Geostatistics of Dependent and Asymptotically Independent Extremes
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Geostatistics of Dependent and Asymptotically Independent Extremes A.C. Davison · R. Huser · E. Thibaud
Received: 1 February 2013 / Accepted: 18 May 2013 / Published online: 18 June 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract Spatial modeling of rare events has obvious applications in the environmental sciences and is crucial when assessing the effects of catastrophic events (such as heatwaves or widespread flooding) on food security and on the sustainability of societal infrastructure. Although classical geostatistics is largely based on Gaussian processes and distributions, these are not appropriate for extremes, for which maxstable and related processes provide more suitable models. This paper provides a brief overview of current work on the statistics of spatial extremes, with an emphasis on the consequences of the assumption of max-stability. Applications to winter minimum temperatures and daily rainfall are described. Keywords Asymptotic independence · Brown–Resnick process · Gaussian process · Generalised Pareto distribution · Max-stable process · Statistics of extremes 1 Introduction In recent years, there has been a major upsurge of research activity in the statistics of extreme events for spatial settings. One reason for this is the realisation among stakeholders (such as climate scientists, environmental engineers, and insurance companies) that in an evolving climate it may be changes in the sizes and frequencies of rare events, rather than changes in the averages, that lead to the most devastating losses of life, damage to infrastructure, and so forth. While it is difficult or even impossible to attribute particular events to the effect of climatic change, the types of events that have long been forecast to increase in frequency by the modeling community— such as heatwaves leading to crop failure and major brush fires, or heavy summer A.C. Davison () · R. Huser · E. Thibaud Ecole Polytechnique Fédérale de Lausanne, EPFL-FSB-MATHAA-STAT, Station 8, 1015 Lausanne, Switzerland e-mail: [email protected] url: http://stat.epfl.ch
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Math Geosci (2013) 45:511–529
rainfall leading to widespread flooding—do indeed seem to be appearing more often than in the recorded past. This motivates attempts to model such events, in order to understand their likely future impacts, and to assess the related risks. Classical geostatistics is a well-developed field surveyed in numerous textbooks (Cressie 1993; Wackernagel 2003; Banerjee et al. 2004; Diggle and Ribeiro 2007; Cressie and Wikle 2011), with much software available and a wide range of user communities corresponding to its many applications. Its basis in Gaussian distributions makes it unsuitable for extremal modeling, however, because the Gaussian density function has an exceptionally light tail and, therefore, can badly underestimate probabilities associated to extreme events. Moreover, the tails of the multivariate Gaussian distribution lead to independent extremes, for any underlying correlation t
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