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Analyse et modélisation de données probabilistes par décomposition de mélange de copules et application à une base de données climatologiques

Abstract : We extend the mixture decomposition of densities methods to the case of data "distribution functions", allowing to classify these functions and to model a probability law for this particular functional data. This law is given by the notion of "distribution of distribution functions" (FDD in French), based on the definition of a distribution function for random variables with values in a probabilistic space. The extensions are realised in associating the FDD to "copulas" functions according to the Sklar's theorem. Copulas joint the multivariate distribution functions (joint) with the univariate distribution functions (margins) for a vector of n random variables. We essentially look at one class of parametric copulas, the Archimedian copulas and we propose three new methods for the estimation of parameters in the multivariate copulas case : with Kendall's rank correlation coefficients, Spearman's coefficient and in maximising the likelihood. The association of the copulas with the FDD, characterises the evolution of the functional data (i.e. the shape of these distribution functions) between different points of the functions inside the classes for each variable, and gives a measure of dependency between the used variables. The methods are first developed for one variable, then several generalisations are proposed for n dimensions. Some theoretical points are discussed, such as the convergence of the algorithm and the fact that the method with copulas is a generalisation of the classical case. An application of the "classification approach" method by copulas is realised on a climate database of the terrestrial atmosphere. The goal is to classify atmospheric "profiles" and to estimate the probability law of these data. The results are compared with those of "classical" methods, showing the performances of our method by mixture decomposition of copulas, and the interest of using probabilistic data.
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https://tel.archives-ouvertes.fr/tel-00002386
Contributor : Mathieu Vrac Connect in order to contact the contributor
Submitted on : Tuesday, February 11, 2003 - 3:18:58 PM
Last modification on : Tuesday, November 10, 2020 - 3:12:09 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 7:50:20 PM

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Mathieu Vrac. Analyse et modélisation de données probabilistes par décomposition de mélange de copules et application à une base de données climatologiques. Interface homme-machine [cs.HC]. Université Paris Dauphine - Paris IX, 2002. Français. ⟨tel-00002386⟩

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