Spatial Extremes Research Articles

Rank-based estimation under asymptotic dependence and independence, with applications to spatial extremes

Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit non-parametric estimation and introduce rank-based M-estimators for parametric models that simultaneously work under asymptotic dependence and asymptotic independence, without requiring prior knowledge on which of the two regimes applies. Asymptotic normality of the proposed estimators is established under weak regularity conditions. We further show how bivariate estimators can be leveraged to obtain parametric estimators in spatial tail models, and again provide a thorough theoretical justification for our approach.

Predicting Future Ranked Statistics and Recorded Values for Some Statistical Distributions

Regional frequency analysis (AFR) brings together a variety of statistical methods aimed at predicting the behavior of extreme hydrological variables at ungauged sites. Regression techniques, geostatistical methods and classification are among the statistical tools frequently encountered in the literature. Methodologies based on these tools lead to regional models that offer a simple, but very useful description of the relationship between extreme hydrological variables and physiometeorological characteristics of a site. These regional models then make it possible to predict the behavior of variables of interest at places where no hydrological information is available. These methods are generally based on restrictive theoretical assumptions, including linearity and normality. These do not reflect the reality of natural phenomena. The general objectives of this paper are to identify the methods affected by these hypotheses, evaluate their impacts and propose improvements aimed at obtaining more realistic and fairer representations. Projection pursuit regression is a non-parametric method similar to generalized additive models and artificial neural networks that are considered in AFR to take into account the non-linearity of hydrological processes. In a comparative study, this paper shows that regression with revealing directions makes it possible to obtain more parsimonious models while preserving the same predictive power as the other nonparametric methods. Canonical Correlation Analysis (ACC) is used to create neighborhoods within which a model (e.g. multiple regression) is used to predict hydrologic variables at ungagged sites on the other hand, ACC strongly depends on the assumptions of normality and linearity. A new methodology for delineating neighborhoods is proposed in this paper and uses revealing direction regression to predict a reference point representing hydrological and physiometeorological information that is relevant to these groupings. The results show that the new methodology generalizes that of ACC, improves the homogeneity of neighborhoods and leads to better performance. In AFR, kriging techniques on transformed spaces are suggested in order to predict extreme hydrological variables. However, a transformation is required so that the hydrological variables of interest derive approximately from a multidimensional normal distribution. This transformation introduces a bias and leads to suboptimal predictions. Solutions have been proposed, but have not been tested in AFR. This paper proposes the approach of spatial copulas and shows that this approach provides satisfactory solutions to the problems encountered with kriging techniques. Max-stable processes are a theoretical formalization of spatial extremes and correspond to a more faithful representation of hydrological processes on the other hand; their characterization of extreme dependence poses technical problems which slow down their adoption. In this paper, the approximate Bayesian calculus is examined as a solution. The results of a simulation study show that the approximate Bayesian computation is superior to the standard approach of compound likelihood. In addition, this approach is more appropriate in order to take into account specification errors.

Maximum composite likelihood estimation for spatial extremes models of Brown–Resnick type with application to precipitation data

AbstractIn this study, we consider the maximum composite likelihood estimator for spatial extremes model class of Brown–Resnick type. The composite likelihood is constructed based on the weighted tail empirical process. It is shown that the proposed estimator is consistent and asymptotically normal under some regularity conditions fulfilled by the model class. We conduct Monte Carlo simulations to evaluate the estimator and apply it to the analysis of a precipitation data set.

Flexible and Fast Spatial Return Level Estimation Via a Spatially Fused Penalty

Spatial extremes are common for climate data as the observations are usually referenced by geographic locations and dependent when they are nearby. An important goal of extremes modeling is to estimate the T-year return level. Among the methods suitable for modeling spatial extremes, perhaps the simplest and fastest approach is the spatial generalized extreme value (GEV) distribution and the spatial generalized Pareto distribution (GPD) that assume marginal independence and only account for dependence through the parameters. Despite the simplicity, simulations have shown that return level estimation using the spatial GEV and spatial GPD still provides satisfactory results compared to max-stable processes, which are asymptotically justified models capable of representing spatial dependence among extremes. However, the linear functions used to model the spatially varying coefficients are restrictive and may be violated. We propose a flexible and fast approach based on the spatial GEV and spatial GPD by introducing fused lasso and fused ridge penalty for parameter regularization. This enables improved return level estimation for large spatial extremes compared to the existing methods. Supplemental files for this article are available online.

