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

Besson, Olivier and Abramovich, Yuri Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case. (2013) IEEE Transactions on Signal Processing, 61 (23). 5819-5829. ISSN 1053-587X

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Official URL: http://dx.doi.org/10.1109/TSP.2013.2285511

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.

Item Type: Article Thanks to IEEE editor. (c) 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. The definitive version is available at http://ieeexplore.ieee.org hal-00904983 International peer-reviewed journal Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)Other partners > WR Systems (USA) DGA/MRIS download Olivier Besson 15 Nov 2013 15:21

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