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Advances in Statistical Modeling of Spatial Extremes

Raphaël Huser, Jennifer L. Wadsworth

arXiv:2007.00774v3stat.ME

TL;DR

Classical spatial-extremes models can be computationally difficult and may not capture the weakening dependence observed as events become more extreme. This review surveys flexible alternatives, including random-scale and conditional models, and concludes that several approaches can bridge asymptotic dependence regimes. It also notes that full-likelihood inference for popular max-stable models remains demanding in relatively low dimensions.

  • Problem

    Classical max-stable and Pareto models impose rigid dependence structures, while environmental data can show weakening dependence at more extreme levels; max-stable full likelihoods are also computationally demanding.

    Source: §2.2 · pp. 13–14, §3.1 · p. 19

  • Method

    The review examines random scale or location mixtures and the conditional spatial extremes model, and also discusses max-mixture and max-id approaches.

    Source: §6 · p. 46

  • Results

    The reviewed random scale or location mixtures and conditional model can bridge asymptotic dependence and independence regimes in a single parametrization.

    Source: §6 · p. 46

  • Takeaways & Limitations

    The review characterizes these alternatives as offering increased flexibility for capturing sub-asymptotic behavior while retaining precisely characterized limiting dependence structures.

    Source: §6 · p. 46

    Full-likelihood inference for popular max-stable models remains computationally demanding even at relatively low dimensions, around D ≈15–20.

    Source: §2.2 · p. 14

Abstract

from arXiv · show

The classical modeling of spatial extremes relies on asymptotic models (i.e., max-stable processes 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.

1 Introduction

Spatial extremes are important for risk assessment, but classical asymptotic models can be computationally difficult and too rigid to capture weakening dependence at finite levels. This review surveys recent flexible spatial models, including approaches that bridge asymptotic dependence classes and a conditional model that can be fitted with likelihood-based methods.

1.1 Motivation

  • The review addresses a gap in coverage of newer spatial-extremes topics by surveying flexible models that can bridge asymptotic dependence classes.It positions itself alongside reviews of classical extremes and sparse multivariate extremes, the latter of which it does not cover.Source: §1.1 · p. 2

1.2 Classical univariate and spatial extreme-value models

  • Spatial extremes support risk assessment across environmental and other fields, while classical max-stable and r-Pareto models face computational burdens and finite-level dependence constraints.Spatial models can borrow strength across locations, but max-stable likelihoods become difficult in high dimensions and spatial block maxima may not represent observed events directly.Source: §1.1 · p. 2, §1.2 · pp. 3–4, §1.3 · p. 5

1.3 Recent developments and paper outline

  • Environmental data often show weakening spatial dependence as events become more extreme, motivating sub-asymptotic models alongside asymptotic max-stable and r-Pareto processes.The conditional spatial extremes approach models behavior given a single site is large; it can represent varied dependence classes, including changes with distance, and supports likelihood-based fitting in reasonably large dimensions.Source: §1.3 · pp. 5–6
  • The paper reviews classical theory and inference, surveys sub-asymptotic and conditional models, illustrates selected approaches in two environmental applications, and closes with research perspectives.The applications concern high-threshold exceedances.Source: §1.3 · p. 6

2 Asymptotic models for spatial extremes

This section reviews marginal GEV and GP limits and their spatial counterparts, max-stable and r-Pareto processes, as models for characterizing spatial extremes and their practical limits in dependence modeling and likelihood-based inference. It also notes that r-Pareto processes allow events to be considered on their original scale when the shape parameter is constant across space.

2.1 Marginal modeling of extremes

  • GEV models describe block maxima and support extrapolation to longer periods, while GP models describe threshold exceedances and remain GP above higher thresholds with the same shape parameter.The GEV is max-stable; the GP scale changes with the threshold.Source: §2.1 · pp. 8–9
  • Spatial models borrow strength across sites to improve marginal estimation and enable prediction at unobserved locations, but conditional independence in latent Gaussian models can underestimate joint extremes.Hierarchical models can incorporate covariates and spatially correlated latent effects; their conditional-independence assumption is often unrealistic for joint extremes.Source: §2.1 · pp. 10–11

2.2 Max-stable processes

  • Max-stable processes are the possible non-degenerate limits of renormalized pointwise maxima, and their heavy-tailed storm amplitudes induce persistent asymptotic dependence.Their spectral construction represents the process as pointwise maxima of independent scale mixtures.Source: §2.2 · pp. 12–13
  • For max-stable models, full-likelihood inference becomes impractical at moderate dimensions, while pairwise likelihood is less efficient and harder to adapt to Bayesian inference.The full-likelihood formula has too many terms when D is roughly greater than 5–10, and implicit event-time methods remain demanding around D = 15–20.Source: §2.2 · pp. 13–15

