Robust Burg Estimation of stationary autoregressive mixtures covariance

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Robust Burg Estimation of stationary autoregressive mixtures covariance


application/pdf Robust Burg Estimation of stationary autoregressive mixtures covariance


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Robust Burg Estimation of stationary autoregressive mixtures covariance Alexis Decurninge, Fr´ed´eric Barbaresco 1 Introduction In many engineering applications, non-Gaussian models are used for example for the modelization of I&Q data of strong clutters in radar such as ground or sea clutters. The family of complex ellip- tically symmetric distributions (which contains a lot of classical distributions such as multivariate Gaussian, multivariate Cauchy distributions and multivariate K-distributions) is a useful generaliza- tion of Gaussian random vectors by keeping the shape and location parameters. For our applications, the location parameter will be zero, so if we suppose that the clutter is ho- mogeneous, the key point is the estimation of the scatter matrix of the clutter through samples x1, .., xN ∈ Cd . The family of elliptical distributions is well adapted for the modelization non- Gaussian samples [7] by its generality given by its non-parametric amplitude part. In this framework, we will aim two kind of robustness for the estimation of the scatter matrix : • a robustness with respect to the amplitude distribution which is often heavy-tailed • a robustness in case of inhomogeneous distribution i.e. the samples are contaminated For general elliptical distributions, many estimators were proposed and among them, a M-estimator [9] proposed by Tyler which was shown to be a maximum likelihood estimator for normalized sam- ples x1 x1 , .., xN xN . Unfortunately, these estimators often lack of the second robustness listed above. We propose then to inspire from robust Huber method for the estimation of the scatter matrix of the samples x1 x1 , .., xN xN that were shown to share a distribution called angular central Gaussian [10]. Furthermore, we will consider stationary signals which adds a Toeplitz structure constraint for the scatter matrix. For Gaussian autoregressive vectors, it is usual to use Burg methods in order to estimate Toeplitz covariance matrices. We propose to adapt these techniques for mixtures of au- toregressive vectors, a big subfamily of the class of elliptical distribution. Estimators that take into account the Toeplitz constraints in the context of general elliptical distributions were proposed by [8]. We will compare our estimators to a geometrical method consisting in computing the median of Gaussian autoregressive models estimated from each sample xi [1] [2][3]. This method was shown to resist to the second robustness aimed thanks to the induced Riemannian geometry of the parame- ters. In the first part, we will explicit the model we consider for the samples. Then, we will present the estimators on normalized samples in the general case. In a third part, we will adapt classical Burg methods for autoregressive mixtures in order to estimate the constrained scatter matrix and we will conclude by presenting some illustrations on simulations. 1 References [1] M. Arnaudon, F. Barbaresco & Yang Le, Riemannian Medians and Means With Applications to Radar Signal Processing, IEEE Journal of Selected Topics in Signal Processing, vol. 7, n. 4, August 2013 [2] F. Barbaresco, Super resolution spectrum analysis regularization: Burg, capon and ago antago- nistic algorithms in Proc. EUSIPCO’96, Trieste, pp. 2005-2008, Sept. 1996 [3] F. Barbaresco, Information Geometry of Covariance Matrix : Cartan-Siegel Homogeneous Bounded Domains, Mostow/Berger fibration and Frechet median, in R. Bhatia & F.Nielsen Ed., ”Matrix Information Geometry”, Springer Lecture Notes in Mathematics, 2012 [4] S. Bausson, F. Pascal, P. Forster, J.P. Ovarlez, P. Larzabal First- and Second-Order Moments of the Normalized Sample Covariance Matrix of Spherically Invariant Random Vectors, IEEE Signal Processing Letters, vol. 14, no. 6, 2007 [5] P.J. Brockwell, R. Dalhaus, Generalized Durbin-Levinson and Burg Algorithms, Journal of Econometrics, 118, pp 129-149, 2003 [6] K. Kreutz-Delgado, The Complex Gradient Operator and the CR Calculus,, 2009 [7] E. Ollila, D. Tyler, V. Koivunen, V. Poor Complex Elliptically Symmetric Distributions : Survey, New Results and Applications, IEEE Transactions on signal processing, vol. 60, no. 11, 2012 [8] G. Pailloux, Estimation Structur´ee de la Covariance du Bruit en D´etection Adaptative PhD Thesis, 2010 [9] D. Tyler A Distribution-Free M-Estimator of Multivariate Scatter, The Annals of Statistics, Vol. 15, no. 1, pp 234-251, 1987 [10] D. Tyler Statistical Analysis for the angular central Gaussian distribution on the sphere, Biometrika, 74, 3, pp 579-589, 1987 2