179 research outputs found
Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case
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 does not depend on this matrix and is fully specified by the matrix dimension and the number of independent training samples . Since this p.d.f. could therefore be pre-calculated for any a priori known , 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 series of two papers 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}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario () while Part 2 deals with the under-sampled scenario ()
Generalized robust shrinkage estimator and its application to STAP detection problem
Recently, in the context of covariance matrix estimation, in order to improve
as well as to regularize the performance of the Tyler's estimator [1] also
called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator
has been introduced in [3]. First, this work extends the results of [3,4] by
giving the general solution of the "shrinkage" fixed-point algorithm. Secondly,
by analyzing this solution, called the generalized robust shrinkage estimator,
we prove that this solution converges to a unique solution when the shrinkage
parameter (losing factor) tends to 0. This solution is exactly the FPE
with the trace of its inverse equal to the dimension of the problem. This
general result allows one to give another interpretation of the FPE and more
generally, on the Maximum Likelihood approach for covariance matrix estimation
when constraints are added. Then, some simulations illustrate our theoretical
results as well as the way to choose an optimal shrinkage factor. Finally, this
work is applied to a Space-Time Adaptive Processing (STAP) detection problem on
real STAP data
Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case
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 and the number of independent training samples , assuming that . Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario () 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 -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
Asymptotic properties of robust complex covariance matrix estimates
In many statistical signal processing applications, the estimation of
nuisance parameters and parameters of interest is strongly linked to the
resulting performance. Generally, these applications deal with complex data.
This paper focuses on covariance matrix estimation problems in non-Gaussian
environments and particularly, the M-estimators in the context of elliptical
distributions. Firstly, this paper extends to the complex case the results of
Tyler in [1]. More precisely, the asymptotic distribution of these estimators
as well as the asymptotic distribution of any homogeneous function of degree 0
of the M-estimates are derived. On the other hand, we show the improvement of
such results on two applications: DOA (directions of arrival) estimation using
the MUSIC (MUltiple SIgnal Classification) algorithm and adaptive radar
detection based on the ANMF (Adaptive Normalized Matched Filter) test
Group Symmetry and non-Gaussian Covariance Estimation
We consider robust covariance estimation with group symmetry constraints.
Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and
Multivariate Generalized Gaussian distribution methods, usually involve
non-convex minimization problems. Recently, it was shown that the underlying
principle behind their success is an extended form of convexity over the
geodesics in the manifold of positive definite matrices. A modern approach to
improve estimation accuracy is to exploit prior knowledge via additional
constraints, e.g., restricting the attention to specific classes of covariances
which adhere to prior symmetry structures. In this paper, we prove that such
group symmetry constraints are also geodesically convex and can therefore be
incorporated into various non-Gaussian covariance estimators. Practical
examples of such sets include: circulant, persymmetric and complex/quaternion
proper structures. We provide a simple numerical technique for finding maximum
likelihood estimates under such constraints, and demonstrate their performance
advantage using synthetic experiments
Bounds for maximum likelihood regular and non-regular DoA estimation in K-distributed noise
We consider the problem of estimating the direction of arrival of a signal embedded in -distributed noise, when secondary data which contains noise only are assumed to be available. Based upon a recent formula of the Fisher information matrix (FIM) for complex elliptically distributed data, we provide a simple expression of the FIM with the two data sets framework. In the specific case of -distributed noise, we show that, under certain conditions, the FIM for the deterministic part of the model can be unbounded, while the FIM for the covariance part of the model is always bounded. In the general case of elliptical distributions, we provide a sufficient condition for unboundedness of the FIM. Accurate approximations of the FIM for -distributed noise are also derived when it is bounded. Additionally, the maximum likelihood estimator of the signal DoA and an approximated version are derived, assuming known covariance matrix: the latter is then estimated from secondary data using a conventional regularization technique. When the FIM is unbounded, an analysis of the estimators reveals a rate of convergence much faster than the usual . Simulations illustrate the different behaviors of the estimators, depending on the FIM being bounded or not
On the convergence of Maronna's -estimators of scatter
In this paper, {we propose an alternative proof for the uniqueness} of
Maronna's -estimator of scatter (Maronna, 1976) for vector observations
under a mild constraint of linear
independence of any subset of of these vectors. This entails in particular
almost sure uniqueness for random vectors with a density as long
as . {This approach allows to establish further relations that demonstrate
that a properly normalized Tyler's -estimator of scatter (Tyler, 1987) can
be considered as a limit of Maronna's -estimator. More precisely, the
contribution is to show that each -estimator converges towards a particular
Tyler's -estimator.} These results find important implications in recent
works on the large dimensional (random matrix) regime of robust -estimation
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