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A simple variogram model with two parameters is presented that includes the power variogram for fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadric model. One parameter controls the sample path roughness of the process. The other parameter allows for a smooth transition between bounded and unbounded variograms, that is, between stationary and intrinsically stationary processes in a Gaussian framework, or between mixing and non-ergodic Brown–Resnick processes when modeling spatial extremes. Copyright

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... Proof. The proof is an adaption of the proof of Proposition 1 in Schlather and Moreva [2017]. By Schlather and Moreva [2017], the function f (g) = (1+g β ) α β is a complete Bernstein function for any 0 < β ≤ 1, 0 < α ≤ 1 and for any conditionally negative definite function g ≥ 0. By Proposition 3.1, the function g(x, y) = A 0 (x − y) α0 + k j=1 A j (c(x) Ij − c(y) Ij ) αj is such a conditionally negative definite function. ...

... The proof is an adaption of the proof of Proposition 1 in Schlather and Moreva [2017]. By Schlather and Moreva [2017], the function f (g) = (1+g β ) α β is a complete Bernstein function for any 0 < β ≤ 1, 0 < α ≤ 1 and for any conditionally negative definite function g ≥ 0. By Proposition 3.1, the function g(x, y) = A 0 (x − y) α0 + k j=1 A j (c(x) Ij − c(y) Ij ) αj is such a conditionally negative definite function. Consequently, (18) is a valid variogram by the same arguments of Schlather and Moreva [2017], that is, one uses that constants are conditionally negative definite functions and the set of conditionally negative definite functions forms a cone. ...

... By Schlather and Moreva [2017], the function f (g) = (1+g β ) α β is a complete Bernstein function for any 0 < β ≤ 1, 0 < α ≤ 1 and for any conditionally negative definite function g ≥ 0. By Proposition 3.1, the function g(x, y) = A 0 (x − y) α0 + k j=1 A j (c(x) Ij − c(y) Ij ) αj is such a conditionally negative definite function. Consequently, (18) is a valid variogram by the same arguments of Schlather and Moreva [2017], that is, one uses that constants are conditionally negative definite functions and the set of conditionally negative definite functions forms a cone. In the case that α = 0, the limiting function is a variogram by the characteristics of conditionally negative functions. ...

In recent years, parametric models for max-stable processes have become a popular choice for modeling spatial extremes because they arise as the asymptotic limit of rescaled maxima of independent and identically distributed random processes. Apart from few exceptions for the class of extremal-t processes, existing literature mainly focuses on models with stationary dependence structures. In this paper, we propose a novel non-stationary approach that can be used for both Brown-Resnick and extremal-t processes - two of the most popular classes of max-stable processes - by including covariates in the corresponding variogram and correlation functions, respectively. We apply our new approach to extreme precipitation data in two regions in Southern and Northern Germany and compare the results to existing stationary models in terms of Takeuchi's information criterion (TIC). Our results indicate that, for this case study, non-stationary models are more appropriate than stationary ones for the region in Southern Germany. In addition, we investigate theoretical properties of max-stable processes conditional on random covariates. We show that these can result in both asymptotically dependent and asymptotically independent processes. Thus, conditional models are more flexible than classical max-stable models.

... When the kind of underlying causative process is unclear, a parametric model that allows for both bounded and unbounded variograms is advantageous. Recently, Schlather and Moreva (2017) proposed a simple model that allows for a smooth transition between a bounded and unbounded variogram: ...

... In this work, we aim at generalizing the univariate variogram model in Schlather and Moreva (2017) to variogram matrices for vector random fields, preserving the flexibility of varying between boundedness and unboundedness for the entries of the variogram matrix by changing the values of parameters in the model. Consider a p-variate random field ZðsÞ ¼ fZ 1 ðsÞ; . . ...

... In this paper, we propose a class of valid parametric variogram matrices, both traditional and pseudo, that is a combination and extension of the main results from Ma (2011a) and Schlather and Moreva (2017), which are connected by the Bernstein function. The entries of this new class of variogram matrices allow for both bounded and unbounded variograms or cross variograms by changing the parameters in each entry independently. ...

