Multivariate Extreme Value Theory Research Articles

Multivariate Extreme Value Theory Based Channel Modeling for Ultra-Reliable Communications

Multivariate Extreme Value Theory Based Channel Modeling for Ultra-Reliable Communications

Multivariate Extreme Value Theory Based Rate Selection for Ultra-Reliable Communications

Multivariate Extreme Value Theory Based Rate Selection for Ultra-Reliable Communications

Joint extremes in precipitation and infectious disease in the USA: A bivariate POT study

Mounting heavy precipitation events (HPEs) caused by the climate change have drawn wide attention. Increased incidences of infectious diseases are known as the common following health impact, while little has been studied about the extremal relationship in between. Therefore, this study aims to investigate the joint extremes of precipitation and infectious disease mortality rate in the USA, using publicly accessible data from the National Centers for Environmental Information and the Centers for Disease Control and Prevention. The study reveals the positive association between heavy precipitations and infectious diseases with slight national and regional differences using multivariate Peaks-Over-Threshold modelling. The strength of extremal dependence is measured by the extreme parameter α from a logistic dependence model in multivariate extreme value theory. The Midwestern USA shows an excessive impact of HPEs on infectious disease mortality (α=0.7524), while the other regions show similar extremal dependence strength with the national one (α values all approximate 0.77). The study also discovered spatial disparities in the extremal dependences for five sub-categories of infectious diseases in each census region, among which mycoses show the strongest extremal dependence with precipitation in almost all regions. These spatial differences of extremal dependence may be attributed to geographic, social-economic factors and the self-inherited characteristics of certain diseases. The findings are expected to assist in developing strategies counteracting extreme risks resulting from weather events and health issues as well. The cutting-edge multivariate Peaks-Over-Threshold (POT) approach employed herein also shows promise for a wide range of extreme risk assessment topics.

Extreme Wind Loading on Flat-Roof-Mounted Solar Arrays with Consideration of Wind Directionality

The assessment of extreme wind loading on solar arrays plays a significant role in ensuring their safe operation under strong winds. Therefore, this paper investigates the extreme wind loading on solar arrays mounted on a flat roof by taking into account the wind directionality effect. The estimation process is conducted by using in situ wind speeds obtained from meteorological stations and wind loading coefficients on solar arrays obtained from wind tunnel tests based on the joint probability distribution of multiple variables and their conditional probabilities. This allows a discussion regarding how the extreme wind loading of solar arrays would be affected by such factors as the uncertainty of wind loading coefficient, the structural orientation of buildings on which solar arrays are mounted, and the directional characteristics of wind speeds. Finally, a comparison among the proposed methods, considering the wind directionality in current wind loading codes, is performed. The extreme wind loading determined by multivariate extreme value theory is found to be comparable to the corresponding estimate calculated according to the independent assumption of directional extreme wind speed. The results of this study provide a valuable reference for the design of wind-resistant solar arrays that takes account of wind directionality effect.

Nonparametric estimation of conditional marginal excess moments

Several risk measures have been proposed in the literature, among them the marginal mean excess, defined as MMEp=E[{Y(1)−Q1(1−p)}+|Y(2)>Q2(1−p)], provided E|Y(1)|<∞, where (Y(1),Y(2)) denotes a pair of risk factors, y+≔max(0,y), Qj the quantile function of Y(j),j∈{1,2}, and p∈(0,1). In this paper we consider a generalization of this measure, where the random variables of main interest (Y(1),Y(2)) are observed together with a random covariate X∈Rd, and where the Y(1) excess is also power transformed. This leads to the concept of conditional marginal excess moment for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the estimator is evaluated by a simulation experiment. We apply also our method on a vehicle insurance customer dataset.

