# Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

1 DEOS - Département Electronique, Optronique et Signal 2 WR Systems USA

Abstract : In the first part of this series of two papers, we extended the expected likelihood approach originally developed in the Gaussian case, to the broader class of complex elliptically symmetric CES distributions and complex angular central Gaussian ACG distributions. More precisely, we demonstrated that the probability density function p.d.f. of the likelihood ratio LR for the unknown actual scatter matrix $\mSigma {0}$ does not depend on the latter: it only depends on the density generator for the CES distribution and is distribution-free in the case of ACG distributed data, i.e., it only depends on the matrix dimension $M$ and the number of independent training samples $T$, assuming that $T \geq M$. Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario $T \leq M$ which deserves a specific treatment since conventional maximum likelihood estimates do not exist. Indeed, inference about the scatter matrix can only be made in the $T$-dimensional subspace spanned by the columns of the data matrix. We extend the results derived under the Gaussian assumption to the CES and ACG class of distributions. Invariance properties of the under-sampled likelihood ratio evaluated at $\mSigma {0}$ are presented. Remarkably enough, in the ACG case, the p.d.f. of this LR can be written in a rather simple form as a product of beta distributed random variables. The regularized schemes derived in the first part, based on the EL principle, are extended to the under-sampled scenario and assessed through numerical simulations.

Keywords : Covariance matrix estimation Elliptically symmetric distributions Expected likelihood Likelihood ratio Regularization

Autor: Olivier Besson - Yuri Abramovich -

Fuente: https://hal.archives-ouvertes.fr/

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