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Advances in statistical modeling of spatial extremes

Advanced Review

Jennifer L. Wadsworth, Raphaël Huser

Published Online: Nov 20 2020

DOI: 10.1002/wics.1537

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Abstract 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

Left: generalized extreme‐value (GEV) density with parameters μ = 0, σ = 1, and ξ = − 0.5 (red), ξ = 0 (black), and ξ = 0.5 (blue). Right: generalized Pareto (GP) density with parameters τ = 1 and ξ = − 0.5 (red), ξ = 0 (black), and ξ = 0.5 (blue)

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Left: model‐based estimate of Pr{max_{1 ≤ j ≤ D} X(s_j) > v} (solid line, estimates based on bootstrapped parameter values in gray), and empirical values from the data (blue crosses). Right: estimates of χ_0.95(s₁, s₂) (darker points) and χ_0.99(s₁, s₂) (lighter points) against distance in coordinates transformed to account for anisotropy (dots), and estimates from fitted model (red lines)

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Pairwise plots of data (black) and simulations (red) conditioning upon the randomly‐selected site 99 exceeding its 0.95 quantile. Pairs of sites are also randomly selected and displayed in the panel heading

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Left: marginal 95% quantiles at the 178 grid locations. Right: estimates of χ_0.95(s₁, s₂) (dark points) and χ_0.99(s₁, s₂) (light points) against distance h = ∥s₁ − s₂∥ in units of latitude. Lines represent kernel smoothed estimates at u = 0.95 (thick line) and u = 0.99 (thin line)

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Top: coefficient χ_u(s₁, s₂) plotted for u = 0.95 (left) and u = 0.99 (right) against the transformed distance . Black dots are empirical estimates for all pairs of stations, while solid curves are the Gaussian copula model (red), the Huser et al. (2017) model with β = 0 (blue) and β > 0 (orange), the Huser and Wadsworth (2019) model (purple), and the r‐Pareto process (green). Bottom: coefficient χ_u(s₁, s₂) (left) for a pair of sites at moderate distance from each other (red dots in Figure 4), and probability Pr{max_{1 ≤ j ≤ D} U_t(s_j) > u} (right), plotted for various thresholds u ∈ (0.8,1). Black and colored curves are as in the top panels. Gray shaded areas are 50, 90, 95% (darker to lighter) pointwise confidence bands for empirical estimates. Vertical dashed lines represent the threshold u = 0.95 used for fitting using the censored likelihood approach

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Left: topographic map of the Netherlands (study region) and neighboring countries, with monitoring stations indicated by dots. Red dots are stations selected to display model diagnostics in Figure 5. Right: extremogram Pr{U_t + h(s₁₄) > 0.95 | U_t(s₁₄) > 0.95} plotted against time lag h = 1, 2, …, 20, for the 14th station s₁₄ (with coordinates 6.575°E, 52.75°N, shown in red on the left panel). The horizontal gray line is a bootstrap 95% upper confidence bound under independence

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Subasymptotic χ (left) and η (right) measures, namely χ_u(s₁, s₂) in (16) and , respectively, plotted with respect to quantile level u ∈ (0, 1), as well as the limiting quantities χ(s₁, s₂) = lim_u → 1 χ_u(s₁, s₂) and η(s₁, s₂) = lim_u → 1 η_u(s₁, s₂) (small dots), for a Gaussian process Y(s) and correlation 0.2 (blue), 0.5 (red), and 0.8 (black) for the random vector {Y(s₁), Y(s₂)}^T

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Top row: independent realizations (gray) from the Gaussian scale mixture X(s) = t{RW(s)}, , with R ∼ Pareto(5) independent of the standard Gaussian process W(s) with correlation function ρ(s₁, s₂) = exp{−(| s₁ − s₂| /0.4)^1.5}, and marginally transformed through t(⋅) such that X(s) is on the unit Pareto scale. Highlighted curves are pointwise maxima (first column, black) and the three largest r‐exceedances based on the risk functionals (second column, red), (third column, purple), (fourth column, blue), and r(X) = X(0.5) (fifth column, orange). The same random seed was used, so some r‐exceedances may be identical in different panels. Bottom row: three independent realizations from the corresponding limiting (extremal‐t) max‐stable process of the form (9) (first column) and r‐Pareto processes of the form (13) (second to fifth columns). For better visualization, processes in the top panels are displayed on a log‐scale, while in the bottom panels, max‐stable processes are plotted on the standard Gumbel scale, and r‐Pareto processes have been transformed through the function t(x) = log{1 + x(e − 1)} such that t(0) = 0 and t(1) = 1. For r‐exceedances and r‐Pareto processes, thicker curves mean larger r(X)

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