Estimation of multivariate critical layers: Applications to rainfall data

Di Bernardino, Elena; Rullière, Didier

Journal de la société française de statistique, Tome 156 (2015) no. 1, pp. 11-50.

Dans un environnement multivarié, le calcul de zones critiques et de périodes de retour associées est un problème difficile. Un cadre théorique possible pour le calcul de ces périodes de retour est essentiellement basé sur la notion de Copule et sur les ensembles de niveau d’une distribution de probabilité multivariée. Dans ce travail, nous proposons une méthodologie rapide et paramétrique pour estimer les zones critiques de distributions multivariées et leurs périodes de retour associées. Le modèle est basé sur des transformations des distributions marginales et sur des transformations de la structure de dépendance au sein de la classe des copules Archimédiennes. La méthodologie est illustrée sur des données réelles de précipitation. Sur ce jeu de données, nous développons également un modèle imbriqué transformé.

Calculating return periods and critical layers (i.e. multivariate quantile curves) in a multivariate environment is a difficult problem. A possible consistent theoretical framework for the calculation of the return period, in a multi-dimensional environment, is essentially based on the notion of copula and level sets of the multivariate probability distribution. In this paper we propose a fast and parametric methodology to estimate the multivariate critical layers of a distribution and its associated return periods. The model is based on transformations of the marginal distributions and transformations of the dependence structure within the class of Archimedean copulas. The model has a tunable number of parameters, and we show that it is possible to get a competitive estimation without any global optimum research. We also get parametric expressions for the critical layers and return periods. The methodology is illustrated on rainfall 5-dimensional real data. On this real data-set we obtain a good quality of estimation and we compare the obtained results with some classical parametric competitors. Finally we provide a simulation study.

Mots clés : Multivariate probability transformations, level sets, estimation copulas, hyperbolic conversion functions, risk assessment, multivariate return periods

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