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5 2 Construction of Covariance Functions and Unconditional Simulation 0 100 200 300 400 500 h Fig. / D d! Here, kind. / . 2. Variograms with exceptional properties can be obtained by sums of cosine functions. However, they do not have any practical relevance. h/ D 1 [4]. Independently, [50] showed that these two properties hold also for bk D 2k . 1 illustrates for ak D 10 3 k 1:1 and bk D 1:11:1 . 9) for some finite non-negative measure F on Œ0; 1/. Here, sd denotes the surface area of the d -dimensional sphere.

Chapter 5 offers a review of the most important methods for the construction of spatial extremes. After a brief illustration of the extreme value theory for univariate and multivariate values, this chapter focuses on spatial max-stable processes, illustrating also the features related to statistical inference and simulation for these processes. Max-stable processes are also contrasted with spatial hierarchical models. The problem of large and complex datasets is thoroughly discussed in Chap. 4, but from the perspective of Bayesian inference for space time point processes, 1 Space-Time Processes and Geostatistics 19 which have become more and more popular, especially thanks to the development of Markov chain Monte Carlo methods, which expand the scope of application of Bayesian methods considerably.

This is indeed true, see Sect. 2. 9) is called spectral turning bands method in geostatistics, see [61]. d 1/-dimensional sphere S . All random d 1 P variables are independent. Again, Z 0 D n 1=2 niD1 Zi yields an approximation to a Gaussian random field for Zi , i D 1; : : : ; n, that are independent and identically distributed according to Z. The value of n should be of order 500 to get good results. 2 shows the performances of the method for the “Gaussian” covariance function. 4. The spectral representation by Bochner and Schoenberg leaves the question open, which discontinuous positive definite functions exist and which are of practical interest.

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