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22 results on '"Escuela de Matematicas [Caracas]"'

1. Invariant density estimation for a reflected diffusion using an Euler scheme

Author

Patrick Cattiaux, José R. León, Clémentine Prieur, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Applied Mathematics, Invariant density, 010102 general mathematics, Mathematical analysis, Estimator, 01 natural sciences, Backward Euler method, Domain (mathematical analysis), discrete approximation for the invariant measure, 010104 statistics & probability, ergodic reflected diffusion process, Euler scheme rate of convergence for large times, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Kernel (statistics), MSC 2010 Primary: 60J60. Secondary: 60H10 Stochastic ordinary differential equations, 37TL40 Invariant measures, 37A25 Ergodicity, mixing, rates of mixing, Euler scheme, 0101 mathematics, Diffusion (business), Mathematics
Abstract

We give an explicit error bound between the invariant density of an elliptic reflected diffusion in a smooth compact domain and the kernel estimator built on the symmetric Euler scheme introduced in [3].

Published
2017
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2. Computational Prediction of Tooth Shades in Direct Restorations with Composite Resins

Author

Zea, Allan, Zea, Aljhadys, Sáenz-Guzmán, Mabel, Zea, Allan, Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Facultad de Odontología, and Facultad de Ingeniería, Universidad Central de Venezuela

Subjects
[SDV.OT]Life Sciences [q-bio]/Other [q-bio.OT], [SDV.MHEP.AHA] Life Sciences [q-bio]/Human health and pathology/Tissues and Organs [q-bio.TO], [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], color matching, color histograms, [SDV.OT] Life Sciences [q-bio]/Other [q-bio.OT], [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], logistic regression, dental composite resins, [SDV.MHEP.AHA]Life Sciences [q-bio]/Human health and pathology/Tissues and Organs [q-bio.TO], restorative dentistry, aesthetics dentistry
Abstract

National audience; The VITA shade guide is a dental instrument that allows accurate and secure determination of tooth color and shade, which constitutes one of the most important but still complex aspects of dental restorations in aesthetics dentistry. In this study, we survey the modern literature in color matching techniques and discuss a logistic regression approach for computational prediction and selection of tooth color in direct restorations using dental composite resins. Test cases were obtained at the school of dentistry of Universidad Central de Venezuela and were further analyzed in a personal computer through the capabilities of the Wolfram language's built-in machine learning package in Mathematica. Altogether, our results exhibit significantly high accuracy rates in front of the classification problem under study, thus emphasizing the impact of this sort of models in current aesthetics dentistry.

Published
2016

3. Estimation for stochastic damping Hamiltonian systems under partial observation. III. Diffusion term

Author

José R. León, Patrick Cattiaux, Clémentine Prieur, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Invariant density, High frequency estimation, Partial observations, 35H10, 01 natural sciences, Hamiltonian system, 010104 statistics & probability, fluctuation-dissipation models, 62M05, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 60F05, Variance estimation, FOS: Mathematics, Ergodic theory, Applied mathematics, 0101 mathematics, Diffusion (business), variance estimation, Mathematics, Central limit theorem, Probability (math.PR), 010102 general mathematics, Estimator, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Term (time), Hypoelliptic diffusion, 60H10, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

This paper is the third part of our study started with Cattiaux, Le\'{o}n and Prieur [Stochastic Process. Appl. 124 (2014) 1236-1260; ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384]. For some ergodic Hamiltonian systems, we obtained a central limit theorem for a nonparametric estimator of the invariant density [Stochastic Process. Appl. 124 (2014) 1236-1260] and of the drift term [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384], under partial observation (only the positions are observed). Here, we obtain similarly a central limit theorem for a nonparametric estimator of the diffusion term., Comment: Published at http://dx.doi.org/10.1214/15-AAP1126 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2016
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4. Recursive Estimation for Stochastic Damping Hamiltonian Systems

Author

Patrick Cattiaux, Clémentine Prieur, José R. León, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), Centre National de la Recherche Scientifique (CNRS), Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Inria Grenoble - Rhône-Alpes, and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Mathematical optimization, Stochastic process, 010102 general mathematics, Mathematical statistics, Nonparametric statistics, Estimator, Variance (accounting), nonparametric estimation, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], 01 natural sciences, hypoelliptic diffusion, partial observations, Term (time), Hamiltonian system, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Ergodic theory, Applied mathematics, 0101 mathematics, Statistics, Probability and Uncertainty, Mathematics, recursive estimation
Abstract

