Sample Mean Versus Sample Fréchet Mean for Combining Complex Wishart Matrices: A Statistical Study

Abstract

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 Fréchet mean can also be found. We derive the expected value of the determinant of the sample Fréchet mean and the expected value of the sample Fréchet mean itself. The population Fréchet 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 Fréchet 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.

Sample Space-time Covariance Matrix Estimation

“…We derive the variance of a sample cross-correlation sequence, which then forms the building block for the sample space-time covariance. Particularisation of our results agree with [28,30,31] and results from spectral estimation such as [32].…”

“…Various attempts have been undertaken for random signals that can be modelled as first order auto-regressive processes [26,27], or generally [28,29]. For the broadband case, analysis has generally been restricted to narrowband signals, such that the spatial covariance matrix is Wishart distributed [30,31]. We derive the variance of a sample cross-correlation sequence, which then forms the building block for the sample space-time covariance.…”

Impact of Space-Time Covariance Estimation Errors on a Parahermitian Matrix EVD

“…Determining the distribution of the sample space-time covariance matrixR[τ ] orR(z) is challenging. In the case where [x][n] is only spatially but not temporally correlated, the instantanious sample covariance matrixR[0] is Wishartdistributed [11], [29]. For such a matrix, the distribution of elements depends both on the size of the sample set, N , and also the entries in the ground truth…”