Conditional Extreme Value Model Research Articles

CovXtreme : MATLAB software for non-stationary penalised piecewise constant marginal and conditional extreme value models

The covXtreme software provides functionality for estimation of marginal and conditional extreme value models, non-stationary with respect to covariates, and environmental design contours. Generalised Pareto (GP) marginal models of peaks over threshold are estimated, using a piecewise-constant representation for the variation of GP threshold and scale parameters on the (potentially multidimensional) covariate domain of interest. The conditional variation of one or more associated variates, given a large value of a single conditioning variate, is described using the conditional extremes model of Heffernan and Tawn (2004), the slope term of which is also assumed to vary in a piecewise constant manner with covariates. Optimal smoothness of marginal and conditional extreme value model parameters with respect to covariates is estimated using cross-validated roughness-penalised maximum likelihood estimation. Uncertainties in model parameter estimates due to marginal and conditional extreme value threshold choice, and sample size, are quantified using a bootstrap resampling scheme. Estimates of environmental contours using various schemes, including the direct sampling approach of Huseby et al. 2013, are calculated by simulation or numerical integration under fitted models. The software was developed in MATLAB for metocean applications, but is applicable generally to multivariate samples of peaks over threshold data. The software and case study data can be downloaded from GitHub, with an accompanying user guide.

Statistical modeling and dependence analysis for tide level via multivariate extreme value distribution method

This paper presents a statistical modeling approach to explore the dependence of extreme values in multi-site tidal water levels (TL) using hourly data from six tidal stations in the Kansai region, Japan. The proposed method utilizes a multi-site conditional extreme value model (CEVM) and a peak over threshold (POT) model with yearly order grouping (YOG) to handle extreme value dependence and marginal modeling of tidal CEVM, respectively. The results demonstrate the effectiveness of the proposed approach in analyzing extreme value dependence among multiple sites, with the YOG method yielding favorable results for extreme value dependence analysis. The extreme dependence effect of TL and the probability of exceedance at multiple sites under extreme conditions are investigated. The paper concludes that the extreme dependence effect should be captured through a multivariate CEVM and provides a more accurate basis for engineering construction and disaster prevention. The novelty of this work lies in its statistical modeling approach that enables accurate multi-site extreme value analysis and provides a more comprehensive understanding of the extreme value dependence of the TLs, which has significant implications for disaster management and prevention.

Linking representations for multivariate extremes via a limit set

AbstractThe study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.

Modeling the Extremes of Bivariate Mixture Distributions With Application to Oceanographic Data

There currently exist a variety of statistical methods for modeling bivariate extremes. However, when the dependence between variables is driven by more than one latent process, these methods are likely to fail to give reliable inferences. We consider situations in which the observed dependence at extreme levels is a mixture of a possibly unknown number of much simpler bivariate distributions. For such structures, we demonstrate the limitations of existing methods and propose two new methods: an extension of the Heffernan–Tawn conditional extreme value model to allow for mixtures and an extremal quantile-regression approach. The two methods are examined in a simulation study and then applied to oceanographic data. Finally, we discuss extensions including a subasymptotic version of the proposed model, which has the potential to give more efficient results by incorporating data that are less extreme. Both new methods outperform existing approaches when mixtures are present.

Assessing compound flooding potential with multivariate statistical models in a complex estuarine system under data constraints

AbstractCompound flooding may result from the interaction of two or more contributing processes, which may not be extreme themselves, but in combination lead to extreme impacts. Here, we use statistical methods to assess compounding effects from storm surge and multiple riverine discharges in Sabine Lake, TX. We employ several trivariate statistical models, including vine‐copulas and a conditional extreme value model, to examine the sensitivity of results to the choice of data pre‐processing steps, statistical model setup, and outliers. We define a response function that represents water levels resulting from the interaction between discharge and surge processes inside Sabine Lake and explore how it is affected by including or ignoring dependencies between the contributing flooding drivers. Our results show that accounting for dependencies leads to water levels that are up to 30 cm higher for a 2% annual exceedance probability (AEP) event and up to 35 cm higher for a 1% AEP event, compared to assuming independence. We also find notable variations in the results across different sampling schemes, multivariate model configurations, and sensitivity to outlier removal. Under data constraints, this highlights the need for testing various statistical modelling approaches, while the choice of an optimal approach remains subjective.

Model‐based inference of conditional extreme value distributions with hydrological applications

AbstractMultivariate extreme value models are used to estimate joint risk in a number of applications, with a particular focus on environmental fields ranging from climatology and hydrology to oceanography and seismic hazards. The semi‐parametric conditional extreme value model of Heffernan and Tawn involving a multivariate regression provides the most suitable of current statistical models in terms of its flexibility to handle a range of extremal dependence classes. However, the standard inference for the joint distribution of the residuals of this model suffers from the curse of dimensionality because, in a d‐dimensional application, it involves a d−1‐dimensional nonparametric density estimator, which requires, for accuracy, a number points and commensurate effort that is exponential in d. Furthermore, it does not allow for any partially missing observations to be included, and a previous proposal to address this is extremely computationally intensive, making its use prohibitive if the proportion of missing data is nontrivial. We propose to replace the d−1‐dimensional nonparametric density estimator with a model‐based copula with univariate marginal densities estimated using kernel methods. This approach provides statistically and computationally efficient estimates whatever the dimension, d, or the degree of missing data. Evidence is presented to show that the benefits of this approach substantially outweigh potential misspecification errors. The methods are illustrated through the analysis of UK river flow data at a network of 46 sites and assessing the rarity of the 2015 floods in North West England.