In this paper, we complete our previous works on the nonparametric estimation of the characteristics (invariant density, drift term, variance term) of some ergodic hamiltonian systems, under partial observations. More precisely, we introduce recursive estimators using the full strength of the ergodic behaviour of the underlying process. We compare the theoretical properties of these estimators with the ones of the estimators we previously introduced in Cattiaux, Leon and Prieur [‘Estimation for Stochastic Damping Hamiltonian Systems under Partial Observation. I. Invariant Density’, Stochastic Processes and their Application, 124(3):1236–1260; ‘Estimation for Stochastic Damping Hamiltonian Systems under Partial Observation. II. Drift term’, ALEA Latin American Journal of Probability and Mathematical Statistics, 11, 359–384; ‘Estimation for Stochastic Damping Hamiltonian Systems under Partial Observation. III. Diffusion term,’ http://hal.archives-ouvertes.fr/hal-01044611].

Published
2015
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5. Inference on the Hurst parameter and variance of diffusions driven by fractional Brownian motion

Author

Corinne Berzin, Alain Latour, José R. León, Inférence Processus Stochastiques ( IPS ), Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, Bickel, P., Diggle, P., Fienberg, S.E., Gather, U., Olkin, I., Zeger, S., Jon Gurstelle, Inférence Processus Stochastiques (IPS), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Universidad Central de Venezuela (UCV), Bickel, P., Diggle, Fienberg, S.E., Gather, U., Olkin, I., Zeger, and S.

Subjects
010102 general mathematics, fractional Brownian motion, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], diffusions, variance, 01 natural sciences, [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Hurst parameter, Inference, AMS: primaire: 60F05, 60F17, 60G22, 65C05, Secondaire 60G12, 60G15, 60H10, 62F12, 33C45, 37M05, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract

International audience; The use of diffusion models driven by fractional noise has become popular for more than two decades. The reasons that produced this situation have been varied in nature. We can mention, among others, those that come from mathematics and other from the applications. With respect to the first group, it should be noted that fractional Brownian motion (fBm) has interesting properties. First, it is self-similar. This property implies that such a process is, from the standpoint of its distribution, invariant with respect to scale transformations. Moreover the fractional noise, the process of increments of the fBm taking in a mesh of equally spaced points, satisfies a strong dependence condition that is a notion away from independence and mixing. Using this last property, it has been possible to model natural phenomena, which exhibit temporal correlations tending to zero so slowly that their sum tends to infinity. With regard to the applications, we should mention that fractional models have become popular for modeling real-life events such as the value assets in financial markets, models of chaos in quantum physics, rivers flow along the time, irregular images, weather events and contaminant transport problems, among others. The fBm is a mean zero Gaussian process with stationary increments and whose covariance function is uniquely determined by the Hurst's parameter, that we denote by $H$ and that is between zero and one. The value $H=\frac12$ is important because the associate process results is the Brownian motion (Bm). The parameter $H$ determines the smoothness of the fBm trajectories. More regular are the trajectories as closer to one is the parameter. The exact opposite happening if $H$ is near zero. In the forties and fifties of the twentieth century, in the study of Bm, the introduction of the stochastic integral by Kiyosi Itô and Paul Levy was the key to the definition of diffusion processes. This important event led to the development of a whole area of probability and mathematics. Similarly, the introduction of several definitions of stochastic integrals with respect to fBm, from the nineties, has led to the definition of pseudo-diffusion processes driven by this noise. As in the case of a Bm the introduction of these processes significantly enriched the theory and the horizon of their applications. In these notes we develop estimation techniques for the parameter $ H $ and the local variance (volatility) of the pseudo-diffusion. The estimation of the two parameters is made simultaneously. We will use the observation of the process in a discrete mesh of points. Then the study of the asymptotics will be done when the mesh's norm tends to zero. We start by defining the second order increments of the process. Using these increments, we build the order $ p $ variations. These variations allow the definition of an estimator of the parameter $H$ for all its range. The reason to work with second order increments, instead of the first order increments, is that variations built through them are asymptotically Gaussian in all the range 0 < H

