Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes simulation highly nontrivial. Algorithms based on finite approximations that are used in practice are often not exact and computationally inefficient. We will present two algorithms for exact simulation of a max-stable process at a finite number of locations. The first algorithm generalizes the approach by Dieker and Mikosch [2015] for Brown-Resnick processes and it is based on simulation from the spectral measure. The second algorithm relies on the idea to simulate only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We study the complexity of both algorithms and prove that the second procedure is always more efficient. Moreover, we provide closed expressions for their implementation that cover the most popular models for max-stable processes and extreme value copulas. For simulation on dense grids, an adaptive design of the second algorithm is proposed.

Modeling of and inference on multivariate data that have been measured in space, such as temperature and pressure, are challenging tasks in environmental sciences, physics and materials science. We give an overview over and some background on modeling with crosscovariance models. The R package RandomFields supports the simulation, the parameter estimation and the prediction in particular for the linear model of coregionalization, the multivariate Matérn models, the delay model, and a spectrum of physically motivated vector valued models. An example on weather data is considered, illustrating the use of RandomFields for parameter estimation and prediction.

Brown-Resnick processes form a flexible class of stationary max-stable processes based on Gaussian random fields. With regard to applications, fast and accurate simulation of these processes is an important issue. In fact, Brown-Resnick processes that are generated by a dissipative flow do not allow for good finite approximations using the definition of the processes. On large intervals we get either huge approximation errors or very long operating times. Looking for solutions of this problem, we give different representations of the Brown-Resnick processes-including random shifting and a mixed moving maxima representation-and derive various kinds of finite approximations that can be used for simulation purposes. Furthermore, error bounds are calculated in the case of the original process by Brown and Resnick (J Appl Probab 14(4): [732][733][734][735][736][737][738][739] 1977). For a one-parametric class of BrownResnick processes based on the fractional Brownian motion we perform a simulation study and compare the results of the different methods concerning their approximation quality. The presented simulation techniques turn out to provide remarkable improvements.

To improve the forecasts of weather extremes, we propose a joint spatial model for the observations and the forecasts, based on a bivariate Brown-Resnick process. As the class of stationary bivariate Brown-Resnick processes is fully characterized by the class of pseudo cross-variograms, we contribute to the theorical understanding of pseudo cross-variograms refining the knowledge of the asymptotic behaviour of all their components and introducing a parsimonious, but flexible parametric model. Both findings are of interest in classical geostatistics on their own. The proposed model is applied to real observation and forecast data for extreme wind gusts at 119 stations in Northern Germany.

The distribution of spatially aggregated data from a stochastic process X may exhibit a different tail behavior than its marginal distributions. For a large class of aggregating functionals we introduce the -extremal coefficient that quantifies this difference as a function of the extremal spatial dependence in X. We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Explicit formulas for the -extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process, which we exploit to develop a method for statistical downscaling. We apply our framework to downscale daily temperature maxima in the south of France from a gridded data set and use our model to generate high resolution maps of the warmest day during the 2003 heatwave.

The efficiency of simulation algorithms for max-stable processes relies on the choice of the spectral representation: different choices result in different sequences of finite approximations to the process. We propose a constructive approach yielding a normalized spectral representation that solves an optimization problem related to the efficiency of simulating max-stable processes. The simulation algorithm based on the normalized spectral representation can be regarded as max-importance sampling. Compared to other simulation algorithms hitherto, our approach has at least two advantages. First, it allows the exact simulation of a comprising class of max-stable processes. Second, the algorithm has a stopping time with finite expectation. In practice, our approach has the potential of considerably reducing the simulation time of max-stable processes.MSC 2010 subject classifications: Primary 60G70; secondary 68U20.where Π is a Poisson point process on (0, ∞) × H with intensity t −2 dt H(df ) andfor all y ∈ K. The functions f in H are the spectral functions of the max-stable process Z. As any max-stable process can be obtained from a max-stable process with standard Fréchet marginals via marginal transformations, we will henceforth assume processes with standard Fréchet marginals, i.e. processes with representation (1.1) and (1.2). According to the spectral representation (1.1), the construction of a max-stable process involves infinitely many points (t, f ) ∈ Π. Nevertheless, since only the maximum over all functions tf counts, the number of points (t, f ) that contribute to Z, i.e. Z(y) = tf (y) for imsart-bj ver.

Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. [2016, Ann. Appl. Stat., to appear] use a Bayesian approach to fit a Brown-Resnick process to extreme temperatures. In this paper, we extend this idea to a methodology that is applicable to general max-stable distributions and that uses full likelihoods. We further provide simple conditions for the asymptotic normality of the median of the posterior distribution and verify them for the commonly used models in multivariate and spatial extreme value statistics. A simulation study shows that this point estimator is considerably more efficient than the composite likelihood estimator in a frequentist framework. From a Bayesian perspective, our approach opens the way for new techniques such as Bayesian model comparison in multivariate and spatial extremes.imsart-ejs ver.

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the one-dimensional case and use a finite number of shape functions satisfying some regularity conditions. For more general shape functions approximation techniques are presented. Our algorithm is applied to the Smith process and the Brown-Resnick process. Finally, we compare our computational results to other approaches. Here, the algorithm for Gaussian processes with transformed marginals turns out to be surprisingly competitive.