# Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach

Abramovich, Yuri and Besson, Olivier Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach. (2012) [Report] (Unpublished)

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## Abstract

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically as likely'' as the a priori unknown actual covariance matrix. This expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this report is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix $\mSigma_{0}$. More precisely, we show 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 $T$. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first chapter of this report is devoted to the conventional scenario ($T \geq M$) while Chapter 2 deals with the under-sampled scenario ($T \leq M$). Although the two parts are closely related, each one is self-contained and could be read independently.

Item Type: Report Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)Other partners > WR Systems (USA) download Olivier Besson 22 Jan 2013 09:40

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