017, 65 (17), pp.4551-4561
ISSN
1941-0476
Publisher
IEEE
Start Page
4551
End Page
4561
Journal / Book Title
IEEE Transactions on Signal Processing
Volume
65
Issue
17
Copyright Statement
© 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Subjects
Science & Technology
Technology
Engineering, Electrical & Electronic
Engineering
Complex Wishart matrix
convex loss functions
extrinsic mean
Frechet mean
intrinsic mean
metrics
multi-variable power spectra
partial coherence
Riemannian distance
Riemannian manifold
risk
POSITIVE-DEFINITE MATRICES
BRAIN CONNECTIVITY
RIEMANNIAN METRICS
PARTIAL COHERENCE
PRECISION MATRIX
SPECTRAL DENSITY
DISTANCES
COVARIANCE
MD Multidisciplinary
Networking & Telecommunications
Publication Status
Published