Max-stable Random Fields Research Articles

A crossinggram for random fields on lattices

The modeling of risk situations that occur in a space framework can be done using max-stable random fields on lattices. Although the summary coefficients for the spatial behavior do not characterize the finite-dimensional distributions of the random field, they have the advantage of being immediate to interpret and easier to estimate. The coefficients that we propose give us information about the tendency of a random field for local oscillations of its values in relation to real valued high levels. It is not the magnitude of the oscillations that is being evaluated, but rather the greater or lesser number of oscillations, that is, the tendency of the trajectories to oscillate. We can observe surface trajectories more smooth over a region according to higher crossinggram value. It takes value in [0,1] and increases with the concordance of the variables of the random field.

About the return period of a catastrophe

Abstract. When a natural hazard event like an earthquake affects a region and generates a natural catastrophe (NatCat), the following questions arise: how often does such an event occur? What is its return period (RP)? We derive the combined return period (CRP) from a concept of extreme value statistics and theory – the pseudo-polar coordinates. A CRP is the (weighted) average of the local RP of local event intensities. Since CRP's reciprocal is its expected exceedance frequency, the concept is testable. As we show, the CRP is related to the spatial characteristics of the NatCat-generating hazard event and the spatial dependence of corresponding local block maxima (e.g., annual wind speed maximum). For this purpose, we extend a previous construction for max-stable random fields from extreme value theory and consider the recent concept of area function from NatCat research. Based on the CRP, we also develop a new method to estimate the NatCat risk of a region via stochastic scaling of historical fields of local event intensities (represented by records of measuring stations) and averaging the computed event loss for defined CRP or the computed CRP (or its reciprocal) for defined event loss. Our application example is winter storms (extratropical cyclones) over Germany. We analyze wind station data and estimate local hazard, CRP of historical events, and the risk curve of insured event losses. The most destructive storm of our observation period of 20 years is Kyrill in 2002, with CRP of 16.97±1.75. The CRPs could be successfully tested statistically. We also state that our risk estimate is higher for the max-stable case than for the non-max-stable case. Max-stable means that the dependence measure (e.g., Kendall's τ) for annual wind speed maxima of two wind stations has the same value as for maxima of larger block size, such as 10 or 100 years since the copula (the dependence structure) remains the same. However, the spatial dependence decreases with increasing block size; a new statistical indicator confirms this. Such control of the spatial characteristics and dependence is not realized by the previous risk models in science and industry. We compare our risk estimates to these.

Stochastic derivative estimation for max-stable random fields

We consider expected performances based on max-stable random fields and we are interested in their derivatives with respect to the spatial dependence parameters of those fields. Max-stable fields, such as the Brown–Resnick and Smith fields, are very popular in spatial extremes. We focus on the two most popular unbiased stochastic derivative estimation approaches: the likelihood ratio method (LRM) and the infinitesimal perturbation analysis (IPA). LRM requires the multivariate density of the max-stable field to be explicit, and IPA necessitates the computation of the derivative with respect to the parameters for each simulated value. We propose convenient and tractable conditions ensuring the validity of LRM and IPA in the cases of the Brown–Resnick and Smith field, respectively. Obtaining such conditions is intricate owing to the very structure of max-stable fields. Then we focus on risk and dependence measures, which constitute one of the several frameworks where our theoretical results can be useful. We perform a simulation study which shows that both LRM and IPA perform well in various configurations, and provide a real case study that is valuable for the insurance industry.

On logarithmically optimal exact simulation of max-stable and related random fields on a compact set