Published
2014
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6. CLT for the zeros of Classical Random Trigonometric Polynomials

Author

José R. León, Jean-Marc Azaïs, Federico Dalmao, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Universidad de la República [Montevideo] (UCUR), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Universidad de la República [Montevideo] (UDELAR)

Subjects
Statistics and Probability, Pure mathematics, Wiener Chaos expansion, Rice Formula, Gaussian, CLT, Of the form, 01 natural sciences, Classical trigonometric polynomials, Random cosines polynomials, 010104 statistics & probability, symbols.namesake, FOS: Mathematics, 0101 mathematics, Mathematics, Central limit theorem, Conjecture, Polynomial chaos, 010102 general mathematics, Probability (math.PR), Wiener Chaos, Variance (accounting), zeros of a random trigonometric polynomial, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Number of zeroes, 60G15, symbols, Statistics, Probability and Uncertainty, Trigonometry, Random variable, Mathematics - Probability
Abstract

We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}\sum_{n=1}^{K} a_n\cos(nt)$, being $(a_n)_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^2K$, $0, Comment: 27 pages. To appear in L'Annales de l'Institut Henri Poincar\'e, Serie B

Published
2014
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7. L^2-deviation for the variance estimator of a fractional diffusion

Author

Berzin , Corinne, Latour , Alain, León , José R., Inférence Processus Stochastiques ( IPS ), Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, Inférence Processus Stochastiques (IPS), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), and Universidad Central de Venezuela (UCV)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], estimation, AMS: primaire: 60F05, 60F17, 60G22, 65C05, Secondaire 60G12, 60G15, 60H10, 62F12, 33C45, 37M05, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], fractional diffusions, Deviation, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], variance, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; We propose punctual estimators for the local variance $\sigma^{2}$ in pseudo-diffusions $X$ driven by a Gaussian noise. We define a kernel estimator of $\sigma^{2}$ based on the quadratic variation of $X$. The consistency and asymptotic normality are shown. Besides, a simulation study is made to assess the performance of those estimators. This study reveals, through various examples, that the estimators approximate very well the true local variance. Then we provide global estimation for the parameter $\sigma^{2}$ in this model that can be considered as a smooth perturbation of a fractional process. Thus we need to study functionals such as the Integrated Square Error (ISE) in order to obtain asymptotics laws to make global hypothesis testing. The Mean Integrated Square Error (MISE) is an usual measure of the precision of a non parametric estimator. A precise asymptotic expansion of the MISE gives optimal bandwith choice. These results with those concerning the ISE lead to contiguity tests for $\sigma$. Estimating the integral of the square of the second derivative of $\sigma^{2}$ also leads to a test of linearity. The proofs are simplified by using the Central Limit Theorem for non-linear functionals that belong to Itô-Wiener's Chaos, of Peccati-Nualart-Tudor.

Published
2013

8. CLT for crossings of random trigonometric polynomials

Author

Jean-Marc Azaïs, José R. León, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Discrete mathematics, Degree (graph theory), 60G15, Gaussian, 010102 general mathematics, Chaos expansion, Interval (mathematics), Covariance, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, symbols.namesake, Crossings of random trigonometric polynomials, symbols, Gravitational singularity, Rice formula, Limit (mathematics), 0101 mathematics, Statistics, Probability and Uncertainty, Trigonometry, Central limit theorem, Mathematics
Abstract

International audience; We establish a central limit theorem for the number of roots of the equation $X_N(t) =u$ when $X_N(t)$ is a Gaussian trigonometric polynomial of degree $N$. The case $u=0$ was studied by Granville and Wigman. We show that for some size of the considered interval, the asymptotic behavior is different depending on whether $u$ vanishes or not. Our mains tools are: a) a chaining argument with the stationary Gaussain process with covariance $\sin(t)/t$, b) the use of Wiener chaos decomposition that explains some singularities that appear in the limit when $u \neq 0$.

