Many environmental processes exhibit weakening spatial dependence as events become more extreme. Well-known limiting models, such as max-stable or generalized Pareto processes, cannot capture this, ...which can lead to a preference for models that exhibit a property known as asymptotic independence. However, weakening dependence does not automatically imply asymptotic independence, and whether the process is truly asymptotically (in)dependent is usually far from clear. The distinction is key as it can have a large impact upon extrapolation, that is, the estimated probabilities of events more extreme than those observed. In this work, we present a single spatial model that is able to capture both dependence classes in a parsimonious manner, and with a smooth transition between the two cases. The model covers a wide range of possibilities from asymptotic independence through to complete dependence, and permits weakening dependence of extremes even under asymptotic dependence. Censored likelihood-based inference for the implied copula is feasible in moderate dimensions due to closed-form margins. The model is applied to oceanographic datasets with ambiguous true limiting dependence structure. Supplementary materials for this article are available online.
Advances in statistical modeling of spatial extremes Huser, Raphaël; Wadsworth, Jennifer L.
Wiley interdisciplinary reviews. Computational statistics,
January/February 2022, 2022-01-00, 20220101, Letnik:
14, Številka:
1
Journal Article
Recenzirano
Odprti dostop
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
Classical modeling of spatial extremes relies on asymptotic models for block maxima (max‐stable processes) or threshold exceedances (r‐Pareto processes). These approaches and more flexible subasymptotic models are presented in this paper.
Abstract A geometric representation for multivariate extremes, based on the shapes of scaled sample clouds in light-tailed margins and their so-called limit sets, has recently been shown to connect ...several existing extremal dependence concepts. However, these results are purely probabilistic, and the geometric approach itself has not been fully exploited for statistical inference. We outline a method for parametric estimation of the limit set shape, which includes a useful non-/semi-parametric estimate as a pre-processing step. More fundamentally, our approach provides a new class of asymptotically motivated statistical models for the tails of multivariate distributions, and such models can accommodate any combination of simultaneous or non-simultaneous extremes through appropriate parametric forms for the limit set shape. Extrapolation further into the tail of the distribution is possible via simulation from the fitted model. A simulation study confirms that our methodology is very competitive with existing approaches and can successfully allow estimation of small probabilities in regions where other methods struggle. We apply the methodology to two environmental datasets, with diagnostics demonstrating a good fit.
Abstract-Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, ...even in moderate dimensions, due to the necessity of repeatedly evaluating the multivariate Gaussian distribution function. In this work, we attempt to achieve truly high-dimensional inference for extremes of spatial processes, while retaining the desirable flexibility in the tail dependence structure, by modifying an established class of models based on scale mixtures Gaussian processes. We show that the desired extremal dependence properties from the original models are preserved under the modification, and demonstrate that the corresponding Bayesian hierarchical model does not involve the expensive computation of the multivariate Gaussian distribution function. We fit our model to exceedances of a high threshold, and perform coverage analyses and cross-model checks to validate its ability to capture different types of tail characteristics. We use a standard adaptive Metropolis algorithm for model fitting, and further accelerate the computation via parallelization and Rcpp. Lastly, we apply the model to a dataset of a fire threat index on the Great Plains region of the United States, which is vulnerable to massively destructive wildfires. We find that the joint tail of the fire threat index exhibits a decaying dependence structure that cannot be captured by limiting extreme value models.
Supplementary materials
for this article are available online.
Perchlorates have been identified on the surface of Mars. This has prompted speculation of what their influence would be on habitability. We show that when irradiated with a simulated Martian UV ...flux, perchlorates become bacteriocidal. At concentrations associated with Martian surface regolith, vegetative cells of Bacillus subtilis in Martian analogue environments lost viability within minutes. Two other components of the Martian surface, iron oxides and hydrogen peroxide, act in synergy with irradiated perchlorates to cause a 10.8-fold increase in cell death when compared to cells exposed to UV radiation after 60 seconds of exposure. These data show that the combined effects of at least three components of the Martian surface, activated by surface photochemistry, render the present-day surface more uninhabitable than previously thought, and demonstrate the low probability of survival of biological contaminants released from robotic and human exploration missions.
Multivariate peaks over thresholds models Rootzén, Holger; Segers, Johan; L. Wadsworth, Jennifer
Extremes (Boston),
03/2018, Letnik:
21, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Multivariate peaks over thresholds modelling based on generalized Pareto distributions has up to now only been used in few and mostly two-dimensional situations. This paper contributes theoretical ...understanding, models which can respect physical constraints, inference tools, and simulation methods to support routine use, with an aim at higher dimensions. We derive a general point process model for extreme episodes in data, and show how conditioning the distribution of extreme episodes on threshold exceedance gives four basic representations of the family of generalized Pareto distributions. The first representation is constructed on the real scale of the observations. The second one starts with a model on a standard exponential scale which is then transformed to the real scale. The third and fourth representations are reformulations of a spectral representation proposed in Ferreira and de Haan (Bernoulli
20
(4), 1717–1737,
2014
). Numerically tractable forms of densities and censored densities are found and give tools for flexible parametric likelihood inference. New simulation algorithms, explicit formulas for probabilities and conditional probabilities, and conditions which make the conditional distribution of weighted component sums generalized Pareto are derived.
The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose ...components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.
The conditional extremes framework allows for event-based stochastic modeling of dependent extremes, and has recently been extended to spatial and spatio-temporal settings. After standardizing the ...marginal distributions and applying an appropriate linear normalization, certain non-stationary Gaussian processes can be used as asymptotically-motivated models for the process conditioned on threshold exceedances at a fixed reference location and time. In this work, we adapt existing conditional extremes models to allow for the handling of large spatial datasets. This involves specifying the model for spatial observations at
d
locations in terms of a latent
m
≪
d
dimensional Gaussian model, whose structure is specified by a Gaussian Markov random field. We perform Bayesian inference for such models for datasets containing thousands of observation locations using the integrated nested Laplace approximation, or INLA. We explain how constraints on the spatial and spatio-temporal Gaussian processes, arising from the conditioning mechanism, can be implemented through the latent variable approach without losing the computationally convenient Markov property. We discuss tools for the comparison of models via their posterior distributions, and illustrate the flexibility of the approach with gridded Red Sea surface temperature data at over 6,000 observed locations. Posterior sampling is exploited to study the probability distribution of cluster functionals of spatial and spatio-temporal extreme episodes.
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 higherdimensional 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.
When assessing the impact of extreme events, it is often not just a single component, but the combined behavior of several components which is important. Statistical modeling using multivariate ...generalized Pareto (GP) distributions constitutes the multivariate analogue of univariate peaks over thresholds modeling, which is widely used in finance and engineering. We develop general methods for construction of multivariate GP distributions and use them to create a variety of new statistical models. A censored likelihood procedure is proposed to make inference on these models, together with a threshold selection procedure, goodness-of-fit diagnostics, and a computationally tractable strategy for model selection. The models are fitted to returns of stock prices of four UK-based banks and to rainfall data in the context of landslide risk estimation. Supplementary materials and codes are available online.
PDF