Spatial deformation for nonstationary extremal dependence

AbstractModeling the extremal dependence structure of spatial data is considerably easier if that structure is stationary. However, for data observed over large or complicated domains, nonstationarity will often prevail. Current methods for modeling nonstationarity in extremal dependence rely on models that are either computationally difficult to fit or require prior knowledge of covariates. Sampson and Guttorp (1992) proposed a simple technique for handling nonstationarity in spatial dependence by smoothly mapping the sampling locations of the process from the original geographical space to a latent space where stationarity can be reasonably assumed. We present an extension of this method to a spatial extremes framework by considering least squares minimization of pairwise theoretical and empirical extremal dependence measures. Along with some practical advice on applying these deformations, we provide a detailed simulation study in which we propose three spatial processes with varying degrees of nonstationarity in their extremal and central dependence structures. The methodology is applied to Australian summer temperature extremes and UK precipitation to illustrate its efficacy compared with a naive modeling approach.

Hierarchical Transformed Scale Mixtures for Flexible Modeling of Spatial Extremes on Datasets With Many Locations

–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.

Conditional modelling of spatio-temporal extremes for Red Sea surface temperatures

Recent extreme value theory literature has seen significant emphasis on the modelling of spatial extremes, with comparatively little consideration of spatio-temporal extensions. This neglects an important feature of extreme events: their evolution over time. Many existing models for the spatial case are limited by the number of locations they can handle; this impedes extension to space–time settings, where models for higher dimensions are required. Moreover, the spatio-temporal models that do exist are restrictive in terms of the range of extremal dependence types they can capture. Recently, conditional approaches for studying multivariate and spatial extremes have been proposed, which enjoy benefits in terms of computational efficiency and an ability to capture both asymptotic dependence and asymptotic independence. We extend this class of models to a spatio-temporal setting, conditioning on the occurrence of an extreme value at a single space–time location. We adopt a composite likelihood approach for inference, which combines information from full likelihoods across multiple space–time conditioning locations. We apply our model to Red Sea surface temperatures, show that it fits well using a range of diagnostic plots, and demonstrate how it can be used to assess the risk of coral bleaching attributed to high water temperatures over consecutive days.

Advances in statistical modeling of spatial extremes

AbstractThe classical modeling of spatial extremes relies on asymptotic models (i.e., max‐stable orr‐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 DataData: Types and Structure > Time Series, Stochastic Processes, and Functional DataStatistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods

Max‐infinitely divisible models and inference for spatial extremes

AbstractFor many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max‐stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max‐infinitely divisible processes, which extend max‐stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max‐infinitely divisible models, which relax the max‐stability property but remain close to some popular max‐stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max‐stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student‐t copula process and its max‐stable limit, even for large block sizes.

"We Have Had It Up To Here": Murder, Civil Disorder, and Civil Rights in a Western Pennsylvanian Industrial Town


 In 1970, citizens of New Castle, Pennsylvania, a small industrial city an hour north of Pittsburgh, responded to the racially motivated murder of a local black Vietnam veteran in that city with vandalism and firebombing that forced the mayor to place the city under a state of emergency for three days. The series of exchanges preceding and following the murder reveals much about that city's history, and how several factors influenced local forms of racism. Existing scholarship has focused on racialized policies and practices in two spatial extremes—large cities and small towns—while this analysis seeks to illustrate how local, regional, and national influences shaped what forms of race-based policies and practices in spaces between these municipal extremes were permissible. Beyond place and space, this research contributes to a different set of conversations about the ways identity and community are articulated through the actions of individuals and groups, and how those understandings are shaped by individual and collective memory. This analysis begins by situating Ronald Mitchell's murder within the historical context of 1970s New Castle, broadens to place New Castle amid much larger and smaller municipalities across the country, and briefly contours some historical forces that shaped racism in policy or practice across time. I illustrate how federal, state, and local authorities responded to crises comparable to that which occurred in response to Mitchell's murder in the 1960s, and highlight how the underlying causes identified during investigations by those bodies manifested throughout the city's history and at the scene of Mitchell's murder. I also explore the role of institutions and memory in shaping knowledge and use of the past and build upon earlier scholarship in asserting their centrality to equitable futures.

