A Comparative Tour through the Simulation Algorithms for Max-Stable Processes

Oesting, Marco; Strokorb, Kirstin

doi:10.1214/20-sts820

Cited by 4 publications

(4 citation statements)

References 38 publications

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“…In Figure 3 we present a simulation of the Brown-Resnick random field for H = 0.5 on Λ 2 50 . The simulation of these fields is complex; see the discussion in the recent overview paper Oesting and Strokorb [39]. For the results in Figure 3 and Table 1 we use the algorithm from Liu et al [35] which leads to a perfect simulation of the field.…”

Section: Example: the Brown-resnick Random Fieldmentioning

confidence: 99%

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Whittle estimation based on the extremal spectral density of a heavy-tailed random field

Damek,

Mikosch,

Zhao

et al. 2022

Preprint

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We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random field and regularly varying max-moving averages.

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Section: Example: the Brown-resnick Random Fieldmentioning

confidence: 99%

“…Often the naive approximation (6.2) is employed, or the simulation technique is not explicitly mentioned. Oesting and Strokorb [39] give an overview on simulation techniques for max-stable processes and point at various problems in this context, in particular when using (6.2).…”

Section: Example: the Brown-resnick Random Fieldmentioning

confidence: 99%

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

Damek,

Mikosch,

Zhao

et al. 2022

Preprint

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“…For the subclass of max-stable models, there are essentially two types of generic simulation algorithm that have been proposed based on variants of model representations stemming from (1) (Oesting and Strokorb, 2019). These algorithms exploit the specific structure of max-stable processes-the so-called spectral representation (de Haan, 1984), whereby on the unit Fréchet scale (i.e., Pr{Z(s) > z} = 1/z, z > 0) the points of the Poisson process {η i ; i = 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Exact Simulation of Max-Infinitely Divisible Processes

Zhong¹,

Huser²,

Opitz³

2021

Preprint

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Max-infinitely divisible (max-id) processes play a central role in extreme-value theory and include the subclass of all max-stable processes. They allow for a constructive representation based on the componentwise maximum of random functions drawn from a Poisson point process defined on a suitable functions space. Simulating from a max-id process is often difficult due to its complex stochastic structure, while calculating its joint density in high dimensions is often numerically infeasible. Therefore, exact and efficient simulation techniques for maxid processes are useful tools for studying the characteristics of the process and for drawing statistical inferences. Inspired by the simulation algorithms for max-stable processes, we here develop theory and algorithms to generalize simulation approaches tailored for certain flexible (existing or new) classes of max-id processes. Efficient simulation for a large class of models can be achieved by implementing an adaptive rejection sampling scheme to sidestep a numerical integration step in the algorithm. We present the results of a simulation study highlighting that our simulation algorithm works as expected and is highly accurate and efficient, such that it clearly outperforms customary approximate sampling schemes. As a byproduct, we also develop here new max-id models, which can be represented as pointwise maxima of general location scale mixtures, and which possess flexible tail dependence structures capturing a wide range of asymptotic dependence scenarios.

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“…The stochastic representation (1) involving an infinite number of processes over which the pointwise maximum is taken also makes simulation cumbersome. While various types of approximate and exact simulation algorithms have been developed for certain max-stable families (Schlather, 2002;Oesting et al, 2012;Thibaud and Opitz, 2015;Dombry et al, 2016;Oesting and Strokorb, 2022), conditional simulation remains computationally laborious (Dombry et al, 2013;Oesting and Schlather, 2013) and does not scale well with the dimension. To construct more "generic" stochastic generators for spatial extremes, it is also possible to directly exploit generative artificialintelligence (AI) techniques from the machine learning literature, such as Generative Adversarial Networks (GANs; see, e.g., Boulaguiem et al, 2022) which bypass the need to specify a parametric dependence structure for extremes at the expense of completely abandoning any known structure, or Variational Autoencoders (VAEs; see, e.g., Lafon et al, 2023;Zhang et al, 2023a), though these recent machine-learning-based approaches are sometimes difficult to train, are always data-hungry, and are often challenging to study from a theoretical perspective, e.g., in terms of the approximation quality of the trained generator, or its ability to accurately reproduce joint tail decay rates.…”

Section: Introductionmentioning

confidence: 99%

Advances in statistical modeling of spatial extremes

Huser

Wadsworth

2020

WIREs Computational Stats

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The classical modeling of spatial extremes relies on asymptotic models (i.e., max-stable or r-Pareto processes) for block maxima or peaks over high thresholds, respectively. However, at finite levels, empirical evidence often suggests that such asymptotic models are too rigidly constrained, and that they do not adequately capture the frequent situation where more severe events tend to be spatially more localized. In other words, these asymptotic models have a strong tail dependence that persists at increasingly high levels, while data usually suggest that it should weaken instead. Another well-known limitation of classical spatial extremes models is that they are either computationally prohibitive to fit in high dimensions, or they need to be fitted using less efficient techniques. In this review paper, we describe recent progress in the modeling and inference for spatial extremes, focusing on new models that have more flexible tail structures that can bridge asymptotic dependence classes, and that are more easily amenable to likelihood-based inference for large datasets. In particular, we discuss various types of random scale constructions, as well as the conditional spatial extremes model, which have recently been getting increasing attention within the statistics of extremes community. We illustrate some of these new spatial models on two different environmental applications. This article is categorized under: Data: Types and Structure > Image and Spatial Data Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods K E Y W O R D S asymptotic dependence and independence, extreme-value theory, max-stable process, Pareto process, random scale mixture

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A Comparative Tour through the Simulation Algorithms for Max-Stable Processes

Cited by 4 publications

References 38 publications

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

Exact Simulation of Max-Infinitely Divisible Processes

Advances in statistical modeling of spatial extremes

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