Published
2013
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9. Some exact formulas for threshold neuron models: correlations and synchrony

Author

Leon, José R., Di Bernardino, Elena, Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Centre d'études et de recherche en informatique et communications (CEDRIC), Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise (ENSIIE)-Conservatoire National des Arts et Métiers [CNAM] (CNAM), and Di Bernardino, Elena

Subjects
Hermite polynomials, count correlations, [STAT.AP]Statistics [stat]/Applications [stat.AP], correlation coe cient, [STAT.AP] Statistics [stat]/Applications [stat.AP], Quantitative Biology::Neurons and Cognition, Mehler's Formula, synchrony, Number of crossing, spike correlations
Abstract

In this paper we study some threshold neuron models with focus on speci c features as correlations and synchrony input of spikes. These models have received much attention recently in neuroscience research. Our aim is in particular to provide exact formulas for these threshold neuron models in order to investigate more deeply the statistical theory behind these stochastic models. Furthermore we prove a bivariate Central Limit Theorem for a two-neurons model for an in nite horizon time, using a coupled harmonic oscillators. We also present simulated examples which illustrate our theoretical results.

Published
2012

10. Chord-distribution functions and Rice formulae. Application to random media

Author

Estrade , Anne, Iribarren , Ileana, Kratz , Marie, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Statistique Appliquée et MOdélisation Stochastique (SAMOS), Université Paris 1 Panthéon-Sorbonne (UP1), Centre d'économie de la Sorbonne (CES), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), ANR mipomodim ANR-05-BLAN-017, UNIVERSIDAD CENTRAL DE VENEZUELA, Université Panthéon-Sorbonne (UP1), Université Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Statistique Appliquée et MOdélisation Stochastique ( SAMOS ), Université Panthéon-Sorbonne ( UP1 ), Centre d'économie de la Sorbonne ( CES ), and Centre National de la Recherche Scientifique ( CNRS ) -Université Panthéon-Sorbonne ( UP1 )

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Palm probability, chord, porous medium, Rice formula, Gaussian process, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], (up/down) crossing, 60K05, 60D05, 60G10, 60G55, 60G70, level-cut process
Abstract

We consider an isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with values 0 or 1 according to whether a regular stationary process $X$ is less or greater than a given level. The lengths of time intervals when $X$ is in a given phase are named chord functions. We are interested into obtaining information on chord functions in order to get most characterizations of the medium such as the volume fractions, the two-point correlation function or the specific surface area. Working with the Palm probability measure and using level crossings techniques, Rice methods in particular, allow to obtain the exact analytical formula of the chord-distribution function as well as the joint distribution function of two successive chords and the two-point correlation function. Also the ``Independent Interval Assumption'' is discussed.

Published
2012

11. Inference for diffusions driven by fractional Brownian motion

Author

Berzin, Corinne, Latour, Alain, León, José R., Inférence Processus Stochastiques ( IPS ), Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, Inférence Processus Stochastiques (IPS), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), and Universidad Central de Venezuela (UCV)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], estimation, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], drift, fractional Brownian motion, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], diffusions, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], variance, AMS: Primaire: 60F05, 60F17, 60G22, 65C05, Secondaire: 60G12, 60G15, 60H10, 62F12, 33C45, 37M05, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; First we state the almost sure convergence for the $k$-power second order increments of the fractional Brownian motion (fBm). Then we give the rate of that convergence, that is convergence in law for the $k$-power variation and more generally for a $g$ functional variation of the fBm, function $g$ being general but centered, including the absolute $k$-power variation. This allows us to propose several estimators of the Hurst parameter $H$ for the fBm, through the observation of one trajectory on a regular grid of points, using classical linear regression. The first one, $\widehat{H}_{k}$, is built using function $g_{k}(x)=\vert {x} \vert ^{k}- E\vert N \vert^{k}$, and the second one, $\widehat{H}_{\log}$, using function $g_{\log}(x) =\log\vert x\vert-E \log \vert N \vert$, leading to unbiased consistent estimators. A Central Limit Theorem (CLT) is also proposed for both estimators. The estimators $\widehat{H}_{k}$ and $\widehat{H}_{\log}$ are linked. That is, if $k(n)$ is a sequence of positive numbers converging to zero with $n$, and if $\widehat{H}_{k(n)}$ denotes the corresponding estimator of the $H$ parameter, we set out that the asymptotic behavior of $\widehat{H}_{k(n)}$ and of $\widehat{H}_{\log}$ are the same. The technics used before lead us to provide simultaneous estimators of parameter $H$ and of the local variance $\sigma$, in four particular simple models all driven by a fBm. As before, a regression model can be written and least square estimators of $H$ and of $\sigma$ are defined. These estimators are still built on the second order increments of the stochastic process solution of the stochastic differential equation. We still enunciate their consistency and a CLT is given for both of them. Furthermore, we consider testing the hypothesis $\sigma_{n}=\sigma$ against an alternative in the four previous models. An evaluation of the asymptotic power of the test is made. Finally simulations of the previous results are proposed.

