Extremal Dependence Structure Research Articles

Spatial Extremal Modelling: A Case Study on the Interplay Between Margins and Dependence

ABSTRACTIt is no secret that statistical modelling often involves making simplifying assumptions when attempting to study complex stochastic phenomena. Spatial modelling of extreme values is no exception, with one of the most common such assumptions being stationarity in the marginal and/or dependence features. If non‐stationarity has been detected in the marginal distributions, it is tempting to try to model this while assuming stationarity in the dependence, without necessarily putting this latter assumption through thorough testing. However, margins and dependence are often intricately connected and the detection of non‐stationarity in one feature might affect the detection of non‐stationarity in the other. This work is an in‐depth case study of this interrelationship, with a particular focus on a spatio‐temporal environmental application exhibiting well‐documented marginal non‐stationarity. Specifically, we compare and contrast four different marginal detrending approaches in terms of our post‐detrending ability to detect temporal non‐stationarity in the spatial extremal dependence structure of a sea surface temperature dataset from the Red Sea.

Convolution Bounds on Quantile Aggregation

Advancing Risk Assessment: New Ways To Compute Quantile Aggregation This issue features a pivotal study on quantile aggregation amid dependence uncertainty, an area critical to finance, risk management, and statistics. The authors introduce “convolution bounds,” derived from a recent inf-convolution formula of quantiles and related risk measures. The obtained analytical tools unify existing results and enhance the understanding of quantile methods by providing general, sharp, and computationally efficient solutions. The results offer insights into the extremal dependence structures, with several implications in risk management and economic analysis applications. For more detailed insights, read the full paper, “Convolution Bounds on Quantile Aggregation” (reference: [insert reference]).

Efficient Modeling of Spatial Extremes over Large Geographical Domains

Various natural phenomena exhibit spatial extremal dependence at short spatial distances. However, existing models proposed in the spatial extremes literature often assume that extremal dependence persists across the entire domain. This is a strong limitation when modeling extremes over large geographical domains, and yet it has been mostly overlooked in the literature. We here develop a more realistic Bayesian framework based on a novel Gaussian scale mixture model, with the Gaussian process component defined by a stochastic partial differential equation yielding a sparse precision matrix, and the random scale component modeled as a low-rank Pareto-tailed or Weibull-tailed spatial process determined by compactly-supported basis functions. We show that our proposed model is approximately tail-stationary and that it can capture a wide range of extremal dependence structures. Its inherently sparse structure allows fast Bayesian computations in high spatial dimensions based on a customized Markov chain Monte Carlo algorithm prioritizing calibration in the tail. We fit our model to analyze heavy monsoon rainfall data in Bangladesh. Our study shows that our model outperforms natural competitors and that it fits precipitation extremes well. We finally use the fitted model to draw inference on long-term return levels for marginal precipitation and spatial aggregates.

Flexible Modeling of Nonstationary Extremal Dependence using Spatially Fused LASSO and Ridge Penalties

Statistical modeling of a nonstationary spatial extremal dependence structure is challenging. Parametric max-stable processes (MSPs) are common choices for modeling spatially indexed block maxima, where an assumption of stationarity is usual to make inference feasible. However, this assumption is unrealistic for data observed over a large or complex domain. We develop a computationally efficient method for estimating extremal dependence using a globally nonstationary but locally stationary MSP construction, with the spatial domain divided into a fine grid of subregions, each with its own dependence parameters. We use LASSO ( L 1 ) or ridge ( L 2 ) penalties to obtain spatially smooth parameter estimates. We then develop a novel data-driven algorithm to merge homogeneous neighboring subregions. The algorithm facilitates model parsimony and interpretability. To make our model suitable for high-dimensional data, we exploit a pairwise likelihood to perform inference and discuss its computational and statistical efficiency. We apply our proposed method to model monthly maximum temperature data at over 1400 sites in Nepal and the surrounding Himalayan and sub-Himalayan regions; we show significant improvements in model fit compared to a stationary model. Furthermore, we demonstrate that the estimated merged partition is interpretable from a geographic perspective and leads to better model diagnostics by adequately reducing the number of parameters.

Improving estimation for asymptotically independent bivariate extremes via global estimators for the angular dependence function

Modelling the extremal dependence of bivariate variables is important in a wide variety of practical applications, including environmental planning, catastrophe modelling and hydrology. The majority of these approaches are based on the framework of bivariate regular variation, and a wide range of literature is available for estimating the dependence structure in this setting. However, such procedures are only applicable to variables exhibiting asymptotic dependence, even though asymptotic independence is often observed in practice. In this paper, we consider the so-called ‘angular dependence function’; this quantity summarises the extremal dependence structure for asymptotically independent variables. Until recently, only pointwise estimators of the angular dependence function have been available. We introduce a range of global estimators and compare them to another recently introduced technique for global estimation through a systematic simulation study, and a case study on river flow data from the north of England, UK.