A multivariate method for evaluating safety from conflict extremes in real time

Several studies have advocated the use of extreme value theory (EVT) traffic conflict models for real-time crash risk prediction using real-time safety indices such as the risk of crash (RC) and return level of a cycle (RLC). This approach provides a logical framework to estimate crash risk by extrapolating from the observed level (i.e., traffic conflict) to the unobserved level (i.e., crash). In these studies, only univariate EVT models that consider only one conflict indicator (e.g. modified time to collision, MTTC) were used which affects the models’ accuracy and precision in estimating crash risk. The use of univariate models is likely due to that existing safety analysis multivariate EVT models have limited capability of delineating the complex dependence structure between multiple conflict indicators for application to real-time safety evaluation. This study proposes a multivariate method for evaluating real-time safety from conflict extremes which consists of novel multivariate EVT models that flexibly integrate multiple conflict indicators and several joint safety indices that comprehensively characterize the safety level of a road facility from multiple dimensions. The proposed approach has several advantages including: 1) it uses four parametric models (tilted Dirichlet, pairwise beta, Husler-Reiss, and extremal-t) for the angular density function for fully describing the dependence level between multiple conflict extremes; and 2) it innovatively develops several important real-time safety indices (e.g., crash risk, joint return levels, and return level concomitant) from the multivariate joint distribution for multidimensionally assessing safety. A seven-step approximate likelihood-based Bayesian inference method for model development is proposed. The proposed model estimation method is applied for cycle-level real-time safety evaluation by combining several conflict indicators at four signalized intersections in the city of Surrey, British Columbia. Three conflict indicators are used: MTTC, post encroachment time (PET), and deceleration rate to avoid a crash (DRAC). Four types of multivariate EVT models were developed. Among these models, for both bivariate and trivariate framework, the Husler-Reiss model has the best goodness-of-fit as it better captures the dependence level among the three conflict indicators. The results indicate that multivariate models identify higher numbers of crash-risk cycles than their corresponding univariate models. Further, most of crash-risk cycles have at least one of joint return levels higher than the threshold (0 for both MTTC and PET, 8.5 m/s2 for DRAC) between a conflict and a collision. For joint return levels from most cycles, one return level exceeds the threshold, while others are lower than the threshold. Under the bivariate framework, all the concomitants of positive return levels are below their own thresholds.

Systemic risk measurement: A limiting threshold copula approach to CoVaR

This article proposes a novel copula-based approach to estimate CoVaR. Copula selection is based on multivariate extreme value theory rather than model fitting results. The Clayton copula as a limiting lower threshold copula is found to be able to provide a useful dependence structure for modelling the joint tail behaviour of financial returns. The out-of-sample testing of one-step quantile forecasts shows that the dynamic Clayton copula significantly outperforms the alternatives. It also shows the GJR-GARCH-Skew t model is a powerful competitor of the GPD-based models. Hence, the method provides a simple and effective way for estimating CoVaR and VaR.

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.

Modeling and simulating spatial extremes by combining extreme value theory with generative adversarial networks

AbstractModeling dependencies between climate extremes is important for climate risk assessment, for instance when allocating emergency management funds. In statistics, multivariate extreme value theory is often used to model spatial extremes. However, most commonly used approaches require strong assumptions and are either too simplistic or over-parameterized. From a machine learning perspective, generative adversarial networks (GANs) are a powerful tool to model dependencies in high-dimensional spaces. Yet in the standard setting, GANs do not well represent dependencies in the extremes. Here we combine GANs with extreme value theory (evtGAN) to model spatial dependencies in summer maxima of temperature and winter maxima in precipitation over a large part of western Europe. We use data from a stationary 2000-year climate model simulation to validate the approach and explore its sensitivity to small sample sizes. Our results show that evtGAN outperforms classical GANs and standard statistical approaches to model spatial extremes. Already with about 50 years of data, which corresponds to commonly available climate records, we obtain reasonably good performance. In general, dependencies between temperature extremes are better captured than dependencies between precipitation extremes due to the high spatial coherence in temperature fields. Our approach can be applied to other climate variables and can be used to emulate climate models when running very long simulations to determine dependencies in the extremes is deemed infeasible.

A Weissman-type estimator of the conditional marginal expected shortfall

The marginal expected shortfall is an important risk measure in finance and actuarial science, which has been extended recently to the case where the random variables of main interest are observed together with a covariate. This leads to the concept of conditional marginal expected shortfall for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the proposed estimator is evaluated with a simulation experiment, and the practical applicability is illustrated on vehicle insurance customer data.