Published
2011

12. Rice formulas and Gaussian waves II

Author

Azaïs, Jean-Marc, Leon, José R., Wschebor, Mario, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Centro de Matemática [Montevideo] (CMAT), Universidad de la República [Montevideo] (UCUR), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Universidad de la República [Montevideo] (UDELAR)

Subjects
60G15 (60G60, 78A10, 78A97, 86A05), ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2011

13. Convergence of the increments of a stochastic integral associated to the stochastic wave equation

Author

Corinne Berzin, Mairene Colina, Universidad Central de Venezuela (UCV), Statistique et Modélisation Stochatisque (SMS), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, Statistique et Modélisation Stochatisque ( SMS ), Laboratoire Jean Kuntzmann ( LJK ), and Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA )

Subjects
Statistics and Probability, Almost sure convergence of increments, Stochastic process, Plane (geometry), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Brownian sheet, Stochastic wave equation, Type (model theory), Wave equation, 01 natural sciences, Stochastic integral, Stochastic integral on the plane, Primary 60H15, Secondary 60F99, 60G44, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Stochastic differential equation, Convergence of random variables, Modelling and Simulation, Modeling and Simulation, Convergence (routing), 0101 mathematics, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics
Abstract

International audience; We study a type of one dimensional wave equation on the plane with non linear random forcing. We are interested in the almost sure behavior of the normalized increments of the solution process associated to this type of wave equation. Also we study the behavior of the normalized increments of some other stochastic integral equation.

Published
2009
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14. Level curves, crossings and specular points for Gaussian models

Author

Kratz, Marie, Leon, José R., Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Centre d'économie de la Sorbonne (CES), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), Statistique Appliquée et MOdélisation Stochastique (SAMOS), Université Paris 1 Panthéon-Sorbonne (UP1), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Centre d'économie de la Sorbonne ( CES ), Université Panthéon-Sorbonne ( UP1 ) -Centre National de la Recherche Scientifique ( CNRS ), Statistique Appliquée et MOdélisation Stochastique ( SAMOS ), Université Panthéon-Sorbonne ( UP1 ), UNIVERSIDAD CENTRAL DE VENEZUELA, Centre National de la Recherche Scientifique (CNRS)-Université Panthéon-Sorbonne (UP1), and Université Panthéon-Sorbonne (UP1)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Hermite polynomials, specular point, CLT, (generalized) Rice formula, AMS 2000: Primary 60G15, Secondary 60G10, 60G60, 60G70, Thirdly 78A97, level curve, co-area formula, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Gaussian fields, crossings
Abstract

preprint; 25 pages; Using generalizations of the well known Rice formula and applying the general method related to m-dependent processes that we settled in earlier works, allow one to obtain representations into the Ito-Wiener Chaos and CLTs for curve-crossings number. This approach not only explains heuristic considerations of Longuet-Higgins on specular points and related problems in the context of sea modelling, but goes far beyond when providing asymptotic results. These results on curve-crossings may also be applied in other fields. One example is the study of the estimator of the natural frequency of a harmonic oscillator.

Published
2007

15. On the second moment of the number of crossings by a stationary Gaussian process

Author

José R. León, Marie Kratz, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Statistique Appliquée et MOdélisation Stochastique (SAMOS), Université Paris 1 Panthéon-Sorbonne (UP1), Centre d'économie de la Sorbonne (CES), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Statistique Appliquée et MOdélisation Stochastique ( SAMOS ), Université Panthéon-Sorbonne ( UP1 ), Centre d'économie de la Sorbonne ( CES ), Université Panthéon-Sorbonne ( UP1 ) -Centre National de la Recherche Scientifique ( CNRS ), UNIVERSIDAD CENTRAL DE VENEZUELA, Université Panthéon-Sorbonne (UP1), and Centre National de la Recherche Scientifique (CNRS)-Université Panthéon-Sorbonne (UP1)