Modelling the VaR Using Integral Probability and Tail Dependence

In this paper, a method is proposed to estimate value at risk (VaR) in a multivariate context by employing the probability integral transformation (PIT) and tail dependence function to model extreme dependence structure. The proposed method involves two vectorial measures of VaR, which use either the distribution function or the survival function. Moreover, properties of these risk measures are analysed, and a connection between them and tail dependence functions is established.

An efficient workflow for modelling high-dimensional spatial extremes

We develop a comprehensive methodological workflow for Bayesian modelling of high-dimensional spatial extremes that lets us describe both weakening extremal dependence at increasing levels and changes in the type of extremal dependence class as a function of the distance between locations. This is achieved with a latent Gaussian version of the spatial conditional extremes model that allows for computationally efficient inference with R-INLA. Inference is made more robust using a post hoc adjustment method that accounts for possible model misspecification. This added robustness makes it possible to extract more information from the available data during inference using a composite likelihood. The developed methodology is applied to the modelling of extreme hourly precipitation from high-resolution radar data in Norway. Inference is performed quickly, and the resulting model fit successfully captures the main trends in the extremal dependence structure of the data. The post hoc adjustment is found to further improve model performance.

Vecchia Likelihood Approximation for Accurate and Fast Inference with Intractable Spatial Max-Stable Models

Max-stable processes are the most popular models for high-impact spatial extreme events, as they arise as the only possible limits of spatially-indexed block maxima. However, likelihood inference for such models suffers severely from the curse of dimensionality, since the likelihood function involves a combinatorially exploding number of terms. In this article, we propose using the Vecchia approximation, which conveniently decomposes the full joint density into a linear number of low-dimensional conditional density terms based on well-chosen conditioning sets designed to improve and accelerate inference in high dimensions. Theoretical asymptotic relative efficiencies in the Gaussian setting and simulation experiments in the max-stable setting show significant efficiency gains and computational savings using the Vecchia likelihood approximation method compared to traditional composite likelihoods. Our application to extreme sea surface temperature data at more than a thousand sites across the entire Red Sea further demonstrates the superiority of the Vecchia likelihood approximation for fitting complex models with intractable likelihoods, delivering significantly better results than traditional composite likelihoods, and accurately capturing the extremal dependence structure at lower computational cost. Supplementary materials for this article are available online.

ENVIRONMENTAL CONTOUR ESTABLISHMENT BASED ON OPENLY AVAILABLE EMODNET WAVE AND WIND DATA FOLLOWING HEFFERNAN AND TAWN METHODOLOGIES

In this paper, I present methodologies and routines for inference of environmental contours based on openly available data from the European EMODnet service, specifically wind and wave timeseries from oil platform 6200146. The conditional marginal distribution models, known as environmental contours, is established based on measured wind and wave heights, at platform 6200146 located in the North Sea. The data is provided through the openly available data platform, EMODnet. The environmental contour model is established by fitting univariate distributions to each data-series independently, whereafter an extremal dependence structure is established following methodologies outlined in (Janet E. heffernan, 2004).

Pricing Multi-Event-Triggered Catastrophe Bonds Based on a Copula–POT Model

The constantly expanding losses caused by frequent natural disasters pose many challenges to the traditional catastrophe insurance market. The purpose of this paper is to develop an innovative and systemic trigger mechanism for pricing catastrophic bonds triggered by multiple events with an extreme dependence structure. Due to the bond’s low cashflow contingencies and the CAT bond’s high return, the multiple-event CAT bond may successfully transfer the catastrophe risk to the huge financial markets to meet the diversification of capital allocations for most potential investors. The designed hybrid trigger mechanism helps reduce the moral hazard and increase the bond’s attractiveness with a lower trigger likelihood, displaying the determinants of the wiped-off coupon and principal by both the magnitude and intensity of the natural disaster events involved. As the trigger indicators resulting from the potential catastrophic disaster might be associated with heavy-tailed margins, nested Archimedean copulas are introduced with marginal distributions modeled by a POT-GP distribution for excess data and common parametric models for moderate risks. To illustrate our theoretical pricing framework, we conduct an empirical analysis of pricing a three-event rainstorm CAT bond based on the resulting losses due to rainstorms in China during 2006–2020. Monte Carlo simulations are carried out to analyze the sensitivity of the rainstorm CAT bond price in trigger attachment levels, maturity date, catastrophe intensity, and numbers of trigger indicators.

