State-space models for maxima precipitation

Naveau, Philippe; Poncet, Paul

Journal de la Société française de statistique & Revue de statistique appliquée, Tome 148 (2007) no. 1, pp. 107-120.

La modélisation des événements climatiques extrêmes est aujourd’hui un champ de recherches particulièrement actif, notamment de par l’importance de leurs impacts économiques et sociaux. Dans cet article nous portons notre attention sur la modélisation statistique des maxima de précipitations, car de telles données sont disponibles aux pas de temps journalier et mensuel sur plusieurs décennies et en de nombreux sites. Notre but est de proposer deux nouveaux modèles à espace d’états dont les fondations probabilistes reposent sur la théorie des valeurs extrêmes (EVT en anglais). Notre premier modèle tire parti des processus max-stables, étudiés entre autres par Davis and Resnick (1989). Il peut être vu comme la transposition du filtre de Kalman linéaire et gaussien à un monde de type Fréchet, où l’addition $a+b$ est remplacée par l’opération maximum $a\vee b=max(a,b)$, et où les bruits sont à queue lourde au lieu d’être gaussiens. Notre second modèle se base sur le modèle de mélange pour les extrêmes proposé par Fougères et al. (2006). Sa flexibilité et sa richesse en termes d’applications en sont un atout essentiel. En plus des interrogations théoriques que suscitent nos modèles, leur principal intérêt est de créer des liens simples et puissants entre la EVT et le domaine de l’assimilation de données. Ce dernier regroupe des techniques statistiques et dynamiques abondamment utilisées dans les études climatiques. Ces procédures nécessitent de combiner d’une part des données issues d’observations et d’autre part les principes dynamiques sous-jacents qui gouvernent le système physique à l’œuvre. C’est pourquoi l’amélioration de notre connaissance des extrêmes et de leur représentation dans le cadre d’un modèle à espace d’états est d’un intérêt tout particulier du point de vue de l’assimilation de données.

A very active research field in atmospheric sciences is centered around the modeling of weather extremes. This is mainly due to the large economic and human impacts of such extreme events. In this paper, we focus on the statistical temporal modeling of precipitation maxima because daily and monthly maxima have been recorded for many decades and at various sites. Our goal is to propose two new state-space models whose distributional foundations lie in Extreme Value Theory (EVT). Our first model takes advantage of max-stable processes, previously studied by Davis and Resnick (1989), among others. It can be viewed as a “translation“ of the gaussian linear Kalman filter into a Fréchet-type world in which the classical addition $a+b$ has been replaced by the max operator $a\vee b=max(a,b)$ and the noise component is from a heavy-tailed distribution instead of being gaussian. Our second state-space model is built from the mixture extremes framework proposed by Fougères et al., (2006). Its strong points are its flexibility and richness with respect to applications. In addition to addressing the theoretical questions brought by our models, the main benefit of introducing them is to propose simple and powerful connections between EVT and data assimilation communities. The latter term “data assimilation” regroups statistical/dynamical techniques extensively used in climate studies. These procedures involve the combination of observational data with the underlying dynamical principles governing the physical system under observation. Hence, improving our knowledge about the representation of extremes in a state-space model framework is of strong interest from a data assimilation point of view.

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