We construct a flexible class of parametric models for both traditional and pseudo variogram matrix (valued functions), where the off-diagonal elements are the traditional cross variograms and pseudo cross variograms, respectively, and the diagonal elements are the direct variograms, based on the method of latent dimensions and the linear model of coregionalization. The entries in the parametric variogram matrix allow for a smooth transition between boundedness and unboundedness by changing the values of parameters, and thus between joint second-order and intrinsically stationary vector random fields, or between multivariate geometric Gaussian processes and multivariate Brown–Resnick processes in spatial extreme analysis.

... where γ α,β (d) is the model by Schlather and Moreva [28] defined as ...

While gradient based methods are ubiquitous in machine learning, selecting the right step size often requires "hyperparameter tuning". This is because backtracking procedures like Armijo's rule depend on quality evaluations in every step, which are not available in a stochastic context. Since optimization schemes can be motivated using Taylor approximations, we replace the Taylor approximation with the conditional expectation (the best $L^2$ estimator) and propose "Random Function Descent" (RFD). Under light assumptions common in Bayesian optimization, we prove that RFD is identical to gradient descent, but with calculable step sizes, even in a stochastic context. We beat untuned Adam in synthetic benchmarks. To close the performance gap to tuned Adam, we propose a heuristic extension competitive with tuned Adam.

... Other non conventional properties of covariance functions have been studied by Alegría (2020) and Alegría et al. (2021), who proposed some modified scale mixtures representations to obtain classes of cross-covariance functions with non-monotonic behaviours (the so-called cross-dimple effect) for vector-valued random fields. In Schlather and Moreva (2017), models that allow for a smooth transition between stationary and intrinsically stationary Gaussian random fields are derived. ...

Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Having covariance functions that allow to specify both local and global properties is certainly on demand. This paper provides a method to find new classes of covariance functions having such properties. We term these models hybrid as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combination of parametric families of measures. In order to illustrate our methodology, we provide new families of covariance functions that are proved to be richer with respect to other well known families that have been proposed by earlier literature. More precisely, we derive a hybrid Cauchy-Mat\'ern model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid Hole-Effect-Mat\'ern model, which is capable of attaining negative values (hole effect), while preserving the local attributes of the traditional Mat\'ern model. Our findings are illustrated through numerical studies with both simulated and real data.

... We considered the Matérn and the Bernstein (Schlather & Moreva, 2017) semi-variogram models for the dependence. Matérn semi-variograms are bounded above, whereas the Bernstein model bridges the two regimes illustrated in Figure 3. ...

Peaks‐over‐threshold analysis using the generalised Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks‐over‐threshold analysis to extremes of functional data. Threshold exceedances defined using a functional r are modelled by the generalised r‐Pareto process, a functional generalisation of the generalised Pareto distribution that covers the three classical regimes for the decay of tail probabilities, and that is the only possible continuous limit for r‐exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalised r‐Pareto processes, discuss model validation and apply the new methodology to extreme European windstorms and heavy spatial rainfall.

... We remind that in this case, the pairwise extremal dependence, as summarized by the extremogram, is linked through the closed form (11) to γ, for which we impose parametric model. We choose (Schlather and Moreva, 2017) ...

The Sihl river, located near the city of Zurich in Switzerland, is under continuous and tight surveillance as it flows directly under the city's main railway station. To issue early warnings and conduct accurate risk quantification, a dense network of monitoring stations is necessary inside the river basin. However, as of 2021 only three automatic stations are operated in this region, naturally raising the question: how to extend this network for optimal monitoring of extreme rainfall events? So far, existing methodologies for station network design have mostly focused on maximizing interpolation accuracy or minimizing the uncertainty of some model's parameters estimates. In this work, we propose new principles inspired from extreme value theory for optimal monitoring of extreme events. For stationary processes, we study the theoretical properties of the induced sampling design that yields non-trivial point patterns resulting from a compromise between a boundary effect and the maximization of inter-location distances. For general applications, we propose a theoretically justified functional peak-over-threshold model and provide an algorithm for sequential station selection. We then issue recommendations for possible extensions of the Sihl river monitoring network, by efficiently leveraging both station and radar measurements available in this region.

... The third example is the variogram used in Gneiting (2002) for modeling the Irish wind data set, namely γ (u) = (1 + a|u| α ) β − 1 with a > 0, 0 < α ≤ 2 and 0 < β ≤ 1, see also Schlather and Moreva (2017) for a detailed presentation of this variogram model. The special case α = β = 1 corresponds to the first example, whilst the limiting case lim β→0 γ (u)/β with α = 2 is formally similar to the second example. ...

Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure. In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase. The difference lies in the way to handle the temporal component. The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure. Both algorithms are scalable as their computational cost is proportional to the number of space-time locations that may be irregular in space and time. They are illustrated and validated through synthetic examples.

Covariance functions are the core of spatial statistics, stochastic processes, machine learning, and many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Covariance functions that allow one to specify both local and global properties are certainly in demand. This paper provides a method for finding new classes of covariance functions having such properties. We refer to these models as hybrid, as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combinations of parametric families of measures. To illustrate our methodology, we provide new families of covariance functions that are proved to be richer than other well-known families proposed in earlier literature. More precisely, we derive a hybrid Cauchy–Matérn model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid hole-effect–Matérn model which is capable of attaining negative values (hole effect) while preserving the local attributes of the traditional Matérn model. Our findings are illustrated through numerical studies with both simulated and real data.

Peaks-over-threshold analysis using the generalized Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks-over-threshold analysis to extremes of functional data. Threshold exceedances defined using a functional $r$ are modelled by the generalized $r$-Pareto process, a functional generalization of the generalized Pareto distribution that covers the three classical regimes for the decay of tail probabilities. This process is the only possible limit for the distribution of $r$-exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalized $r$-Pareto processes, discuss model validation, and use the new methodology to study extreme European windstorms and heavy spatial rainfall.

Simulation and Analysis of Random Fields

This book arises as the natural continuation of the International Spring School "Advances and Challenges in Space-Time modelling of Natural Events," which took place in Toledo (Spain) in March 2010. This Spring School above all focused on young researchers (Master students, PhD students and post-doctoral researchers) in academics, extra-university research and the industry who are interested in learning about recent developments, new methods and applications in spatial statistics and related areas, and in exchanging ideas and findings with colleagues.

In order to incorporate the dependence between the spatial random fields of
observed and forecasted maximal wind gusts, we propose to model them jointly by
a bivariate Brown-Resnick process. As there is a one-to-one correspondence
between bivariate Brown-Resnick processes and pseudo cross-variograms, station-
ary Brown-Resnick processes can be characterized by properties of the
underlying pseudo cross-variogram. We particularly focus on the investigation
of their asymp- totic behaviour and introduce a flexible parametric model both
being interesting in classical geostatistics on their own. The model is applied
to real observation and forecast data for 110 stations in Northern Germany. The
resulting post-processed forecasts are verified

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

We develop classification results for max--stable processes, based on their spectral representations. The structure of max--linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max--stable processes based on the notion of co--spectral functions. In particular, we discuss the spectrally continuous--discrete, the conservative--dissipative, and positive--null decompositions. For stationary max--stable processes, the latter two decompositions arise from connections to non--singular flows and are closely related to the classification of stationary sum--stable processes. The interplay between the introduced decompositions of max--stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative. A result on general Gaussian processes with stationary increments and continuous paths is obtained. Comment: 40 pages. Minor changes. Technical Report 487, Department of Statistics, University of Michigan

Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance $\sigma^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}\delta_{U_i}$ be a Poisson point process on the real line with intensity $e^{-y} dy$. We show that the law of the random family of functions $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$, is translation invariant. In particular, the process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable process with standard Gumbel margins. The process $\eta$ arises as a limit of a suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary Gaussian processes as $n\to\infty$ if and only if $W$ is a (nonisotropic) fractional Brownian motion on ${\mathbb{R}}^d$. Under suitable conditions on $W$, the process $\eta$ has a mixed moving maxima representation. Comment: Published in at http://dx.doi.org/10.1214/09-AOP455 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Martin and Walker ((1997) J. Appl. Prob. 34, 657-670) proposed the power-law ρ(υv) = c|υ|-β, |υ| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(υ) = (1 + |υ|c\α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

Multivariate data measured in space, such as temperature and pressure or the content of two metals in geological deposits, requires models that allow to incorporate spatial and cross-dependence of observations. We introduce some novel bivariate models, the powered exponential (or stable) covariance model and the Cauchy covariance model with flexible smoothness, variance, scale and cross-correlation parameters. In addition, we present a bundle of compactly supported bivariate covariance models obtained by the cut-off embedding technique from covariance functions with Euclid's hat scale mixture representation. Finally, we show that the circulant embedding algorithm always works if all covariance components have compact support. On the way we extend the circulant embedding method even for univariate models.