Estimating POT second-order parameter for bias correction

The stable tail dependence function provides a full characterisation of the extremal dependence structures. Unfortunately, the estimation of the stable tail dependence function often suffers from significant bias, whose scale relates to the Peaks-Over-Threshold (POT) second-order parameter. For this second-order parameter, this paper introduces a penalised estimator that discourages it from being too close to zero. This paper then establishes this estimator's asymptotic consistency, uses it to correct the bias in the estimation of the stable tail dependence function, demonstrates its desirable empirical properties in the estimation of the extremal dependence structures, and illustrates it with an application to liability claim data.

Pairwise counter-monotonicity

We systematically study pairwise counter-monotonicity, an extremal notion of negative dependence. A stochastic representation and an invariance property are established for this dependence structure. We show that pairwise counter-monotonicity implies negative association, and it is equivalent to joint mix dependence if both are possible for the same marginal distributions. We find an intimate connection between pairwise counter-monotonicity and risk sharing problems for quantile agents. This result highlights the importance of this extremal negative dependence structure in optimal allocations for agents who are not risk averse in the classic sense.

Modelling Dependency Structures of Carbon Trading Markets between China and European Union: From Carbon Pilot to COVID-19 Pandemic

The exploration of the dependency structure of the Chinese and EU carbon trading markets is crucial to the construction of a globally harmonized carbon market. In this paper, we studied the characteristics of structural interdependency between China’s major carbon markets and the European Union (EU) carbon market before and after the launch of the national carbon emissions trading scheme (ETS) and the occurrence of the new coronavirus (COVID-19) by applying the C-vine copula method, with the carbon trading prices of the EU, Beijing, Shanghai, Guangdong, Shenzhen and Hubei as the research objects. The study shows that there exists a statistically significant dependence between the EU and the major carbon markets in China and their extremal dependences and dependence structures are different at different stages. After the launch of the national carbon ETS, China has become more independent in terms of interdependency with the EU carbon market, and is more relevant between domestic carbon markets. Most importantly, we found that the dependence between the EU and Chinese carbon markets has increased following the outbreak of COVID-19, and tail dependency structures existed before the launch of the national carbon ETS and during the outbreak of the COVID-19. The results of this study provide a basis for the understanding of the linkage characteristics of carbon trading prices between China and the EU at different stages, which in turn can help market regulators and investors to formulate investment decisions and policies.

Flexible modeling of multivariate spatial extremes

We develop a novel multi-factor copula model for multivariate spatial extremes, which is designed to capture the different combinations of within- and cross-field extremal dependence structures between different spatial processes. Our proposed model, which can be seen as a multi-factor copula model, can capture all possible distinct combinations of extremal dependence structures within each individual spatial process while allowing flexible cross-field extremal dependence structures for both upper and lower tails. We show how to perform Bayesian inference for the proposed model using a Markov chain Monte Carlo algorithm based on carefully designed block proposals with an adaptive step size. In our real data application, we apply our model to study the upper and lower extremal dependence structures of the daily maximum air temperature (TMAX) and daily minimum air temperature (TMIN) from the state of Alabama in the southeastern United States. The fitted multivariate spatial model is found to provide a good fit in the lower and upper joint tails, both in terms of the spatial dependence structure within each individual process, as well as in terms of the cross-field dependence structure. Our results suggest that the TMAX and TMIN processes are quite strongly spatially dependent over the state of Alabama, and moderately cross-dependent. From a practical perspective, this implies that it may be worthwhile to model them jointly when interest lies in computing spatial risk measures that involve both quantities.

Structure Learning for Extremal Tree Models

AbstractExtremal graphical models are sparse statistical models for multivariate extreme events. The underlying graph encodes conditional independencies and enables a visual interpretation of the complex extremal dependence structure. For the important case of tree models, we develop a data-driven methodology for learning the graphical structure. We show that sample versions of the extremal correlation and a new summary statistic, which we call the extremal variogram, can be used as weights for a minimum spanning tree to consistently recover the true underlying tree. Remarkably, this implies that extremal tree models can be learned in a completely non-parametric fashion by using simple summary statistics and without the need to assume discrete distributions, existence of densities or parametric models for bivariate distributions.

Asymptotic Dependence Modelling of the BRICS Stock Markets

With the use of empirical data, this paper focuses on solving financial and investment issues involving extremal dependence of 10 pairwise combinations of the 5 BRICS (Brazil, Russia, India, China, and South Africa) stock markets. Daily closing equity indices from 5 January 2010 to 6 August 2018 are used in the study. Unlike previous literature, we use bivariate point process and conditional multivariate extreme value models to investigate the extremal dependence of the stock market returns. However, it is observed that the point process was able to model many more extreme observations or exceedances that contribute to the likelihood estimation. It gives more information than the threshold excess method of the CMEV model. This study shows varying levels of low extremal dependence structure whose outcomes are highly beneficial to investors, portfolio managers and other market participants interested in maximising investment returns and financial gains.

