Conditional simulation of max-stable processes

Since many environmental processes are spatial in extent, a single extreme event may affect several locations, and the spatial dependence must be taken into account in an appropriate way. This paper proposes a framework for conditional simulation of max-stable processes and gives closed forms for the regular conditional distributions of Brown--Resnick and Schlather processes. We test the method on simulated data and present applications to extreme rainfall around Zurich and extreme temperatures in Switzerland. The proposed framework provides accurate conditional simulations and can handle problems of realistic size. Copyright 2013, Oxford University Press.

[1]  Anthony C. Davison,et al.  Statistical Modelling of Spatial Extremes , 2012 .

[2]  L. Haan,et al.  Extreme value theory : an introduction , 2006 .

[3]  Laurens de Haan,et al.  Stationary max-stable fields associated to negative definite functions. , 2008, 0806.2780.

[4]  Jun S. Liu,et al.  Covariance Structure and Convergence Rate of the Gibbs Sampler with Various Scans , 1995 .

[5]  P. Naveau,et al.  Variograms for spatial max-stable random fields , 2006 .

[6]  L. de Haan,et al.  A Spectral Representation for Max-stable Processes , 1984 .

[7]  M. Penrose.  Semi-Min-Stable Processes , 1992 .

[8]  Marco Oesting,et al.  Simulation of Brown–Resnick processes , 2012 .

[9]  Yizao Wang,et al.  Conditional sampling for spectrally discrete max-stable random fields , 2010, Advances in Applied Probability.

[10]  A. Genz.  Numerical Computation of Multivariate Normal Probabilities , 1992 .

[11]  Laurens de Haan,et al.  Spatial extremes: Models for the stationary case , 2006 .

[12]  M. Schlather.  Simulation and Analysis of Random Fields , 2001 .

[13]  K. Weintraub.  Sample and Ergodic Properties of Some Min-Stable Processes , 1991 .

[14]  Mathieu Ribatet,et al.  Spatial extremes: Max-stable processes at work , 2013 .

[15]  Martin Schlather,et al.  Models for Stationary Max-Stable Random Fields , 2002 .

[16]  B. Mandelbrot.  Fractal Geometry of Nature , 1984 .

[17]  C. Zhou,et al.  On spatial extremes: With application to a rainfall problem , 2008, 0807.4092.

[18]  M. Schlather,et al.  Conditional sampling for max-stable processes with a mixed moving maxima representation , 2012, Extremes.

[19]  Jonathan A. Tawn,et al.  A dependence measure for multivariate and spatial extreme values: Properties and inference , 2003 .

[20]  Sidney I. Resnick,et al.  Prediction of Stationary Max-Stable Processes , 1993 .

[21]  Marc G. Genton,et al.  On the likelihood function of Gaussian max-stable processes , 2011 .

[22]  Sidney I. Resnick,et al.  Extreme values of independent stochastic processes , 1977 .

[23]  Sidney I. Resnick,et al.  Basic properties and prediction of max-ARMA processes , 1989, Advances in Applied Probability.

[24]  M. Beniston.  The 2003 heat wave in Europe: A shape of things to come? An analysis based on Swiss climatological data and model simulations , 2004 .

[25]  S. Padoan,et al.  Likelihood-Based Inference for Max-Stable Processes , 2009, 0902.3060.

[26]  Gordon A. Fenton,et al.  Simulation and analysis of random fields , 1990 .

[27]  Clément Dombry,et al.  Regular conditional distributions of continuous max-infinitely divisible random fields , 2013 .

[28]  Anthony C. Davison,et al.  Geostatistics of extremes , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  A. Davison,et al.  Statistical Modeling of Spatial Extremes , 2012, 1208.3378.

[30]  Zakhar Kabluchko,et al.  Ergodic properties of max-infinitely divisible processes , 2009, 0905.4196.