Subjects
Statistics and Probability, 60G70, Covariance function, Generalization, Gaussian processes, Second moment of area, AMS Primary 60G15, Secondary 60G10, 60G70, Finite variance, 01 natural sciences, 010104 statistics & probability, symbols.namesake, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], spectral moment, FOS: Mathematics, level curve, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], Differentiable function, 0101 mathematics, Gaussian process, Mathematics, Discrete mathematics, Hermite polynomials, Probability (math.PR), 010102 general mathematics, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], 16. Peace & justice, [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], crossings, Geman condition, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60G15, symbols, 60G15 (Primary) 60G10, 60G70 (Secondary), Statistics, Probability and Uncertainty, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], 60G10, Mathematics - Probability
Abstract

Cram\'{e}r and Leadbetter introduced in 1967 the sufficient condition \[\frac{r''(s)-r''(0)}{s}\in L^1([0,\delta],dx),\qquad \delta>0,\] to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function $r$. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition., Comment: Published at http://dx.doi.org/10.1214/009117906000000142 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2006
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16. Piecewise Fractional Brownian Motion

Author

Rachid Harba, E. Perrin, I. Iribarren, Rachid Jennane, Département Images, Robotique, Automatique et Signal [Orléans] (IRAUS), Laboratoire pluridisciplinaire de recherche en ingénierie des systèmes, mécanique et énergétique (PRISME), Université d'Orléans (UO)-Institut National des Sciences Appliquées - Centre Val de Loire (INSA CVL), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université d'Orléans (UO)-Institut National des Sciences Appliquées - Centre Val de Loire (INSA CVL), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), Laboratoire Creatis, Compte Général, Centre de Recherche et d'Application en Traitement de l'Image et du Signal (CREATIS), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École Supérieure Chimie Physique Électronique de Lyon-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), and Jennane, Rachid

Subjects
[SPI.OPTI] Engineering Sciences [physics]/Optics / Photonic, [SDV.IB.IMA]Life Sciences [q-bio]/Bioengineering/Imaging, Gaussian, [INFO.INFO-IM] Computer Science [cs]/Medical Imaging, fractional Brownian motion, 02 engineering and technology, Omega, Nonlinear Sciences::Adaptation and Self-Organizing Systems, Mathematics::Probability, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, 0202 electrical engineering, electronic engineering, information engineering, Calculus, Brownian motion, Mathematics, [SDV.MHEP] Life Sciences [q-bio]/Human health and pathology, self-similarity, Mathematical analysis, [PHYS.MECA.ACOU]Physics [physics]/Mechanics [physics]/Acoustics [physics.class-ph], [SPI.ELEC]Engineering Sciences [physics]/Electromagnetism, [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [PHYS.PHYS.PHYS-MED-PH]Physics [physics]/Physics [physics]/Medical Physics [physics.med-ph], symbols, Piecewise, [SDV.IB]Life Sciences [q-bio]/Bioengineering, 020201 artificial intelligence & image processing, [PHYS.MECA.ACOU] Physics [physics]/Mechanics [physics]/Acoustics [physics.class-ph], Fractal, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, symbols.namesake, [INFO.INFO-IM]Computer Science [cs]/Medical Imaging, Electrical and Electronic Engineering, Gaussian process, [SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing, [SPI.ACOU]Engineering Sciences [physics]/Acoustics [physics.class-ph], [SDV.IB] Life Sciences [q-bio]/Bioengineering, [SPI.ACOU] Engineering Sciences [physics]/Acoustics [physics.class-ph], [PHYS.PHYS.PHYS-MED-PH] Physics [physics]/Physics [physics]/Medical Physics [physics.med-ph], Fractional Brownian motion, Stochastic process, [SPI.ELEC] Engineering Sciences [physics]/Electromagnetism, ACL, Computer Science::Software Engineering, 020206 networking & telecommunications, [SDV.IB.IMA] Life Sciences [q-bio]/Bioengineering/Imaging, Gaussian noise, Signal Processing, [SPI.OPTI]Engineering Sciences [physics]/Optics / Photonic, [SDV.MHEP]Life Sciences [q-bio]/Human health and pathology
Abstract