2.3 r-Pareto processes

  • r-Pareto processes generalize spatial exceedances by conditioning on a homogeneous risk functional that defines event severity, and they have a threshold-stability property.The choice of risk functional can represent maxima, minima, spatial integrals, or values at selected sites.Source: §2.3 · p. 16
  • r-Pareto likelihoods require fewer exponent-function derivatives than max-stable likelihoods, enabling higher-dimensional inference, but multivariate distribution calculations and normalization constants can remain burdensome.de Fondeville and Davison (2018) proposed Quasi-Monte Carlo calculations and gradient scoring to address these costs; the gradient score avoids normalization constants.Source: §2.3 · p. 18
  • r-Pareto processes allow events to be considered on their original scale, provided the shape parameter ξ(s) is constant across space.Source: §2.3 · p. 19

3 Sub-asymptotic models for spatial extremes

Classical asymptotic models can miss weakening dependence as events become more extreme, motivating models that flexibly capture sub-asymptotic tails. The section reviews inverted max-stable and max-mixture models, random scale and location mixtures, and max-infinitely divisible models.

3.1 Asymptotic dependence classes

  • Asymptotic max-stable and Pareto processes remain dependent at increasingly extreme levels unless exactly independent, limiting their ability to represent spatially localized severe events.For Gaussian processes, fixed correlation also fixes the joint-tail decay rate, constraining sub-asymptotic behavior.Source: §3.1 · pp. 19–20

3.2 Inverted max-stable processes and max-mixture models

  • Inverted max-stable processes provide positively associated asymptotic independence models, with tail dependence determined by the max-stable process's extremal coefficient.Their likelihood inference commonly uses pairwise likelihood, as for max-stable processes.Source: §3.2 · p. 22
  • Max-mixtures combine max-stable and inverted max-stable components, allowing dependence class to vary with distance when the max-stable component has finite-range dependence.The mixture can improve sub-asymptotic fit, but is often heavily parameterized and its mixture parameter is difficult to estimate.Source: §3.2 · p. 23

3.3 Random scale mixtures, and related models

  • Pareto-tailed Gaussian scale mixtures are asymptotically dependent, while Weibull-tailed mixtures are asymptotically independent; the tail parameter adds flexibility at fixed correlation.Combining the tail forms yields an asymptotically independent model with an asymptotically dependent submodel on the parameter-space boundary.Source: §3.3 · p. 25
  • Gaussian location mixtures span dependence classes according to the mixing-tail regime: Pareto or Weibull tails with β < 1 give perfect dependence, exponential tails give dependence, and Weibull tails with β > 1 give independence.These models can be viewed as random scale mixtures after exponentiation, a monotone transformation that preserves dependence.Source: §3.3 · p. 26
  • Because a shared random scale persists across space, random scale mixtures cannot reach complete independence at long distances and are usually realistic only over relatively small domains.Their asymptotic dependence class is the same for every site pair.Source: §3.3 · pp. 27–28

3.4 Max-infinitely divisible processes

  • Weibull-tailed max-infinitely divisible models are asymptotically independent yet converge to an extremal-t max-stable model as β decreases to zero, with β controlling distance from that boundary.This provides flexibility for finite block maxima.Source: §3.4 · p. 30

4 Conditional spatial extremes model

The conditional spatial extremes model addresses high-dimensional inference and persistent long-range dependence in random scale constructions. It models the process conditional on an extreme at a site, allowing flexible dependence, including asymptotic independence and distance-varying dependence.

4.1 Background

  • The conditional spatial extremes model was introduced to address computational and long-domain limitations, and can capture both asymptotic dependence and independence, including changes in class with distance.It characterizes the process conditional on an extreme at a selected location, extending the conditional extreme-value approach to spatial processes.Source: §4.1 · pp. 30–31

4.2 Model

  • Distance-dependent normalization can yield non-degenerate conditional limits under asymptotic independence where a distance-invariant normalization may fail.The model specifies these normalizations as functions of displacement from the conditioning site.Source: §4.1 · p. 32, §4.2 · p. 33

4.3 Inference

  • Inference combines likelihoods across conditioning sites using a composite likelihood; it avoids censored likelihoods but can be slower because exceedances at multiple sites are counted repeatedly.For large numbers of locations, inference can combine a subset of conditioning sites.Source: §4.3 · p. 35

5 Environmental applications

The Dutch wind application compares flexible sub-asymptotic models with asymptotic alternatives, while the Irish temperature application uses a conditional spatial extremes model to assess dependence and estimate island-wide exceedance risks.

