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Séminaire exceptionnel# Bruit, Non-linéaire et Information - 19 juin à 14h30

2012-06-06

A l'occasion de la soutenance de HDR de Steeve ZOZOR, un séminaire exceptionnel est organisé le 19 juin en salle Mont-Blanc de 14h30 à 18h. 5 intervenants exposeront leur domaine de recherche.

14h30-15h05 *Co-evolving agents subject to local versus nonlocal barycentric interactions* par Pr. Max Hongler (LPM, EPFL, Lausanne, Suisse)

15h10-15h45 *Extracting correlation structure from large random matrices* par Pr. Alfred Hero (University of Michigan)

15h50-16h25 *A Probabilistic Approach to the Umbral and Operational Calculu*s, par Pr. Christophe Vignat (LSS, Universite d'Orsay & EPFL)

Pause

16h45-17h20 *Entropic moments and Renyi moments of hypergeometric polynomials: Asymptotics and open problems*, par Pr. Jesus Dehesa (Universidad de Granada, Espana)

17h25-18h00 *Generalized Cramér-Rao inequalitiesand characterization of q-Gaussians*, par Pr. Jean-François Bercher (ESIEE, Universite Paris Est)**Résumés** :

*Co-evolving agents subject to local versus nonlocal barycentric interactions* (M. Hongler)

The mean-field dynamics of a collection of stochastic agents with local versus nonlocal interactions is studied via analytically soluble models. The nonlocal interactions result from a barycentric modulation of the observation range of the agents.

Our modeling framework is based on a discrete two-velocity Boltzmann dynamics which can be analytically discussed. Depending on the interaction range, we analytically observe a transition from a purely diffusive regime without definite pattern to a flocking evolution represented by a solitary wave traveling with constant velocity.

*Extracting correlation structure from large random matrices* (A. Hero)

Random matrices arise in many areas of engineering, social sciences, and natural sciences. For example, when rows of the random matrix record successive samples of a multivariate response the sample correlation between the columns can reveal important dependency structure in the multivariate response, e.g., stars, hubs and triangles of co-dependency. However, when the number of samples is nite and the number p of columns increases such exploration becomes futile due to a phase transition phenomenon: spurious discoveries will eventually dominate. In this presentation I will present theory for predicting these phase transitions and present Poisson limit theorems that can be used to predict finite sample behavior of correlation structure. The theory has application to areas including bioinformatics, portfolio selection, social networks, and computer vision.*A Probabilistic Approach to the Umbral and Operational Calculus* (C. Vignat)

This talk will introduce a probabilistic approach to the Operational calculus by Nieto and Truax, and of the Umbral calculus by Blissard. In the first part, the power exponential derivation operators of Nieto and Truax will be introduced and interpreted as expectation operators over a suitably chosen random variable. In a second part, the Umbral calculus will be introduced and, in the case of the Hermite and the Bernoulli umbrae, an equivalent expectation operator will be derived. It will be shown that this probabilistic approach allows to simplify considerably some computations that involve special functions such as the Hermite or the Bernoulli polynomials.

*Entropic moments and Rényi moments of hypergeometric polynomials: Asymptotics and open problems *(J. S. Dehesa)

The hypergeometric polynomials yn(x) control the physical solutions of numerous quantum and classical wave equations. Moreover, the macroscopic properties of atomic and molecular systems are known to be determined, according to the Density Functional Theory of Hohenberg and Kohn, by means of the qth-order entropic moments and Renyi entropies of the Rakhmanov probability density of these polynomials. In this talk we will discuss two analytical methods to calculate these quantities. One is based on the multivariate Bell polynomials of combinatorics, and another one uses some linearization formulas of powers of hypergeometric polynomials whose coefficients are expressed in terms of some generalised functions of Lauricella and Srivastava-Daoust types. In addition, the nth- and qth-asymptotics of the entropic moments and Rényi entropies of the four canonical systems of orthogonal hypergeometric polynomials are determined and some open problems are given.

*Generalized Cramér-Rao inequalities and characterization of q-Gaussians* (J.-F. Bercher)

In this talk, we will discuss Cramer-Rao inequalities in the context of nonextensive statistics and in estimation theory. We will give characterizations of generalized q-Gaussian distributions, and introduce generalized versions of Fisher information. Topics will include (i) the derivation of new extended Cramér-Rao inequalities for the estimation of a parameter, involving general q-moments of the estimation error, (ii) the derivation of Cramér-Rao inequalities saturated by generalized q-Gaussian distributions, (iii) the definition of generalized Fisher informations, (iv) the identification and interpretation of some prior results, and finally, (v) the suggestion of new estimation methods.

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