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


Frédéric Barbaresco
Optimal matching between curves in a manifold
Drone Tracking Using an Innovative UKF
Jean-Louis Koszul et les structures élémentaires de la Géométrie de l’Information
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics
Session Geometrical Structures of Thermodynamics (chaired by Frédéric Barbaresco, François Gay-Balmaz)
Opening and closing sessions (chaired by Frédéric Barbaresco, Frank Nielsen, Silvère Bonnabel)
GSI'17-Closing session
GSI'17 Opening session
Démonstrateur franco-britannique "IRM" : gestion intelligente et homéostatique des radars multifonctions
Principes & applications de la conjugaison de phase en radar : vers les antennes autodirectives
Nouvelles formes d'ondes agiles en imagerie, localisation et communication
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Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics
Reparameterization invariant metric on the space of curves
Probability density estimation on the hyperbolic space applied to radar processing
SEE-GSI'15 Opening session
Lie Groups and Geometric Mechanics/Thermodynamics (chaired by Frédéric Barbaresco, Géry de Saxcé)
Opening Session (chaired by Frédéric Barbaresco)
Invited speaker Charles-Michel Marle (chaired by Frédéric Barbaresco)
Koszul Information Geometry & Souriau Lie Group 4Thermodynamics
MaxEnt’14, The 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
Koszul Information Geometry & Souriau Lie Group Thermodynamics
Robust Burg Estimation of stationary autoregressive mixtures covariance
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Oral session 7 Quantum physics (Steeve Zozor, Jean-François Bercher, Frédéric Barbaresco)
Opening session (Ali Mohammad-Djafari, Frédéric Barbaresco)
Tutorial session 1 (Ali Mohammad-Djafari, Frédéric Barbaresco, John Skilling)
Prix Thévenin 2014
SEE/SMF GSI’13 : 1 ère conférence internationale sur les Sciences  Géométriques de l’Information à l’Ecole des Mines de Paris
Synthèse (Frédéric Barbaresco)
POSTER SESSION (Frédéric Barbaresco)
ORAL SESSION 16 Hessian Information Geometry II (Frédéric Barbaresco)
Information/Contact Geometries and Koszul Entropy
Geometric Science of Information - GSI 2013 Proceedings
Médaille Ampère 2007


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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