We consider the random field \begin{equation*}M(t)=\mathop{\mathrm{sup}}_{n\geq1}\{-\log A_{n}+X_{n}(t)\},\qquad t\in T,\end{equation*} for a set $T\subset\mathbb{R}^{m}$, where $(X_{n})$ is an i.i.d. sequence of centered Gaussian random fields on $T$ and $0<A_{1}<A_{2}<\cdots$ are the arrivals of a general renewal process on $(0,\infty)$, independent of $(X_{n})$. In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs $c(d)=c(\{t_{1},\ldots,t_{d}\})$ function evaluations to sample $X_{n}$ at $d$ locations $t_{1},\ldots,t_{d}\in T$. We provide an algorithm which samples $M(t_{1}),\ldots,M(t_{d})$ with complexity $O(c(d)^{1+o(1)})$ as measured in the $L_{p}$ norm sense for any $p\ge1$. Moreover, if $X_{n}$ has an a.s. converging series representation, then $M$ can be a.s. approximated with error $\delta$ uniformly over $T$ and with complexity $O(1/(\delta\log(1/\delta))^{1/\alpha})$, where $\alpha$ relates to the Hölder continuity exponent of the process $X_{n}$ (so, if $X_{n}$ is Brownian motion, $\alpha=1/2$).

Spatial Risk Measures and Rate of Spatial Diversification

An accurate assessment of the risk of extreme environmental events is of great importance for populations, authorities and the banking/insurance/reinsurance industry. Koch (2017) introduced a notion of spatial risk measure and a corresponding set of axioms which are well suited to analyze the risk due to events having a spatial extent, precisely such as environmental phenomena. The axiom of asymptotic spatial homogeneity is of particular interest since it allows one to quantify the rate of spatial diversification when the region under consideration becomes large. In this paper, we first investigate the general concepts of spatial risk measures and corresponding axioms further and thoroughly explain the usefulness of this theory for both actuarial science and practice. Second, in the case of a general cost field, we give sufficient conditions such that spatial risk measures associated with expectation, variance, value-at-risk as well as expected shortfall and induced by this cost field satisfy the axioms of asymptotic spatial homogeneity of order 0, −2, −1 and −1, respectively. Last but not least, in the case where the cost field is a function of a max-stable random field, we provide conditions on both the function and the max-stable field ensuring the latter properties. Max-stable random fields are relevant when assessing the risk of extreme events since they appear as a natural extension of multivariate extreme-value theory to the level of random fields. Overall, this paper improves our understanding of spatial risk measures as well as of their properties with respect to the space variable and generalizes many results obtained in Koch (2017).

A central limit theorem for functions of stationary max-stable random fields on [formula omitted

Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on Rd. Then, we show that appropriate functions of the Brown–Resnick random field with a power variogram and of the Smith random field satisfy the central limit theorem. Another strong motivation for our work lies in the fact that central limit theorems for random fields on Rd have been barely considered in the literature. As an application, we briefly show the usefulness of our results in a risk assessment context.

On generalized max-linear models in max-stable random fields

Abstract In practice, it is not possible to observe a whole max-stable random field. Therefore, we propose a method to reconstruct a max-stable random field in C([0, 1]k) by interpolating its realizations at finitely many points. The resulting interpolating process is again a max-stable random field. This approach uses a generalized max-linear model. Promising results have been established in the k = 1 case of Falk et al. (2015). However, the extension to higher dimensions is not straightforward since we lose the natural order of the index space.

Clustering of high values in random fields

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with \(\mathbb {Z}^{2}\), and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of \(\mathbb {R}^{2}\). We show that, under a suitable coordinatewise mixing condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levels. The general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.

Analysis of Random Fields UsingCompRandFld

Statistical analysis based on random fields has become a widely used approach in order to better understand real processes in many fields such as engineering, environmental sciences, etc. Data analysis based on random fields can be sometimes problematic to carry out from the inferential prospective. Examples are when dealing with: large dataset, counts or binary responses and extreme values data. This article explains how to perform, with the R package CompRandFld, challenging statistical analysis based on Gaussian, binary and max-stable random fields. The software provides tools for performing the statistical inference based on the composite likelihood in complex problems where standard likelihood methods are difficult to apply. The principal features are illustrated by means of simulation examples and an application of Irish daily wind speeds.