Starting from fractional Brownian motion (fBm) of unique parameter H, a piecewise fractional Brownian motion (pfBm) of parameters H/sub o/, H/sub i/, and /spl gamma/ is defined. This new process has two spectral regimes: It behaves like an fBm of parameter H/sub o/ for low frequencies |/spl omega/

Published
2005

17. Convergence in fractional models

Author

Berzin , Corinne, León , José R., Laboratoire de Statistiques et Analyse des Données (LABSAD), Université Pierre Mendès France - Grenoble 2 (UPMF), Universidad Central de Venezuela (UCV), Project 'Modelaje Estocástico Aplicado' of the Agenda Petróleo of FONACIT Venezuela, Laboratoire de Statistiques et Analyse des Données ( LABSAD ), Université Pierre Mendès France - Grenoble 2 ( UPMF ), Escuela de Matematicas [Caracas], and UNIVERSIDAD CENTRAL DE VENEZUELA

Subjects
Level crossings, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], local time, fractional Brownian motion, limit theorem, Primary 60F05, Secondary 60G15, 60G18, 60H10, 62F03, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], rate of convergence
Abstract

http://www.math.washington.edu/~ejpecp/; International audience; We consider a fractional Brownian motion with Hurst parameter strictly between 0 and 1. We are interested in the asymptotic behaviour of functionals of the increments of this and related processes and we propose several probabilistic and statistical applications.

Published
2005
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18. Fast and Exact Synthesis for 1-D Fractional Brownian Motion and Fractional Gaussian Noises

Author

R. Harba, Rachid Jennane, E. Perrin, I. Iribarren, Département Images, Robotique, Automatique et Signal [Orléans] (IRAUS), Laboratoire pluridisciplinaire de recherche en ingénierie des systèmes, mécanique et énergétique (PRISME), Université d'Orléans (UO)-Institut National des Sciences Appliquées - Centre Val de Loire (INSA CVL), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université d'Orléans (UO)-Institut National des Sciences Appliquées - Centre Val de Loire (INSA CVL), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), Escuela de Matematicas [Caracas], Universidad Central de Venezuela (UCV), and Jennane, Rachid

Subjects
synthesis, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, Gaussian, fractional Brownian motion, 02 engineering and technology, 01 natural sciences, Gaussian random field, Combinatorics, Fast algorithms, 010104 statistics & probability, symbols.namesake, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, 0202 electrical engineering, electronic engineering, information engineering, 0101 mathematics, Electrical and Electronic Engineering, Gaussian process, Circulant matrix, Brownian motion, [SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing, Mathematics, Fractional Brownian motion, Covariance matrix, Applied Mathematics, ACL, Mathematical analysis, 020206 networking & telecommunications, Gaussian noise, Signal Processing, symbols, fractional Gaussian noises, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing
Abstract

In this letter, it is shown that fast and exact fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) signals can be synthesized by the circulant embedding method (CEM). CEM consists in embedding the N/spl times/N covariance matrix of the stationary fGn process in a larger 2M/spl times/2M circulant matrix such that M /spl ges/N-1. CEM is exact, since second-order statistics of the generated data are those of the Gaussian fGn. CEM is fast, since the optimal case M=N-1 can be reached. Fast and exact fBm sequences can be easily recovered from fGn ones.

Published
2002

19. nth-order fractional Brownian motion and fractional Gaussian noises

Author

Aline Bonami, C. Berzin-Joseph, E. Perrin, Rachid Harba, I. Iribarren, Laboratoire d'électronique, signaux, images (LESI), Université d'Orléans (UO), Laboratoire de Statistiques et Analyse des Données (LABSAD), Université Pierre Mendès France - Grenoble 2 (UPMF), Universidad Central de Venezuela (UCV), Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Laboratory of Electronics, Signals and Images ( LESI ), Université d'Orléans ( UO ), Laboratoire d'Electronique, Signaux, Images, Laboratoire de Statistiques et Analyse des Données ( LABSAD ), Université Pierre Mendès France - Grenoble 2 ( UPMF ), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, Mathématiques - Analyse, Probabilités, Modélisation - Orléans ( MAPMO ), and Université d'Orléans ( UO ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Mathematical optimization, Covariance function, Gaussian, fractional Brownian motion, 02 engineering and technology, Upper and lower bounds, symbols.namesake, signal synthesis, Mathematics::Probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0202 electrical engineering, electronic engineering, information engineering, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], Electrical and Electronic Engineering, Brownian motion, Mathematics, 1/f processes, Fractional Brownian motion, self-similarity, Estimation theory, Mathematical analysis, Estimator, 020206 networking & telecommunications, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], Gaussian noise, Signal Processing, symbols, 020201 artificial intelligence & image processing, fractional Gaussian noises, parameter estimation
Abstract