We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max-stable processes based on the notion of co-spectral functions. In particular, we discuss the spectrally continuous-discrete, the conservative-dissipative, and the positive-null decompositions. For stationary max-stable processes, the latter two decompositions arise from connections to nonsingular flows and are closely related to the classification of stationary sum-stable processes. The interplay between the introduced decompositions of max-stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative.

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

We discuss necessary and sufficient conditions for power-law and polynomial models to be correlation functions on bounded domains. These results date back to unpublished work by Matheron (1974) and generalize the findings of Gneiting (1999).

Covariance functions and variograms are the most important ingredients in the classical approaches to geostatistics. We give an overview over the approaches how models can be obtained. Variant types of scale mixtures turn out to be the most important way of construction. Some of the approaches are closely related to simulation methods of unconditional Gaussian random field, for instance the turning bands and the random coins. We discuss these methods and complement them by an overview over further methods.

This paper explores the use of visualization through animations, coined visuanimation, in the field of statistics. In particular, it illustrates the embedding of animations in the paper itself and the storage of larger movies in the online supplemental material. We present results from statistics research projects using a variety of visuanimations, ranging from exploratory data analysis of image data sets to spatio-temporal extreme event modelling; these include a multiscale analysis of classification methods, the study of the effects of a simulated explosive volcanic eruption and an emulation of climate model output. This paper serves as an illustration of visuanimation for future publications in Stat. Copyright © 2015 John Wiley & Sons, Ltd.

General PrinciplesVariogram Cloud and Sample VariogramMathematical Properties of the VariogramRegularization and Nugget EffectVariogram ModelsFitting a Variogram ModelVariography in Presence of a DriftSimple Applications of the VariogramComplements: Theory of Variogram Estimation and Fluctuation

Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed
stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently,
inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability
of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely
used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class,
we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information
below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the
methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically
simplify the likelihood. The Stephenson–Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood.
We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein-Uhlenbeck processes.

We propose an approach to spatial modeling of extreme rainfall, based on
max-stable processes fitted using partial duration series and a censored
threshold likelihood function. The resulting models are coherent with
classical extreme-value theory and allow the consistent treatment of
spatial dependence of rainfall using ideas related to those of classical
geostatistics. We illustrate the ideas through data from the Val Ferret
watershed in the Swiss Alps, based on daily cumulative rainfall totals
recorded at 24 stations for four summers, augmented by a longer series
from nearby. We compare the fits of different statistical models
appropriate for spatial extremes, select that best fitting our data, and
compare return level estimates for the total daily rainfall over the
stations. The method can be used in other situations to produce
simulations needed for hydrological models, and in particular, for the
generation of spatially heterogeneous extreme rainfall fields over
catchments.

A method for simulating a stationary Gaussian process on a fine rectangular grid in [0, 1] ⊂ℝ is described. It is assumed that the process is stationary with respect to translations of ℝ, but the method does not require the process to be isotropic. As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform. However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied. It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on ℝ, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on ℝ, is strictly positive. The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer. An approximate procedure is also proposed to cover cases where it is not feasible to apply the procedure in its exact form.

Fractional Brownian surfaces are commonly used as models for landscapes and other physical processes in space. This work shows how to simulate fractional Brownian surfaces on a grid efficiently and exactly by embedding them in a periodic Gaussian random field and using the fast Fourier transform. Periodic embeddings are given that are proven to yield positive definite covariance functions and hence yield exact simulations for all possible densities of the simulation grid. Numerical results show these embeddings can sometimes be made more efficient in practice. Further numerical results show how the ideas developed for simulating fractional Brownian surfaces can be used for simulating other Gaussian random fields. The simulation methodology is used to study the behavior of a simple estimator of the parameters of a fractional Brownian surface.

Regularization with radial basis functions is an effective method in many machine learning applications. In recent years classes of radial basis func-tions with compact support have been proposed in the approximation theory literature and have become more and more popular due to their computational advantages. In this paper we study the statistical properties of the method of reg-ularization with compactly supported basis functions. We consider three popular classes of compactly supported radial basis functions. In the setting of estimat-ing a periodic function in a white noise problem, we show that regularization with (periodized) compactly supported radial basis functions is rate optimal and adapts to unknown smoothness up to an order related to the radial basis function used. Due to results on equivalence of the white noise model with many impor-tant models including regression and density estimation, our results are expected to give insight on the performance of such methods in more general settings than the white noise model.