Stochastic simulation of streamflow and spatial extremes: a continuous, wavelet-based approach

Abstract. Stochastically generated streamflow time series are used for various water management and hazard estimation applications. They provide realizations of plausible but as yet unobserved streamflow time series with the same temporal and distributional characteristics as the observed data. However, the representation of non-stationarities and spatial dependence among sites remains a challenge in stochastic modeling. We investigate whether the use of frequency-domain instead of time-domain models allows for the joint simulation of realistic, continuous streamflow time series at daily resolution and spatial extremes at multiple sites. To do so, we propose the stochastic simulation approach called Phase Randomization Simulation using wavelets (PRSim.wave) which combines an empirical spatio-temporal model based on the wavelet transform and phase randomization with the flexible four-parameter kappa distribution. The approach consists of five steps: (1) derivation of random phases, (2) fitting of the kappa distribution, (3) wavelet transform, (4) inverse wavelet transform, and (5) transformation to kappa distribution. We apply and evaluate PRSim.wave on a large set of 671 catchments in the contiguous United States. We show that this approach allows for the generation of realistic time series at multiple sites exhibiting short- and long-range dependence, non-stationarities, and unobserved extreme events. Our evaluation results strongly suggest that the flexible, continuous simulation approach is potentially valuable for a diverse range of water management applications where the reproduction of spatial dependencies is of interest. Examples include the development of regional water management plans, the estimation of regional flood or drought risk, or the estimation of regional hydropower potential. Highlights. Stochastic simulation of continuous streamflow time series using an empirical, wavelet-based, spatio-temporal model in combination with the parametric kappa distribution. Generation of stochastic time series at multiple sites showing temporal short- and long-range dependence, non-stationarities, and spatial dependence in extreme events. Implementation of PRSim.wave in R package PRSim: Stochastic Simulation of Streamflow Time Series using Phase Randomization.

Assessing the risk of disruption of wind turbine operations in Saudi Arabia using Bayesian spatial extremes

Saudi Arabia has been seeking to reduce its dependence on oil by diversifying its energy portfolio, including the largely underused energy potential from wind. However, extreme winds can possibly disrupt the wind turbine operations, thus preventing the stable and continuous production of wind energy. In this study, we assess the risk of disruptions of wind turbine operations, based on return levels with a hierarchical spatial extreme modeling approach for wind speeds in Saudi Arabia. Using a unique Weather Research and Forecasting dataset, we provide the first high-resolution risk assessment of wind extremes under spatial non-stationarity over the country. We account for the spatial dependence with a multivariate intrinsic autoregressive prior at the latent Gaussian process level. The computational efficiency is greatly improved by parallel computing on subregions from spatial clustering, and the maps are smoothed by fitting the model to cluster neighbors. Under the Bayesian hierarchical framework, we measure the uncertainty of return levels from the posterior Markov chain Monto Carlo samples, and produce probability maps of return levels exceeding the cut-out wind speed of wind turbines within their lifetime. The probability maps show that locations in the South of Saudi Arabia and near the Red Sea and the Persian Gulf are at very high risk of disruption of wind turbine operations.

Graphical Models for Extremes

SummaryConditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk of rare events through asymptotically justified limit models such as max-stable and multivariate Pareto distributions. Statistical modelling in this field has been limited to moderate dimensions so far, partly owing to complicated likelihoods and a lack of understanding of the underlying probabilistic structures. We introduce a general theory of conditional independence for multivariate Pareto distributions that enables the definition of graphical models and sparsity for extremes. A Hammersley–Clifford theorem links this new notion to the factorization of densities of extreme value models on graphs. For the popular class of Hüsler–Reiss distributions we show that, similarly to the Gaussian case, the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices. New parametric models can be built in a modular way and statistical inference can be simplified to lower dimensional marginals. We discuss learning of minimum spanning trees and model selection for extremal graph structures, and we illustrate their use with an application to flood risk assessment on the Danube river.

Composite likelihood methods for histogram-valued random variables

Symbolic data analysis has been proposed as a technique for summarising large and complex datasets into a much smaller and tractable number of distributions—such as random rectangles or histograms—each describing a portion of the larger dataset. Recent work has developed likelihood-based methods that permit fitting models for the underlying data while only observing the distributional summaries. However, while powerful, when working with random histograms this approach rapidly becomes computationally intractable as the dimension of the underlying data increases. We introduce a composite-likelihood variation of this likelihood-based approach for the analysis of random histograms in K dimensions, through the construction of lower-dimensional marginal histograms. The performance of this approach is examined through simulated and real data analysis of max-stable models for spatial extremes using millions of observed datapoints in more than $$K=100$$ dimensions. Large computational savings are available compared to existing model fitting approaches.

Causal Mechanism of Extreme River Discharges in the Upper Danube Basin Network

SummaryExtreme hydrological events in the Danube river basin may severely impact human populations, aquatic organisms and economic activity. One often characterizes the joint structure of extreme events by using the theory of multivariate and spatial extremes and its asymptotically justified models. There is interest, however, in cascading extreme events and whether one event causes another. We argue that an improved understanding of the mechanism underlying severe events is achieved by combining extreme value modelling and causal discovery. We construct a causal inference method relying on the notion of the Kolmogorov complexity of extreme conditional quantiles. Tail quantities are derived by using multivariate extreme value models, and causal-induced asymmetries in the data are explored through the minimum description length principle. Our method CausEV for causality for extreme values uncovers causal relationships between summer extreme river discharges in the upper Danube basin and finds significant causal links between the Danube and its Alpine tributary Lech.