Dependence bounds for the difference of stop-loss payoffs on the difference of two random variables

This paper considers the difference of stop-loss payoffs where the underlying is a difference of two random variables. The goal is to study whether the comonotonic and countermonotonic modifications of those two random variables can be used to construct upper and lower bounds for the expected payoff, despite the fact that the payoff function is neither convex nor concave. The answer to the central question of the paper requires studying the crossing points of the cumulative distribution function (cdf) of the original difference with the cdf's of its comonotonic and countermonotonic transforms. The analysis is supplemented with a numerical study of longevity trend bonds, using different mortality models and population data. The numerical study reveals that for these mortality-linked securities the three pairs of cdf's generally have unique pairwise crossing points. Under symmetric copulas, all crossing points can reasonably be approximated by the difference of the marginal medians, but this approximation is not necessarily valid for asymmetric copulas. Nevertheless, extreme dependence structures can give rise to bounds if the layers of the bond are selected to hedge tail risk. Further, the dependence uncertainty spread can be low if the layers are selected to hedge median risk, and, subject to a trade-off, to hedge tail risk as well.

Multivariate Regional Frequency Analysis Using Conditional Extreme Values Approach

AbstractFloods can be characterized by various correlated variables such as peak flow, flood volume, duration, and time to peak. Hence, flood risk can be better assessed by performing frequency analysis in a multivariate (rather than a univariate) framework. However, multivariate approaches involve sophisticated analyses that require considerably more data than univariate approaches. In a univariate framework, flood risk assessment at an ungauged or data‐sparse location/site is generally performed through regional frequency analysis (RFA), based on information pooled from a group of sites resembling the target site. However, RFA has received little attention in the multivariate framework. The available literature recommends the use of index‐flood approach (IFA) for multivariate RFA (MRFA), even though IFA has theoretical shortcomings. Another issue is that marginals of all the flood‐related variables may not exhibit extreme behavior simultaneously. Conventionally used at‐site multivariate models are not suitable for describing the dependence structure of extremes in the regions of the support of the joint distribution where only some variables exhibit extreme behavior. This article proposes a conditional extreme values approach (CEA) to address the aforementioned issues in MRFA. Its effectiveness is demonstrated through Monte Carlo simulation experiments and a case study on watersheds from a flood‐prone region in India. Results indicate that the proposed approach can reliably predict the joint distribution of multiple flood‐related variables (and thus flood risk) at ungauged/sparsely gauged sites by effectively capturing the regional dependence structure between those variables.

Asymptotic theory for the inference of the latent trawl model for extreme values

AbstractThis article develops statistical inference methods and their asymptotic theory for the latent trawl model for extremes, which captures serial dependence in the time series of exceedances above a threshold. We review two methods based on pairwise likelihood and show that they underestimate the serial dependence in the extremes. We propose two generalized method of moments procedures based on auto‐covariance matching to overcome this shortcoming. Out of those four inference approaches, two are single‐stage strategies while the others have two stages, and we provide central limit theorems in the sense of weakly approaching sequences of distributions for all of them. This additional flexibility ensures good behavior between the estimators and estimates of the limiting distribution. In an empirical illustration using London air pollution data, we find that the two‐stage auto‐covariance matching scheme yields a high‐quality inference. It comprises two interpretable steps and correctly captures the serial dependence structure of extremes while performing on par with other methods in terms of marginal fit.

Modeling Concurrent Hydroclimatic Extremes With Parametric Multivariate Extreme Value Models

AbstractEstimating the dependence structure of concurrent extremes is a fundamental issue for accurate assessment of their occurrence probabilities. Identifying the extremal dependence behavior is also crucial for scientific understanding of interactions between the variables of a multidimensional environmental process. This study investigates the suitability of parametric multivariate extreme value models to correctly represent and estimate the dependence structure of concurrent extremes. Probabilistic aspects of multivariate extreme value theory with point process representation are discussed and illustrated with application to the concurrence of rainfall deficits, soil moisture deficits, and high temperatures. Application is concerned with the investigation of extremal behavior and risk assessment in Marathwada, a drought‐prone region of Maharashtra state, India. To characterize the multivariate extremes, marginal distributions are specified first and transformed into unit Fréchet margins. Standardized distributions are represented by a Poisson point process and coordinates of data points are transformed to pseudo‐polar coordinates to make the dependence form more explicit. The extremal dependence structure is described through angular densities on the unit simplex. Strong dependence is observed between soil moisture deficits and high temperatures, whereas rainfall deficits are mildly dependent on these two variables. Overall, a weak dependence is observed between the variables considered. Estimated extremal dependence is further used to compute probabilities of a few critical extreme combinations. Results demonstrate the ability of parametric multivariate models to characterize the complex dependence structure of concurrent extremes. These models can provide a powerful new perspective for appropriate statistical analysis of dependent hydroclimatic extremes in higher dimensions.