The circulant embedding technique allows for the fast and exact simulation of stationary and intrinsically stationary Gaussian random fields. The method uses periodic embeddings and relies on the fast Fourier transform. However, exact simulations require that the periodic embedding is nonnegative definite, which is frequently not the case for two-dimensional simulations. This work considers a suggestion by Michael Stein, who proposed nonnegative definite periodic embeddings based on suitably modified, compactly supported covariance functions. Theoretical support is given to this proposal, and software for its implementation is provided. The method yields exact simulations of planar Gaussian lattice systems with up to 10 6 lattice points for wide classes of processes, including those with powered exponential, Matérn and Cauchy covariances.

We study the sample path regularity of a second-order random field (Xt)t[set membership, variant]T where T is an open subset of . It is shown that the conditions on its covariance function, known to be equivalent to mean square differentiability of order k, imply that the sample paths are a.s.Â in the local Sobolev space . We discuss their necessity, and give additional conditions for the sample paths to be in a local Sobolev space of fractional order [mu]. This finally allows, via Sobolev embeddings, to draw conclusions about a.s.Â continuous differentiability of the sample paths.

Let\(\varphi (r) = \sqrt {r^2 + c^2 } \) or\(\varphi (r) = 1/\sqrt {r^2 + c^2 } \). Givenf:ℛ
n
→ℛ, we establish convergence orders of interpolation
where the cardinal functionx withx(j)=δ0j
is a linear combination of integer shifts
, of a fast decaying function$$\psi (x) = \mathop \Sigma \limits_{j \in N} \mu _i \varphi (||x - j||),x \in R^n ,$$
N ⊂F″being a finite subset. Ifn is odd and\(\varphi (r) = \sqrt {r^2 + c^2 } \), we find convergence orders ofh
n+1, whereas for\(\varphi (r) = 1/\sqrt {r^2 + c^2 } \), convergence orders ofh
n−1 are obtained whenn≥3 is odd.

Modelling spatio-temporal processes has become an important issue in current
research. Since Gaussian processes are essentially determined by their second
order structure, broad classes of covariance functions are of interest. Here, a
new class is described that merges and generalizes various models presented in
the literature, in particular models in Gneiting (J. Amer. Statist. Assoc. 97
(2002) 590--600) and Stein (Nonstationary spatial covariance functions (2005)
Univ. Chicago). Furthermore, new models and a multivariate extension are
introduced.

We prove that a stationary max--infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesaro summable to 0). These criteria are applied to some classes of max--infinitely divisible processes. Comment: 15 pages

We discuss necessary and sufficient conditions for power-law and polynomial models to be correlation functions on bounded domains. These results date back to unpublished work by Matheron (1974) and generalize the findings of Gneiting (1999).

abstract This letter considers stationary local approximations to intrinsically stationary random functions on . For any continuous variogram [gamma] and any ball in , there exists a covariance function with compact support whose respective variogram is arbitrarily close to [gamma] when restricted to the ball.

. Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid## This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over m +1 equispaced points on a line can be produced at the cost of an initial FFT of length 2m with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an (m+1)×(m+1) Toeplitz correlation matrix R can be embedded in a nonnegative definite 2M×2M circulant matrix S, exact realizations of the normal multivariate y #N(0,R) can be generated via FFTs of length 2M . Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which M = m is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated. Key words. geostatistics, ...

Fractal behavior and long-range dependence have been observed in an
astonishing number of physical systems. Either phenomenon has been modeled by
self-similar random functions, thereby implying a linear relationship between
fractal dimension, a measure of roughness, and Hurst coefficient, a measure of
long-memory dependence. This letter introduces simple stochastic models which
allow for any combination of fractal dimension and Hurst exponent. We
synthesize images from these models, with arbitrary fractal properties and
power-law correlations, and propose a test for self-similarity.

Représentation stationnaires et représentations minimales pour les F.A.I, Note Géostatistique no. 125 N-377, Ecole des Mines de Paris, Fontainebleau, France

- Matheron G

- Wackernagel

Représentation stationnaires et représentations minimales pour les F.A.I, Note Géostatistique no. 125 N-377

- G Matheron

Représentation stationnaires et représentations minimales pour les F.A.I Note Géostatistique no. 125 N-377 Ecole des Mines de

- Matherong