An Alternative to PCA for Estimating Dominant Patterns of Climate Variability and Extremes, with Application to U.S. and China Seasonal Rainfall

Floods and droughts are driven, in part, by spatial patterns of extreme rainfall. Heat waves are driven by spatial patterns of extreme temperature. It is therefore of interest to design statistical methodologies that allow the rapid identification of likely patterns of extreme rain or temperature from observed historical data. The standard work-horse for the rapid identification of patterns of climate variability in historical data is Principal Component Analysis (PCA) and its variants. But PCA optimizes for variance not spatial extremes, and so there is no particular reason why the first PCA spatial pattern should identify, or even approximate, the types of patterns that may drive floods, droughts or heatwaves, even if the linear assumptions underlying PCA are correct. We present an alternative pattern identification algorithm that makes the same linear assumptions as PCA, but which can be used to explicitly optimize for spatial extremes. We call the method Directional Component Analysis (DCA), since it involves introducing a preferred direction, or metric, such as “sum of all points in the spatial field”. We compare the first PCA and DCA spatial patterns for U.S. and China winter and summer rainfall anomalies, using the sum metric for the definition of DCA in order to focus on total rainfall anomaly over the domain. In three out of four of the examples the first DCA spatial pattern is more uniform over a wide area than the first PCA spatial pattern and as a result is more obviously relevant to large-scale flooding or drought. Also, in all cases the definitions of PCA and DCA result in the first PCA spatial pattern having the larger explained variance of the two patterns, while the first DCA spatial pattern, when scaled appropriately, has a higher likelihood and greater total rainfall anomaly, and indeed is the pattern with the highest total rainfall anomaly for a given likelihood. The first DCA spatial pattern is arguably the best answer to the question: what single spatial pattern is most likely to drive large total rainfall anomalies in the future? It is also simpler to calculate than PCA. In combination PCA and DCA patterns yield more insight into rainfall variability and extremes than either pattern on its own.

Exploration and Inference in Spatial Extremes Using Empirical Basis Functions

Statistical methods for inference on spatial extremes of large datasets are yet to be developed. Motivated by standard dimension reduction techniques used in spatial statistics, we propose an approach based on empirical basis functions to explore and model spatial extremal dependence. Based on a low-rank max-stable model, we propose a data-driven approach to estimate meaningful basis functions using empirical pairwise extremal coefficients. These spatial empirical basis functions can be used to visualize the main trends in extremal dependence. In addition to exploratory analysis, we describe how these functions can be used in a Bayesian hierarchical model to model spatial extremes of large datasets. We illustrate our methods on extreme precipitations in eastern USA. Supplementary materials accompanying this paper appear online

Improved Return Level Estimation via a Weighted Likelihood, Latent Spatial Extremes Model

Uncertainty in return level estimates for rare events, like the intensity of large rainfall events, makes it difficult to develop strategies to mitigate related hazards, like flooding. Latent spatial extremes models reduce the uncertainty by exploiting spatial dependence in statistical characteristics of extreme events to borrow strength across locations. However, these estimates can have poor properties due to model misspecification: Many latent spatial extremes models do not account for extremal dependence, which is spatial dependence in the extreme events themselves. We improve estimates from latent spatial extremes models that make conditional independence assumptions by proposing a weighted likelihood that uses the extremal coefficient to incorporate information about extremal dependence during estimation. This approach differs from, and is simpler than, directly modeling the spatial extremal dependence; for example, by fitting a max-stable process, which is challenging to fit to real, large datasets. We adopt a hierarchical Bayesian framework for inference, use simulation to show the weighted model provides improved estimates of high quantiles, and apply our model to improve return level estimates for Colorado rainfall events with 1% annual exceedance probability. Supplementary materials accompanying this paper appear online.

Sampling sup‐normalized spectral functions for Brown–Resnick processes

Sup‐normalized spectral functions form building blocks of max‐stable and Pareto processes and therefore play an important role in modelling spatial extremes. For one of the most popular examples, the Brown–Resnick process, simulation is not straightforward. In this paper, we generalize two approaches for simulation via Markov chain Monte Carlo methods and rejection sampling by introducing new classes of proposal densities. In both cases, we provide an optimal choice of the proposal density with respect to sampling efficiency. The performance of the procedures is demonstrated in an example.

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.