Estimation of the expected shortfall given an extreme component under conditional extreme value model

For two risks, X and Y, the Marginal Expected Shortfall (MES) is defined as $\mathbb {E}[Y\mid X>F_{X}^{\leftarrow }(1-p)]$ , where FX is the distribution function of X and p is small. In this paper we establish consistency and asymptotic normality of an estimator of MES on assuming that (X, Y ) follows a Conditional Extreme Value (CEV) model. The theoretical findings are supported by simulation studies. Our procedure is applied to some financial data.

Transition kernels and the conditional extreme value model

Classical multivariate extreme value modelling assumes that the joint distribution belongs to a multivariate domain of attraction and this assumption requires that each marginal distribution be individually attracted to a univariate extreme value distribution. The Heffernan and Tawn (J R Stat Soc Ser B (Stat Methodol) 66(3):497–546, 2004) alternative extremal model for multivariate data does not require all the components belong to an extremal domain of attraction but assumes instead the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components is extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation of the probability of certain risk regions. The original focus on conditional distributions has technical drawbacks but is a natural assumption in many contexts. The technical drawbacks are overcome by relying on convergence of measures and the theory of extended regular variation Heffernan and Resnick (Ann Appl Probab 17(2):537–71, 2007); Das and Resnick (Extremes 14(1):29–61, 2000a); Das et al. (Adv Appl Probab 45(1):139–163, 2013). We compare the two approaches and describe in what way relying on variational limit properties of conditional distributions restricts the class of limit approximations.

Empirical Bayes estimation for the conditional extreme value model

A new estimation strategy for estimating the parameters of the Heffernan and Tawn conditional extreme value model is proposed. The technique makes use of empirical Bayes estimation for the conditional likelihood that otherwise does not have a simple closed‐form expression. The approach is tested on simulations from different types of extreme dependence (and independence) structures, as well as for two real data cases consisting of precipitation analysis conditional on extreme temperature in Boulder, Colorado, and Los Angeles, California, USA. The strategy generally has good coverage when informative priors are used for one of the parameters, except for the independence case where the coverage is low until the sample size reaches about 50. Results for the precipitation and temperature data are found to be consistent with the semi‐non‐parametric strategy. The presented model can be potentially applied in a wide variety of science fields, especially in earth, environment and climate sciences. Copyright © 2014 John Wiley & Sons, Ltd.

Spatial extreme value analysis to project extremes of large‐scale indicators for severe weather

Concurrently high values of the maximum potential wind speed of updrafts (Wmax) and 0–6 km wind shear (Shear) have been found to represent conducive environments for severe weather, which subsequently provides a way to study severe weather in future climates. Here, we employ a model for the product of these variables (WmSh) from the National Center for Atmospheric Research/United States National Center for Environmental Prediction reanalysis over North America conditioned on their having extreme energy in the spatial field in order to project the predominant spatial patterns of WmSh. The approach is based on the Heffernan and Tawn conditional extreme value model. Results suggest that this technique estimates the spatial behavior of WmSh well, which allows for exploring possible changes in the patterns over time. While the model enables a method for inferring the uncertainty in the patterns, such analysis is difficult with the currently available inference approach. A variation of the method is also explored to investigate how this type of model might be used to qualitatively understand how the spatial patterns of WmSh correspond to extreme river flow events. A case study for river flows from three rivers in northwestern Tennessee is studied, and it is found that advection of WmSh from the Gulf of Mexico prevails while elsewhere, WmSh is generally very low during such extreme events. © 2013 The Authors. Environmetrics published by JohnWiley & Sons, Ltd.

Extreme Risk of Public Timber REITs during the Global Financial Crisis

A conditional volatility model with extreme value statistics is employed to assess the time-varying risk inherent in three real estate investment trusts (REITs) that have specialized in owning or managing timberlands. Among the four alternative approaches, the conditional extreme value model provides the best risk estimates. The estimated risk of individual timber REITs varies with time. During the volatile period between 2008 and 2009, the absolute values of daily Value-at-Risk (VaR) estimates indicate that an investor could lose up to 13% of an investment over one day with a 99% confidence level.

Detecting a conditional extreme value model

In classical extreme value theory probabilities of extreme events are estimated assuming all the components of a random vector to be in a domain of attraction of an extreme value distribution. In contrast, the conditional extreme value model assumes a domain of attraction condition on a sub-collection of the components of a multivariate random vector. This model has been studied in Heffernan and Tawn (JRSS B 66(3):497–546, 2004), Heffernan and Resnick (Ann Appl Probab 17(2):537–571, 2007), and Das and Resnick (2009). In this paper we propose three statistics which act as tools to detect this model in a bivariate set-up. In addition, the proposed statistics also help to distinguish between two forms of the limit measure that is obtained in the model.