Wishart distributions for decomposable graphs

When considering a graphical Gaussian model ${\mathcal{N}}_G$ Markov with respect to a decomposable graph $G$, the parameter space of interest for the precision parameter is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. The parameter space for the scale parameter of ${\mathcal{N}}_G$ is the cone $Q_G$, dual to $P_G$, of incomplete matrices with submatrices corresponding to the cliques of $G$ being positive definite. In this paper we construct on the cones $Q_G$ and $P_G$ two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior.Comment: Published at http://dx.doi.org/10.1214/009053606000001235 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gaussian approximation of Gaussian scale mixture

For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance between $Z\sqrt{V}$ and $Z\sqrt{t_0}$ in the $L^2(\R)$ sense, is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.Comment: 13 page

An Expectation Formula for the Multivariate Dirichlet Distribution

AbstractSuppose that the random vector (X1, …, Xq) follows a Dirichlet distribution on Rq+ with parameter (p1, …, pq)∈Rq+. For f1, …, fq>0, it is well-known that E(f1X1+…+fqXq)−(p1+…+pq)=f−p11…f−pqq. In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let Ωr denote the cone of all r×r positive-definite real symmetric matrices. For x∈Ωr and 1⩽j⩽r, let detjx denote the jth principal minor of x. For s=(s1, …, sr)∈Rr, the generalized power function of x∈Ωr is the function Δs(x)=(det1x)s1−s2(det2x)s2−s3…(detr−1x)sr−1−sr(detrx)sr; further, for any t∈R, we denote by s+t the vector (s1+t, …, sr+t). Suppose X1, …, Xq∈Ωr are random matrices such that (X1, …, Xq) follows a multivariate Dirichlet distribution with parameters p1, …, pq. Then we evaluate the expectation E[Δs1(X1)…Δsq(Xq)Δs1+…+sq+p((a+f1X1+…+fqXq)−1)], where a∈Ωr, p=p1+…+pq, f1, …, fq>0, and s1, …, sq each belong to an appropriate subset of Rr+. The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the Xj's are singular. Our derivation utilizes the framework of symmetric cones, so that our results are valid for multivariate Dirichlet distributions on all symmetric cones

Structural interpretation of the modification of soda-lime-silica glass properties by P2O5

The influence of P2O5 addition on soda-lime-silica glass properties is discussed on the basis of a structural model. It involves two effects: Insertion of phosphate molecular complexes into the silicate network interstices, and increase in silicate network condensation. The latter results from the scavenging of sodium and calcium cations by P2O5. The properties considered in this paper are high-temperature viscosity and liquidus temperature. The P2O5 additions are in the 0.8 to 3.0 mol% range. The silica content of the glasses varies from 60 to 67 mol%, and the ratio CaO/(CaO+Na2O) from 0 to 0.85. Phosphate molecular complexes decrease the liquidus temperature and the high-temperature viscosity increasing the network condensation results in an increase in the viscosity, but its effect on the liquidus temperature depends on whether the glass composition lies on an upward or downward liquidus surface of the phase diagram

Spectroscopy of High Energy BL Lac Objects with X-shooter on the VLT

Context. The study of BL Lac objects (BLL) detected in gamma-rays gives insights on the acceleration mechanisms in play in such systems and is also a valuable tool to constrain the density of the extragalactic background light. As their spectra are dominated by the non-thermal emission of the jet and the spectral features are weak and narrow in the optical domain, measuring their redshift is challenging. However such a measure is fundamental as it allows a firm determination of the distance and luminosity of the source, and therefore a consistent model of its emission. Aims. Measurement of the redshift of BLL detected in gamma-rays and determination of global properties of their host galaxies. Methods. We observed a sample of eight BLL (KUV 00311-1938, PKS 0447-439, PKS 0301-243, BZB J0238-3116, BZB J0543-5532, BZB J0505+0415, BZB J0816-1311 and RBS 334) with the X-shooter spectrograph installed at the ESO Very Large Telescope in order to take advantage of its unprecedented wavelength coverage and of its resolution about 5 times higher than generally used in such studies. We extracted UVB to NIR spectra that we then corrected for telluric absorption and calibrated in flux. We systematically searched for spectral features. When possible, we determined the contribution of the host galaxy to the overall emission. Results. Of the eight BLL, we measured the redshift of five of them and determined lower limits for two through the detection of intervening systems. All seven of these objects have redshifts greater than 0.2. In two cases, we refuted redshift values reported in other publications. Through careful modelling, we determined the magnitude of the host galaxies. In two cases, the detection of emission lines allowed to provide hints on the overall properties of the gas in the host galaxies.Comment: 19 pages, 10 figures, 7 tables, in press on A&

