Bivariate Extreme Value Models Research Articles

New estimation methods for extremal bivariate return curves

AbstractIn the multivariate setting, estimates of extremal risk measures are important in many contexts, such as environmental planning and structural engineering. In this paper, we propose new estimation methods for extremal bivariate return curves, a risk measure that is the natural bivariate extension to a return level. Unlike several existing techniques, our estimates are based on bivariate extreme value models that can capture both key forms of extremal dependence. We devise tools for validating return curve estimates, as well as representing their uncertainty, and compare a selection of curve estimation techniques through simulation studies. We apply the methodology to two metocean data sets, with diagnostics indicating generally good performance.

Application of Bivariate Extreme Value models to describe the joint behavior of temporal and speed related surrogate measures of safety

Limitations of historical crash data and the advantages of surrogate measures of safety have already been stressed by various authors. To describe nearness-to-collision, mostly time-based indicators are applied, and there is a consensus among researchers that speed-related indicators are needed to account for the severity dimension. There have been attempts to fit bivariate Extreme Value models to predict the number of crashes, however modeling crash frequency together with severity has received much less attention.The aim of this paper is to apply Extreme Value models to various pairs of temporal and speed-related indicators in order to investigate the dependence structure between them as well as to construct probability based risk levels and examine them in relation to severity levels. Bivariate threshold excess models were fitted to a dataset of left-turning and straight moving vehicle interactions recorded in a signalized intersection. The dependence structure between variable pairs were thoroughly investigated; it was concluded that temporal and speed related indicators were found independent, which means that road users getting close to each other in time does not necessarily come with high speeds. Therefore these indicators should be combined in order to properly predict severity; a temporal indicator on its own is not enough to make inferences about the severity of events. Using the calculated joint probability of events risk levels were constructed illustrating the events of equal probability.

From univariate to bivariate extreme value models: Approaches to integrate traffic conflict indicators for crash estimation

This study develops bivariate extreme value models to integrate traffic conflict indicators for crash estimation. Based on video data collected from four sites of two signalized intersections, the automated traffic conflict analysis system was used to extract the TTC and PET between left-turn vehicles and through vehicles. Bivariate Generalized Extreme Value (BGEV) and Bivariate Generalized Pareto (BGP) models that jointly used the two conflict indicators were then developed, and the number of crashes were derived from the estimated model parameters. Univariate Generalized Extreme Value (UGEV) and Univariate Generalized Pareto (UGP) models were also applied using individual conflict indicators. The developed bivariate models and univariate models were evaluated by comparing model estimated crashes to observed left-turn crashes. The results show that the BGP model performed the best, followed by the BGEV model, UGP model, and UGEV model. It is found that crash estimates from univariate models based on PET and TTC are underestimated and overestimated, respectively. The bivariate models integrating the two indicators improve the crash estimation accuracy and precision. The BGP model outperforming the BGEV model is likely due to the former allowing more efficient use of the collected data. Overall, it is concluded that the bivariate extreme value modeling approach that is capable of integrating traffic conflict indicators with clear boundaries between traffic conflicts and crashes is more promising for crash estimation. Moreover, with the emerging automated traffic conflict analysis and connected vehicle techniques that facilitate the indicators extraction, the bivariate approach can be readily applied to provide accurate crash estimations for proactive road safety analysis.

Validating the bivariate extreme value modeling approach for road safety estimation with different traffic conflict indicators

A range of conflict indicators have been developed for traffic conflict observation. The various conflict indicators have been shown in earlier studies to be of different and sometimes independent nature. Therefore, there is a need to combine different indicators to gain better understanding of the underlying severity of traffic events and for more reliable safety analysis. This study proposes a bivariate extreme value model to integrate different traffic conflict indicators for road safety estimation, and the model is validated with actual crash data. Based on video data collected from four signalized intersections in two Canadian cities, computer vision techniques were utilized to identify rear-end traffic conflicts using several indicators. The conflict indicators included: time to collision (TTC), modified time to collision (MTTC), post encroachment time (PET), and deceleration to avoid crash (DRAC). Then bivariate extreme value models were developed for combinations of each two indicators, and the numbers of crashes were estimated from the models and compared to the observed crashes. The results show that most of the estimated crashes are in the range of 95% Poisson confidence interval of observed crashes, which indicates that the bivariate extreme value model is a promising tool for road safety estimation. Moreover, the accuracy of estimated crashes are different for different indicator combinations. The results show that the estimates of TTC&PET are the most accurate, followed by TTC&MTTC, TTC&DRAC, PET&MTTC, PET&DRAC and MTTC&DRAC. A further correlation analysis suggests that a combination of two independent conflict indicators leads to better crash estimation performance.

Retracted: Understanding the relationship of tourism demands connecting with economy and tourism stock index in an extreme case of Thailand bivariate extreme value copula approach

Interestingly, there is no confirmation that the Bivariate Extreme Value models and Bivariate Extreme Value Copula models are employed to explore tourism variables, especially studying in Thailand. This paper aims to statistically clarify extreme cases in tourism demands connecting with a macroeconomic level and micro tourism factor. Three quarterly indexes, including Thailand economic expansion rates, CENTEL stock indexes, and tourist arrivals were collected during 2003 to 2015. Empirically, the results showed extreme value copula models can better capture the dependence structure between three economic components than using the bivariate extreme value models. In addition, the extreme dependent results among the three comparable variables can be implied that the special tourism policies for extreme situations should not be neglected. It is reasonable to simultaneously adjusted tourism policies to both normal cases and critical situations during the whole year.

