An extended Gaussian max-stable process model for spatial extremes

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Abstract

The extremes of environmental processes are often of interest due to the damage that can be caused by extreme levels of the processes. These processes are often spatial in nature and modelling the extremes jointly at many locations can be important. In this paper, an extension of the Gaussian max-stable process is developed, enabling data from a number of locations to be modelled under a more flexible framework than in previous applications. The model is applied to annual maximum rainfall data from five sites in South-West England. For estimation we employ a pairwise likelihood within a Bayesian analysis, incorporating informative prior information.

Introduction

Multivariate extreme value theory provides a way to quantify the joint extreme behaviour of several variables. Many environmental processes are spatial in nature and modelling multivariate data can be of great importance. Methods for modelling such data, based on a multivariate generalisation of the classical approach to univariate extremes, were proposed by Tawn, 1988, Tawn, 1990 and Smith et al. (1990). Multivariate extensions of the threshold exceedance approach were developed by Coles and Tawn (1991) and Joe et al. (1991).

Smith (1990) suggests a procedure using the theory of max-stable processes for modelling data which are collected on a grid of points in space. This approach can be considered as an infinite dimensional extension of multivariate extreme value theory. The extension has the advantage that it can be used to consider problems concerning aggregation of the process over the whole region, and interpolation to anywhere within the region. Another advantage of this approach is that models based on the resulting family of multivariate extreme value distributions are workable even for a large number of grid points. Other applications of max-stable processes are given in Coles (1993), Coles and Walshaw(1994) and Coles and Tawn (1996a). Coles (1993) gives a class of max-stable process models which can utilise all data exceeding pre-defined thresholds. Max-stable processes are used by Coles and Walshaw(1994) to model the directional dependence of extreme wind speeds, and by Coles and Tawn (1996a) to develop a model for spatially aggregated rainfall extremes.

Bayesian techniques can be used to incorporate information other than the data into the model in the form of prior distributions. These techniques have the potential to be very useful in improving estimation in extreme value problems, since extreme data are naturally scarce. Applications of Bayesian techniques to univariate problems have been considered by Coles and Powell (1996) and Coles and Tawn (1996b), and applications to multivariate problems have been considered by Smith and Walshaw (2003) and Smith (2005).

The Gaussian max-stable process, introduced by Smith (1990), suffers from a lack of flexibility, since the Gaussian storm profile that generates the process has a constant variance matrix across all points in the space of interest. In this paper the model is extended to a more flexible framework and is applied to annual maximum rainfall data from five locations in South-West England. For estimation, Smith (1990) suggests a collection of somewhat ad hoc techniques, due to the difficulty arising from the fact that the joint distribution of the process at more than two sites is unknown. The approach we take here is to estimate our extended model using a pairwise likelihood within a Bayesian analysis. The pairwise likelihood, which is the product of the likelihoods for all pairs of sites, comes from a general class of composite likelihoods introduced by Lindsay (1988). Hjort (1993) was the first to consider using a product of bivariate likelihoods. Hjort (1993) called the approach quasi-likelihood but since then the name pairwise likelihood has been adopted. In the Bayesian analysis we use informative priors for the marginal site parameters, based on expert prior information given in Coles and Tawn (1996a) . Prior distributions for parameters relating to the dependence structure of the process are specified to be non-informative.

Section snippets

Extreme value theory

Let Y1,Y2, be an independent and identically distributed (IID) sequence of random variables. Classical univariate extreme value theory is concerned with the limiting behaviour of Mn=max{Y1,,Yn} as n, after a linear normalisation of Mn. The key result of univariate extreme value theory states that the limiting distribution of Mn is the generalised extreme value (GEV) distribution:G(z)=exp-1+ξz-μσ-1/ξ,for {z:1+ξ(z-μ)/σ>0} and with -<μ<,σ>0 and -<ξ<.

For multivariate extremes, we focus on

Theory

A stochastic process {Yt,tT}, where T is an arbitrary index set, is a max-stable process (de Haan, 1984) if the following property holds:

if Yt(1),,Yt(n) are n independent copies of the process, then there exists constants ant>0 and bnt such that {Yt*,tT} is identical in law to {Yt,tT}, where Yt*=maxi=1,,nYt(i)-bntant,tT.

This definition has the consequence that the distribution of the vector (Yt1,,Ytd) for any points t1,,td in the index set T is of multivariate extreme value form, as

Overview

The data considered here are annual maximum rainfall data from five sites in South-West England, recorded over the period 1914–1988. A satellite image of the site region is given in Fig. 1. We use longitudes and latitudes to specify the site locations t1,,t5. These data have also been considered by Coles (1993), Coles and Tawn (1996a), Coles and Tawn (1996b), Smith and Walshaw (2003) and Smith (2005). We jointly model the annual maximum rainfall at these sites using the extended Gaussian

Results

Markov chain Monte Carlo (MCMC) techniques are used to simulate from a Markov chain whose equilibrium distribution is the posterior distribution of (m,Σ-1,w,λ). The MCMC scheme used is a Gibbs sampler with a Metropolis step for each parameter.

The MCMC sampler was run for 200 000 iterations and convergence was obtained after about 10 000 iterations. The estimates in this section are based on the remaining 190 000 iterations. Various convergence diagnostics, such as those described by Heidelberger

Discussion

In this paper an extension to the Gaussian max-stable process model was given. This model allowed for greater flexibility in the shape of the storm profile by allowing the covariance structure to vary across different locations in the space of interest. The model was applied to annual maximum rainfall data from five locations in South-West England by assuming a pairwise likelihood and using a Bayesian approach. Under the pairwise likelihood methodology the model could easily be applied to more

References (24)

  • J. Hüsler et al.

    Maxima of normal random vectors: between independence and complete dependence

    Statist. Probab. Lett.

    (1989)
  • S.G. Coles

    Regional modelling of extreme storms via max-stable processes

    J. Roy. Statist. Soc. B

    (1993)
  • S.G. Coles et al.

    Bayesian methods in extreme value modelling: a review and new developments

    Internat. Statist. Rev.

    (1996)
  • S.G. Coles et al.

    Modelling multivariate extreme events

    J. Roy. Statist. Soc. B

    (1991)
  • S.G. Coles et al.

    Modelling extremes of the areal rainfall process

    J. Roy. Statist. Soc. B

    (1996)
  • S.G. Coles et al.

    A Bayesian analysis of extreme rainfall data

    Appl. Statist.

    (1996)
  • S.G. Coles et al.

    Directional modelling of extreme wind speeds

    Appl. Statist.

    (1994)
  • L. de Haan

    A spectral representation for max-stable processes

    Ann. Probab.

    (1984)
  • P.J. Diggle et al.

    Analysis of Longitudinal Data

    (1994)
  • Geweke, J., 1992. Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In: Bernardo,...
  • P. Heidelberger et al.

    Simulation run length control in the presence of an initial transient

    Oper. Res.

    (1983)
  • Hjort, N.L., 1993. A quasi-likelihood method for estimating parameters in spatial covariance functions. Technical...
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