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.

Spatio-temporal variability of dry and wet spells and their influence on crop yields

Spatial modelling of extreme dry (wet) spells is relevant to design agricultural insurances and establish risk management strategies. We characterised dry (wet) spells during the period 1971–2010 in 133 rainfall stations and investigated their impact on crop yields across Belgium. A day was considered as a dry (wet) day when the total amount of rainfall is below (above) 0.2 mm. A two-way ANOVA demonstrated significant differences between low and high yields across 12 agricultural regions in the number of consecutive dry (wet) days, rainfall amount during the wet days and their total during the crop growing season. Generalised Extreme Value distributions were fitted to the maxima series using maximum likelihood to estimate the distribution parameters. The number of consecutive dry (wet) days and rainfall amount was considered during the cropping season between March and October. We modelled maxima observed at rainfall stations by means of max-stable processes, an extension of multivariate extreme value theory. Spatial extreme modelling was performed through pairwise likelihood inference by a madogram and enabled 20-year return levels to be simulated across regions based on geographic coordinates and altitude as covariates. Spatial return levels confirmed the exceptionality of 2016 and 2018 as extreme wet and dry years with high return periods. The 2018 drought resulted in yield reductions of 74% for maize and 42% for potato, while the 2016 rainfall induced yield losses of up to 39% for winter cereals, 45% for maize and 53% for potato. The max-stable models provided a good fit to the joint behaviour of extremes in the number of consecutive dry (wet) days during the growing season as an indication for crop stress. The method allows for spatio-temporal characterisation of generalised extreme value distributions across regions, and provides a tool for weather based insurances and risk management.

Truncated pair-wise likelihood for the Brown-Resnick process with applications to maximum temperature data

Max-stable processes are a natural extension of multivariate extreme value theory important for modeling the spatial dependence of environmental extremes. Inference for max-stable processes observed at several spatial locations is challenging due to the intractability of the full likelihood function. Composite likelihood methods avoid these difficulties by combining a number of low-dimensional likelihood objects, typically defined on pairs or triplets of spatial locations. This work develops a new truncation procedure based on ℓ1-penalty to reduce the complexity and computational cost associated with the composite likelihood function. The new method is shown to achieve a favorable trade-off between computational burden and statistical efficiency by dropping a number of noisy sub-likelihoods. The properties of the new method are illustrated through numerical simulations and an application to real extreme temperature data.

Systemic risk allocation using the asymptotic marginal expected shortfall

This paper defines asymptotic marginal expected shortfall (AMES) for banks within a financial system and provides corresponding estimation method based on multivariate extreme value theory. The estimation method does not assume a specific dependence structure among bank equity returns. Both theoretical AMES and the estimator possess additive property and thus can serve as a tool to allocate system-wide risk to individual institutions. We apply the AMES to 30 global systemically important financial institutions (G-SIFIs). We show that the AMES outperforms the MES in predicting extreme losses during extreme systemic events. By taking the AMES as the reference point for allocating systemic risk to individual institutions, we show that an allocation according to simple bank characteristics such as size and individual risk can be imperfect. The allocation unfairness of individual risk or size across all the G-SIFIs has increased since 2008.

Multivariate extremes and max-stable processes: discussion of the paper by Zhengjun Zhang

This discussion reviews the paper by Zhengjun Zhang in the context of broader research on multivariate extreme value theory and max-stable processes.

Estimation of extreme wind pressure coefficient in a zone by multivariate extreme value theory

Knowledge on the design value of extreme wind pressure coefficients (EWPC) of a specific zone of buildings is essential for the wind-resistant capacity of claddings. This paper presents a method to estimate the representative EWPC introducing the multivariate extreme value model. The spatial correlations of the extreme wind pressures at different locations can be consider through the multivariate extreme value. The moving average method is also adopted in this method, so that the measured point pressure can be converted to wind pressure of an area. The proposed method is applied to wind tunnel test results of a large flat roof building. Comparison with existing methods shows that it can give a good estimation for all target zones with different sizes.