5.1 Dutch wind speed data

  • For Dutch winds, weak temporal extremal dependence at lag 1 rapidly vanished at longer lags and higher thresholds across stations.The analysis used 1,594 non-missing winter days at 30 stations, with days treated as independent replicates for fitting.Source: §5.1 · p. 36, Figure 4 · p. 37
  • The Dutch comparison fitted Gaussian scale-mixture variants and the Huser–Wadsworth hybrid alongside r-Pareto and Gaussian-copula models, using censored likelihood at threshold probability 0.95.Anisotropy was estimated first and then used to transform the station coordinates for the model fits.Source: §5.1 · pp. 37–38
  • The Huser and Wadsworth (2019) model had the best overall BIC, with the Gaussian scale mixture close behind; both improved on the Gaussian copula and r-Pareto process.The r-Pareto process fit was worse than the Gaussian copula, and diagnostics showed it overestimated dependence at high quantiles and underestimated it at lower quantiles.Source: §5.1 · pp. 39–40, Table 1 · p. 39

5.2 Irish temperature data

  • The conditional framework permits either asymptotic dependence class, but the choice must be made manually using likelihood values and goodness-of-fit diagnostics.The Irish data showed some weakening dependence at higher quantiles, consistent with either asymptotic independence or sub-asymptotic asymptotic dependence.Source: §5.2 · pp. 41–42
  • For Ireland, the selected conditional model had 10 parameters; composite likelihood used 30 of 178 conditioning sites while retaining information from all 178 locations.Each likelihood component conditioned on a site exceeding the 0.95 Laplace quantile.Source: §5.2 · pp. 42–43
  • In Ireland, estimated dependence remained positive across the island, with χ0.95(s1, s2) estimates around or above 0.3 and stronger dependence roughly southwest to northeast.The fitted model's α estimates were at least about 0.74 over the island, alongside β estimates near 1.Source: §5.2 · p. 43
  • Irish conditional-model simulations appeared consistent with the data, and fitted dependence estimates agreed satisfactorily with empirical estimates; modeled exceedance probabilities also agreed in-sample and permit extrapolation beyond observed values.The probability estimates extend to thresholds beyond the empirical distribution's upper endpoint.Source: §5.2 · pp. 43–44, Figure 8 · p. 45
  • The estimated 100-year return level for the island-wide maximum corresponds approximately to the 0.99998 Laplace quantile and the 543-year return level at an individual site.The island-wide return period is defined from the probability that at least one location exceeds the level on a summer day.Source: §5.2 · p. 45

6 Conclusion

The review covers spatial models designed to capture sub-asymptotic tail behavior, including approaches that bridge asymptotic dependence and independence. It also identifies trade-offs and open challenges in interpretation, computation, and modeling large domains.

6 Conclusion

  • Random scale or location mixtures and the conditional spatial extremes model can bridge asymptotic dependence and independence, while max-mixture and max-id approaches are also briefly discussed.Most reviewed models use common random factors and therefore cannot capture complete independence at large distances; the conditional approach avoids this limitation but is more heavily parameterized and harder to interpret unconditionally.Source: §6 · p. 46
  • Sub-asymptotic models are generally easier to handle in high dimensions than asymptotic counterparts, but likelihood-based fitting remains dimension-limited.Flexible random scale models and the Huser–Wadsworth model with censored likelihood apply up to about D = 30–50; the uncensored conditional model is currently limited to about D = 300–500.Source: §6 · p. 46
  • Truly higher-dimensional problems require sparse models, but the cited sparse multivariate Pareto framework does not apply to the paper’s asymptotic-independence or hybrid models.The review calls for relatively parsimonious models combining flexible tail and spatial decay, intuitive unconditional interpretation, and feasible large-dimensional inference.Source: §6 · pp. 46–47
  • Most extreme-value applications assume stationarity and isotropy, while large domains may require models that accommodate nonstationarity and flexible joint tail structures.The environmental applications incorporate geometric anisotropy, but the review identifies large-domain modeling as an area needing further research.Source: §6 · p. 47

Funding Information

The work was supported by a KAUST Office of Sponsored Research award.

Funding Information

  • The work was supported by the King Abdullah University of Science and Technology Office of Sponsored Research under Award No. OSR-CRG2017-3434.Source: Funding Information · p. 47
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