Etude de l'influence de la température sur la compressibilité et la résistance à la rupture de matériaux granulaires

9 pagesLes poudres sont couramment utilisées sous forme de comprimés. Le comportement de ces poudres est sensible à différents paramètres tels que la température et l'humidité. La plupart des études de comportement ont porté sur l'influence des paramètres contrôlant la compression mais très rarement sur l'influence des paramètres extérieurs. Aussi avons nous choisi d'étudier l'influence de la température au cours du cycle de compression sur la compressibilité et sur les caractéristiques finales des comprimés obtenus à partir d'un composé organique fixé. Des expériences ont été réalisées avec une presse uniaxiale LLOYDS. Un système de chauffage et de régulation en température a été adapté sur des matrices de compression uniaxiale. Un suivi de la porosité et des émissions acoustiques en cours de compression a été réalisé. Les relations de ces caractéristiques avec les mesures de la résistance à la rupture et l'évolution des microstructures des comprimés obtenus sont discutées

Variation of the Western Equatorial Pacific ocean, 1986-1988

Twenty one oceanographic sections made along 165°E during 1984-1988 provide a unique picture of the 1986-1987 El Nino and the subsequent La Nina in the Western Equatorial Pacific. The mean of six cruises from January 1984 throught June 1986, a relatively normal period, provides a reference with which the later sections are compared... Changes in the stratification along 165°E were corresponddingly large, reflecting both the geostrophic balance of the strong zonal currents and the changes in the volume of warm water in the Western Equatorial Pacific. The anomaly of warm water volume corresponded closely to the time integral of the warm water transport across 165°E. Local wind forcing and remotely forced waves were both important causes of the transport fluctuations. Winds, precipitation, and currents were all important factors determining the depth of the surface mixed layer and the thickness of the underlying barrier layer. The way in which these factors interact is a strong function of latitude. (D'après résumé d'auteur

Anti-windup Design for Linear Discrete-time Systems Subject to Actuator Additive Faults and Saturations

International audienceIn this paper a method is proposed to design an anti-windup scheme for discrete time linear systems with input saturations and actuator additive failures. This method provides a fault tolerant system reconfiguration mechanism with a control law which compensates for the estimated actuator additive faults and maintains the overall system stability in spite of actuator saturations. The design approach is derived from the solution of linear matrix inequalities (LMI) to guarantee the stability regions. For that purpose the fault tolerant control method is based on a linear quadratic regulator (LQR) and a fault estimator for compensation purposes. This method was tested in realistic simulations with the software Carins (CNES) on a pressure and mass flow rate model of a cryogenic test bench cooling circuit

Cryogenic Liquid Rocket Engine Test Bench Fault-Tolerant Control System: Cooling System Application

International audienceA nonlinear fault-tolerant control strategy relying on quantitative physics-based models for a cryogenic combustion bench operation is proposed in this paper. The aim is to improve the reliability of a cryogenic bench operation in the transients and to allow to converge to a wider range of operating points. The fault detection is performed with residual-based methods. The residual is generated by an unknown input observer with an unscented transformed which also allows to reconstruct the unknown input. Then the goal is to provide a fault-tolerant system reconfiguration mechanism with a control law which compensates for the estimated actuator additive faults to maintain the overall system stability. For that purpose we use a model predictive control method on an equivalent system with the reconstructed unknown input. An error feedback and a fault compensation control law is designed in order to minimize an infinite horizon cost function within the framework of linear matrix inequalities. The model and the estimation part were validated on real data from Mascotte test bench (ONERA/CNES), and the reconfiguration control law was validated in realistic simulations