Quantitative assessment and visualization of flight risk induced by coupled multi-factor under icing conditions

Aircraft icing has been proven to be one of the most serious threats to flight safety. During the analysis of flight risk under icing conditions, quantitative assessment and visualization of flight risk are quite essential as they provide safe manipulation strategies in intricate conditions. However, they are rarely studied. Since the icing flight accidents are the result of the coupling of multiple unfavorable factors, in present study, we have proposed a method to quantitatively assess flight risk induced by multi-factor coupling under icing conditions by Monte-Carlo simulation and multivariate extreme value theory. The results demonstrate that the flight risk probability increases with the rise of unfavorable factors. Besides, a flight risk visualization method named flight safety window has been presented to build the flight risk distribution cloud maps in different complex conditions. The cloud maps show that the icing would give rise to atrophy of the safety scope, and the consequence would be even more severe when coupled with other more unfavorable factors. The proposed methods in this study would be useful in flight risk analysis under icing conditions and can enhance the pilot's situational awareness in selecting correct strategies within the safety zone to avoid unsafe manipulation.

Vertical profiles of wind gust statistics from a regional reanalysis using multivariate extreme value theory

Abstract. Many applications require wind gust estimates at very different atmospheric height levels. For example, the renewable energy sector is interested in wind and gust predictions at the hub height of a wind power plant. However, numerical weather prediction models typically only derive estimates for wind gusts at the standard measurement height of 10 m above the land surface. Here, we present a statistical post-processing method to derive a conditional distribution for hourly peak wind speed as a function of height. The conditioning variables are taken from the COSMO-REA6 regional reanalysis. The post-processing method was trained using peak wind speed observations at five vertical levels between 10 and 250 m from the Hamburg Weather Mast. The statistical post-processing method is based on a censored generalized extreme value (cGEV) distribution with non-homogeneous parameters. We use a least absolute shrinkage and selection operator to select the most informative variables. Vertical variations of the cGEV parameters are approximated using Legendre polynomials, such that predictions may be derived at any desired vertical height. Further, the Pickands dependence function is used to assess dependencies between gusts at different heights. The most important predictors are the 10 m gust diagnostic, the barotropic and the baroclinic mode of absolute horizontal wind speed, the mean absolute horizontal wind at 700 hPa, the surface pressure tendency, and the lifted index. Proper scores show improvements of up to 60 % with respect to climatology, especially at higher vertical levels. The post-processing model with a Legendre approximation is able to provide reliable predictions of gusts' statistics at non-observed intermediate levels. The strength of dependency between gusts at different levels is non-homogeneous and strongly modulated by the vertical stability of the atmosphere.

Oil rig protection against wind and wave in Lebanon

The oil rigs that will be installed in the Lebanese offshore are floating rigs. This type can be affected by environmental risks such as extreme wind speed and wave height. We study these risks in Beddawi region in the northern Lebanon during the winter season by applying the extreme value theory to real data. We estimate at a first stage the return levels associated to return periods of 50, 100, and 500 years for each risk separately using the classical univariate theory. At a second stage, we use the logistic parametric model and the bivariate copula functions in multivariate extreme value theory to estimate the dependence between extreme wind speed and wave height as well as joint exceedance probabilities and joint return levels to take into consideration the risk of these two environmental factors simultaneously.

Spatial Dependence and Space-Time Trends in Extreme Events

The statistical theory of extremes is extended to observations that are non-stationary and not independent. The non-stationarity over time and space is controlled via the scedasis (tail scale) in the marginal distributions. Spatial dependence stems from multivariate extreme value theory. We establish asymptotic theory for both the weighted sequential tail empirical process and the weighted tail quantile process based on all observations, taken over time and space. The results yield two statistical tests for homoscedasticity in the tail, one in space and one in time. Further, we show that the common extreme value index can be estimated via a pseudo-maximum likelihood procedure based on pooling all (non-stationary and dependent) observations. Our leading example and application is rainfall in Northern Germany.