[Identification de variables aléatoires multi-modales par mélange de décompositions sur la chaos polynômial]
Une méthodologie est proposée pour l'identification d'une variable aléatoire multi-modale à partir d'échantillons. La variable aléatoire est vue comme un mélange fini de variables aléatoires uni-modales. Une représentation fonctionnelle de la variable aléatoire est utilisée. Elle peut être interprétée comme un mélange de décompositions sur le chaos polynômial. Après une séparation adaptée des échantillons en sous-ensembles d'échantillons uni-modaux, les coefficients de la décomposition sont identifiés en utilisant une technique de projection empirique. Cette procédure d'identification permet une représentation générique d'une large classe de variables aléatoires multi-modales avec une décomposition sur chaos polynômial généralisé de faible degré.
A methodology is introduced for the identification of a multi-modal real-valued random variable from a collection of samples. The random variable is seen as a finite mixture of uni-modal random variables. A functional representation of the random variable is used, which can be interpreted as a mixture of polynomial chaos expansions. After a suitable separation of samples into sets of uni-modal samples, the coefficients of the expansion are identified by using an empirical projection technique. This identification procedure allows for a generic representation of a large class of multi-modal random variables with low-order generalized polynomial chaos representations.
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Mots-clés : Statistique, Quantification d'incertitudes, Identification, Densité multi-modale, Chaos Polynômial, Modèle de mélange fini, Méthodes spectrales stochastiques
Anthony Nouy 1
@article{CRMECA_2010__338_12_698_0,
author = {Anthony Nouy},
title = {Identification of multi-modal random variables through mixtures of polynomial chaos expansions},
journal = {Comptes Rendus. M\'ecanique},
pages = {698--703},
publisher = {Elsevier},
volume = {338},
number = {12},
year = {2010},
doi = {10.1016/j.crme.2010.09.003},
language = {en},
}
Anthony Nouy. Identification of multi-modal random variables through mixtures of polynomial chaos expansions. Comptes Rendus. Mécanique, Volume 338 (2010) no. 12, pp. 698-703. doi : 10.1016/j.crme.2010.09.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.09.003/
[1] Stochastic Finite Elements: A Spectral Approach, Springer, Berlin, 1991
[2] Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Archives of Computational Methods in Engineering, Volume 16 (2009) no. 3, pp. 251-285
[3] Fast numerical methods for stochastic computations: A review, Communications in Computational Physics, Volume 5 (2009), pp. 242-272
[4] Stochastic finite elements: Computational approaches to stochastic partial differential equations, Zamm-Zeitschrift fur Angewandte Mathematik und Mechanik, Volume 88 (2008) no. 11, pp. 849-873
[5] The homogeneous chaos, Am. J. Math., Volume 60 (1938), pp. 897-936
[6] The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, The Annals of Mathematics, Second Series, Volume 48 (1947) no. 2, pp. 385-392
[7] Solution of stochastic partial differential equations using Galerkin finite element techniques, Computer Methods in Applied Mechanics and Engineering, Volume 190 (2001), pp. 6359-6372
[8] Uncertainty propagation using Wiener–Haar expansions, Journal of Computational Physics, Volume 197 (2004) no. 1, pp. 28-57
[9] Multi-element generalized polynomial chaos for arbitrary propability measures, SIAM Journal on Scientific Computing, Volume 28 (2006) no. 3, pp. 901-928
[10] Physical systems with random uncertainties: Chaos representations with arbitrary probability measure, SIAM Journal on Scientific Computing, Volume 26 (2004) no. 2, pp. 395-410
[11] Maximum likelihood estimation of stochastic chaos representations from experimental data, International Journal for Numerical Methods in Engineering, Volume 66 (2006) no. 6, pp. 978-1001
[12] Identification of random shapes from images through polynomial chaos expansion of random level-set functions, International Journal for Numerical Methods in Engineering, Volume 79 (2009) no. 2, pp. 127-155
[13] On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data, Journal of Computational Physics, Volume 217 (2006) no. 1, pp. 63-81
[14] Identification of bayesian posteriors for coefficients of chaos expansions, Journal of Computational Physics, Volume 229 (2010) no. 9, pp. 3134-3154
[15] A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering, Volume 196 (2007) no. 45-48, pp. 4521-4537
[16] Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms, Computer Methods in Applied Mechanics and Engineering, Volume 197 (2008), pp. 4718-4736
[17] A. Nouy. Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Archives of Computational Methods in Engineering (2010), , in press. | DOI
[18] Finite Mixture Models, Wiley, New York, 2000
[19] The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM Journal on Scientific Computing, Volume 24 (2002) no. 2, pp. 619-644
[20] The Bayesian Choice, Springer Verlag, New York, 1994
[21] Bayesian Data, Analysis, Chapman and Hall, Boca Raton, FL, 2003
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