Extreme Value Analysis of Multivariate High-Frequency Wind Speed Data

In this article we analyze the extremal behavior of wind speed with a measurement frequency of 8 Hz, measured on three meteorological masts in Denmark. In the first part of this article we set up a conditional model for the time series consisting of threshold exceedances from maxima per second for two consecutive days. The model directly captures the nonstationary nature of wind speed during the day. Conditional on previous wind speed values with recorded exceedance, we assume a Markov-like structure for exceedances, where the conditional distribution follows the generalized Pareto distribution. In addition, we analyze the dependence structure in large wind speed values between different masts by using bivariate extreme value models. The initial motivation for this research was in the context of renewable energy. Specifically, the extremal dynamics of wind speed at small time scales plays a critical role for designing and locating turbines on wind farms.

Statistical models for bivariate extremal analysis of a spatial process

In the present paper different dependence measures are examined to investigate the structure of a spatial process of storm surges in the Dutch part of the North Sea. Four different dependence measures are implemented to the data: (1) the correlation coefficient as well as the Spearman (rank) correlation of the sample, (2) the transformation of the variables to determine the marginal distributions, (3) the dependence measures (χ, ) and (4) the coefficient of tail dependence η. Two different approaches for modelling the threshold exceedances are examined, namely the implementation of bivariate extreme value models and the use of more general point process characterizations of extremal dependence. The strength of dependence appears to vary with the location as well as with the separation distance. It is also observed that the point process approach estimates are in general lower than those using the bivariate threshold exceedance models.

Medindo a Influência do Mercado dos EUA sobre as Interdependências Observadas na América Latina

A modelagem da estrutura de dependência entre os extremos pode ser feita via modelos paramétricos diferenciáveis que são classificados em duas famílias: logística e mista, que por sua vez contêm modelos simétricos e assimétricos. Os modelos bivariados são muito úteis em aplicações práticas da teoria de valores extremos, em particular na área financeira. Considerando a forte influência do mercado norte americano nos demais mercados financeiros, investigamos se a estrutura de dependência entre os mercados latino americanos se altera após a retirada dessa influência comum. Para isso, usamos um modelo de regressão polinomial com erros GARCH(1,1) e ajustamos modelos bivariados de valores extremos aos resíduos padronizados.

On Pickands coordinates in arbitrary dimensions

Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins. In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral δ-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks–over–threshold models are established, which are based on Pickands coordinates.

A characterization of the rate of convergence in bivariate extreme value models

It is well known that the rate of convergence of the extremes in an iid sample of univariate random variables is determined by the distance of the underlying distribution from a generalized Pareto distribution. We extend this result to higher dimensions.

Improving financial risk assessment through dependency

Understanding dependency between financial markets is crucial when measuring globally integrated exposures to risk. To this end the first step may be the investigation of the joint behaviour of their most representative indexes. We fit by parametric and nonparametric methods bivariate extreme value models on the component wise maxima and minima computed monthly from several pairs of indexes representing the North American, Latin American, and Emerging markets. We analyse the role of the asymmetric models, finding which market drives the dependency, and express the degrees of dependence using measures of linear and nonlinear dependency such as the linear correlation coefficient ρ and the measure τ based on the dependence function. We discuss the interpretation of τ as a conditional probability that a crash occurs in a market given that a catastrophic event has occurred in some other market. We assess risks by computing probabilities associated with joint extreme events and by computing joint risk measures. We show empirically that the joint Value-at-Risk may be severely under-estimated if independence is assumed between markets. To take into account the clustering of extreme events we compute the bivariate extremal index and incorporate this information in the analysis.

Robust weighted likelihood estimators with an application to bivariate extreme value problems

AbstractThe authors achieve robust estimation of parametric models through the use of weighted maximum likelihood techniques. A new estimator is proposed and its good properties illustrated through examples. Ease of implementation is an attractive property of the new estimator. The new estimator downweights with respect to the model and can be used for complicated likelihoods such as those involved in bivariate extreme value problems. New weight functions, tailored for these problems, are constructed. The increased insight provided by our robust fits to these bivariate extreme value models is exhibited through the analysis of sea levels at two East Coast sites in the United Kingdom.

Extremal limit laws for a class of bivariate Poisson vectors

It is well known that conventional extreme value limit laws break down for the Poisson distribution: no normalization can be found to avoid degeneracy of the limit law of sample maxima. Anderson et al. (Ann. Appl. Probab. 7 (1997) 953) tackled this problem with a triangular array argument, letting both the sample size and Poisson mean grow at appropriate rates. This leads to a Gumbel limit law for sample maxima. In applications, this means that it may be appropriate to model extremes of Poisson processes using standard extreme value models and techniques. This paper extends the limit results to a class of bivariate Poisson distributions. Suitably normalized, and with a degree of dependence that is also permitted to grow at a suitable rate, we find that the limit distribution corresponds to the class of bivariate extreme value models that would have arisen, had the population been bivariate normal, cf. Hüsler and Reiss (Statist. Probab. Lett. 7 (1989) 283). This adds weight to the argument that, for practical applications involving Poisson variables, even in the presence of dependence, standard extreme value models can be applied, despite the degeneracy that arises by applying the usual asymptotic argument.