Random tessellations associated with max-stable random fields

With any max-stable random process $\eta$ on $\mathcal{X}=\mathbb{Z}^d$ or $\mathbb{R}^d$, we associate a random tessellation of the parameter space $\mathcal{X}$. The construction relies on the Poisson point process representation of the max-stable process $\eta$ which is seen as the pointwise maximum of a random collection of functions $\Phi=\{\phi_i, i\geq 1\}$. The tessellation is constructed as follows: two points $x,y\in \mathcal{X}$ are in the same cell if and only if there exists a function $\phi\in\Phi$ that realizes the maximum $\eta$ at both points $x$ and $y$, i.e. $\phi(x)=\eta(x)$ and $\phi(y)=\eta(y)$. We characterize the distribution of cells in terms of coverage and inclusion probabilities. Most interesting is the stationary case where the asymptotic properties of the cells are strongly related to the ergodic properties of the non-singular flow generating the max-stable process. For example, we show that: i) the cells are bounded almost surely if and only if $\eta$ is generated by a dissipative flow; ii) the cells have positive asymptotic density almost surely if and only if $\eta$ is generated by a positive flow.

Statistical Inference for Max-Stable Processes in Space and Time

SummaryMax-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several families of max-stable random fields have been proposed in the literature. One such representation is based on a limit of normalized and rescaled pointwise maxima of stationary Gaussian processes that was first introduced by Kabluchko and co-workers. This paper deals with statistical inference for max-stable space–time processes that are defined in an analogous fashion. We describe pairwise likelihood estimation, where the pairwise density of the process is used to estimate the model parameters. For regular grid observations we prove strong consistency and asymptotic normality of the parameter estimates as the joint number of spatial locations and time points tends to ∞. Furthermore, we discuss extensions to irregularly spaced locations. A simulation study shows that the method proposed works well for these models.

Some new classes of stationary max-stable random fields

We present two new classes of stationary max-stable random fields. For the first class, we use the spectral representation due to Schlather (2002) and assume that the stationary process used in the representation is proportional to a power of a max-stable random field. We derive the finite dimensional distributions, explain the relationship between distributions of both max-stable random fields and give sufficient conditions for the sample paths to be γ-Hölder continuous functions for some γ∈(0,1). For the second class, we consider a multiplicative factor model and a Poisson–Voronoï tessellation of Rd to construct new max-stable random fields. We provide explicit expressions for the pairwise distribution function.

Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions

We establish characterization results for the ergodicity of stationary symmetric $\alpha$-stable (S$\alpha$S) and $\alpha$-Frechet random fields. We show that the result of Samorodnitsky [Ann. Probab. 33 (2005) 1782-1803] remains valid in the multiparameter setting, that is, a stationary S$\alpha$S ($0<\alpha<2$) random field is ergodic (or, equivalently, weakly mixing) if and only if it is generated by a null group action. Similar results are also established for max-stable random fields. The key ingredient is the adaption of a characterization of positive/null recurrence of group actions by Takahashi [Kodai Math. Sem. Rep. 23 (1971) 131-143], which is dimension-free and different from the one used by Samorodnitsky.

Regular conditional distributions of continuous max-infinitely divisible random fields

This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$ given observations $\{\eta(t_i)=y_i,\ 1\leq i\leq k\}$. Our starting point is the point process representation of max-infinitely divisible processes by Giné, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process, introduce the notions of extremal function, sub-extremal function and hitting scenario associated to the constraints and derive the associated distributions. This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This formula extends a recent result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. This paper offers new tools and perspective or prediction in extreme value theory together with numerous potential applications.

Conditional sampling for spectrally discrete max-stable random fields

Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.

Conditional sampling for spectrally discrete max-stable random fields

Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.

Extremal shot noises, heavy tails and max-stable random fields

We consider the extremal shot noise defined by M(y) = sup{mh(y − x); (x, m) ∈ � }, whereis a Poisson point process on R d × (0, +∞) with intensity λ dxG (dm) and h : R d →( 0, +∞) is a measurable function. Extremal shot noises naturally appear in extreme value theory as a model for spatial extremes and serve as basic models for annual maxima of rainfall or for coverage field in telecommunications. In this work, we examine their properties such as boundedness, regularity and ergodicity. Connections with max-stable random fields are established: we prove a limit theorem when the distribution G is heavy-tailed and the intensity of points λ goes to infinity. We use a point process approach strongly connected to the Peak Over Threshold method used in extreme value theory. Properties of the limit max-stable random fields are also investigated.

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

Convergence of Distributions Generated by Stationary Stochastic Processes

Convergence of Distributions Generated by Stationary Stochastic Processes