Statistical Modeling of Spatial Extremes

Abstract

The areal modeling of the extremes of a natural process such as rainfall or temperature is important in environmental statistics; for example, understanding extreme areal rainfall is crucial in flood protection. This article reviews recent progress in the statistical modeling of spatial extremes, starting with sketches of the necessary elements of extreme value statistics and geostatistics. The main types of statistical models thus far proposed, based on latent variables, on copulas and on spatial max-stable processes, are described and then are compared by application to a data set on rainfall in Switzerland. Whereas latent variable modeling allows a better fit to marginal distributions, it fits the joint distributions of extremes poorly, so appropriately-chosen copula or max-stable models seem essential for successful spatial modeling of extremes.Comment: Published in at http://dx.doi.org/10.1214/11-STS376 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A dependence modelling study of extreme rainfall in Madeira Island

“…A review of spatial extremes methods based on latent variables, copulas and spatial max-stable processes was presented by Davison et al (2012), who refer that appropriately chosen copula or maxstable models seem to be essential for the modelling of spatial extremes. The ability to describe and model the dependence between variables, regardless of their marginal distribution functions, is the major advantage of the copula functions approach.…”

Modeling extreme rainfall A comparative study of spatial extreme value models

“…Recent studies exploiting hierarchical modeling for extreme precipitation are for instance Apputhurai and Stephenson (2013) and Dyrrdal et al (2015). Davison et al (2012) introduced a simple hierarchical structure for spatial extremes, called the latent variable model (LVM). Note that the usual lack of clear spatial pattern for the shape ξ(·) when dealing with precipitation data, jointly with the difficulty of estimating this parameter lead in this paper to consider ξ(·) ≡ ξ 0 .…”

“…The covariates we use in this paper are the constant, the longitude, latitude and altitude. Note that Davison et al (2012) consider the exponential form ρ ε· (h) = δ · exp(−h/λ · ) to model the correlation of the latent Gaussian processes. We follow this choice to be consistent with the choices made in Sections 2.2.1 and 2.2.3 when defining the EGP and ETP.…”

“…From Davison et al (2012), one knows that the LVM is particularly appealing when the estimation of the marginal distributions is of interest, as it focuses on modeling the GEV parameters. The main drawback is that the dependence structure of extremes is not considered since conditional independence is assumed.…”

“…Since precipitation phenomena have a spatial feature and data are generally observed at several stations, the dedicated setting to handle this question is that of (conditionally) max-stable spatial models. Detailed and helpful reviews on these models are Cooley et al (2012), Davison et al (2012) or .…”