A generalization of fractional Brownian motion (fBm) of parameter H in ]0, 1[ is proposed. More precisely, this work leads to nth-order fBm (n-fBm) of H parameter in ]n-1, n[, where n is any strictly positive integer. They include fBm for the special case n=1. Properties of these new processes are investigated. Their covariance function are given, and it is shown that they are self similar. In addition, their spectral shape is assessed as 1/f/sup /spl alpha// with /spl alpha/ belonging to ]1; +/spl infin/[, providing a larger framework than classical fBm. Special interest is given to their nth-order stationary increments, which extend fractional Gaussian noises. The covariance function and power spectral densities are calculated. The properties and signal processing tasks such as a Cholesky-type synthesis technique and a maximum likelihood estimation method of the H parameter are presented. The results show that the estimator is efficient (unbiased and reaches the Cramer-Rao lower bound) for a large majority of tested values.

Published
2001
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20. Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes

Author

JoséR. León, Marie Kratz, Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), and Universidad Central de Venezuela (UCV)

Subjects
Statistics and Probability, Hermite spline, Gaussian processes, extremes, 01 natural sciences, Hermite polynomial expansion, Matrix polynomial, Cubic Hermite spline, 010104 statistics & probability, symbols.namesake, Hermite interpolation, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Modelling and Simulation, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, Gaussian process, Mathematics, Hermite polynomials, Polynomial chaos, Applied Mathematics, 010102 general mathematics, Mathematical analysis, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], crossings, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, symbols, Maxima, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract

We propose a new method to get the Hermite polynomial expansion of crossings of any level by a stationary Gaussian process, as well as the one of the number of maxima in an interval, under some assumptions on the spectral moments of the process.

Published
1997
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21. Estimación de la matriz de los momentos espectrales de orden 2 de un campo aleatorio Gaussiano a partir de las curvas de nivel

Author

Berzin , Corinne, Iribarren , Ileana, Laboratoire de probabilités et statistiques d'Orsay, Université Paris-Sud - Paris 11 (UP11), Departamento de Matematica (IVIC), Instituto Venezolano de Investigaciones Cientificas (IVIC), Enrique Cabaña, Universidad de la República, Montevideo, Uruguay, Équipe de Probabilités et Statistiques ( ÉQUIPE DE PROBABILITéS ET STATISTIQUES ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), Escuela de Matematicas [Caracas], UNIVERSIDAD CENTRAL DE VENEZUELA, and Enrique Cabaña, Universidad de la República, Montevideo, Uruguay

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract

International audience; Extendiendo la idea del caso uniparamétrico de definir estimadores del segundo momento espectral a partir del número de cruces del proceso con una barrera, se definen estimadores de la matriz de los momentos espectrales de segundo orden basados en funcionales definidos sobre los conjuntos de nivel del campo. Se obtienen propiedades asintóticas y se da una expresión para la varianza asintótica.

Published
1988

22. Estimation de la matrice des moments spectraux d'ordre 2 d'un champ aléatoire gaussien à l'aide des courbes de niveau

Author

Berzin, Corinne, Iribarren, Ileana, Laboratoire de probabilités et statistiques d'Orsay, Université Paris-Sud - Paris 11 (UP11), Departamento de Matematica (IVIC), Instituto Venezolano de Investigaciones Cientificas (IVIC), Équipe de Probabilités et Statistiques ( ÉQUIPE DE PROBABILITéS ET STATISTIQUES ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), Escuela de Matematicas [Caracas], and UNIVERSIDAD CENTRAL DE VENEZUELA

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract

International audience; On propose des estimateurs pour la matrice des moments spectraux du second ordre d'un champ aléatoire Gaussien bi-indexé stationnaire et centré. Les estimateurs sont définis à l'aide de fonctionnelles sur les courbes de niveau du champ. On étudie ses propriétés statistiques et on donne une expression de la matrice de covariances asymptotique.

Published
1988

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