Generalized Pareto Distribution Model Research Articles

Analysis of the upper tail of the short-term extreme tension distribution of mooring line by the peaks-over-threshold method

The consistency of the upper tail of the extreme response distribution plays an important role in the accuracy of the extreme response prediction. Although the peaks-over-threshold method has a high data utilization of the response time series, the consistency of the upper tail of the extreme response distribution was not thoroughly investigated, especially for the inherent randomness of the short-term sea state. This study aims to estimate the upper tail of the short-term extreme tension distribution of the mooring line considering the short-term variability by the peaks-over-threshold method. The consistency of the extreme tension distribution tail is quantified by a relative error compared with the global maximum method, and the theoretical optimal threshold is defined by minimizing the largest relative error among the tail quantiles for different thresholds. Results from a case study for the mooring line of the fish cage indicate that the effect of the random seed is significant for consistency in the prediction of the distribution tail of the short-term extreme tension. And the generalized Pareto distribution model combined with the L-moments method is recommended for the theoretical optimal threshold. These findings provide some useful suggestions for the preliminary design of the mooring line for offshore structures.

Spatio-temporal dependence modelling of extreme rainfall in South Africa: A Bayesian integrated nested Laplace approximation technique

The spatial and spatio-temporal dependence modeling to extreme value distributions have been used to analyze the extremes of daily maximum rainfall data across selected weather stations in South Africa combining generalized Pareto distribution (GPD) with the flexible Bayesian Latent Gaussian Model (LGM). The paper demonstrated the spatio-temporal GPD model for applications in extreme rainfall data that capture systematic variation through spatial and spatio-temporal modeling framework, in which the temporal constitutes the week and month as random separately. The paper uses the Bayesian integrated Nested Laplace approximation (INLA) algorithm to estimate marginal posterior means of the parameters and hyper-parameters for Bayesian spatio-temporal models. The Bayesian inferences using INLA technique were applied to obtain prediction of the return levels at each station, which incorporate uncertainty due to model estimation, as well as the randomness that is inherent in the processes.

Modeling GPDA and Its Application in Deep Mineral Prediction in the Jiguanzui Cu-Au Mining Area in Hubei, China

Geochemical anomalies are the basis of mineral deposit prediction. Through the study of the characteristics of geochemical anomalies, we found that their distribution was consistent with a generalized Pareto distribution (GPD). In the present study, we designed a model for geochemical anomaly extraction via a GPD. In the designed GPD model, we used the kurtosis method to estimate the threshold value of the GPD. Furthermore, a principal component analysis (PCA) was used to extract comprehensive information of different geochemical elements in which minerals are enriched. On this basis, a new algorithm named the GPDA model was designed for deep mineral prediction by using the GPD and PCA, and the methods of the GPDA for selecting parameters were studied. The study data for Ba, Pb, As, Cu, Au, Mo, Co, and Zn originated from 26 exploration lines of the Jiguanzui Cu-Au mining area in Hubei, China. The proposed GPDA model was applied to deep mineral prediction in the study area. We estimated the parameters of the GPDA model, and the thresholds of Ba, Pb, As, Cu, Au, Mo, Co, and Zn were 457.8612, 56.1823, 28.8454, 910.1272, 89.4283, 34.5267, 84.9445, and 121.4863, respectively. The comprehensive information threshold value was 0.4551. The comprehensive abnormal distribution area of geochemical element contents was obtained from thresholds. The results showed that the method used to identify abnormal areas was consistent with the range of ore bodies identified by actual engineering exploration, demonstrating that the GPDA model was effective. Finally, we predicted that there was a new blind ore body located at a depth of about 1100 m below ground between drill holes KZK10 and KZK11. The results have important theoretical and practical significance for deep ore prospecting.

Can jumps improve the futures margin level? An empirical study based on an SE-SVCJ-GPD model

In addition to the characteristics of leptokurtic fat-tailed distribution, financial sequences also exhibit typical volatility and jumps. Moreover, jumps exhibit self-exciting and clustering characteristics under extreme events. However, studies on dynamic margin levels often ignore jumps. In this study, we combine the self-exciting stochastic volatility with correlated jumps (SE-SVCJ) model with a generalized Pareto distribution (GPD) to measure the optimal margin level for the stock index futures market. Value at risk (VaR) is estimated and forecasted using the SE-SVCJ-GPD, SVCJ-GPD, and generalized autoregressive conditional heteroskedasticity with GPD (GARCH-GPD) models. SE-SVCJ-GPD can undertake more risks in the long or short trading position of stock index futures contracts. Moreover, the backtesting experiment results show that the SE-SVCJ-GPD model provides a more accurate margin level forecast than the other methods in both positions. This study’s findings have practical significance and theoretical value for assessing the level of risk and taking corresponding risk-prevention measures.

Modelling of extreme uniform temperature for high-speed railway bridge piers using maximum entropy and field monitoring

Due to the atmospheric temperature and solar radiation, the effects of temperature variation in bridge structures should be considered. Such variation induces notable deformations and movements, jeopardizing the safety of bridges and high-speed trains operations. However, the temperature action is a random process and its distribution is difficult to determine. The existing methods for analyzing structural temperatures are insufficient to meet the precision requirement. Therefore, the accurate prediction of extreme structural temperatures relates to the accurate evaluation on the bridge safety. This paper proposes a robust and accurate model for predicting the extreme structural temperature of a bridge. A field experiment, spanning over 2 years, was carried out on a high-speed railway bridge; and the long-term (56-years) atmospheric temperature data was adopted. Probabilistic models for the structural temperature were established using the Maximum Entropy (MaxEnt) model and the Generalized Pareto distribution (GPD) model; and the predictions on the uniform structural temperature ( T u) with 50 and 100 years return periods are presented. Additionally, the performance between the MaxEnt model and the GPD model is compared, based on the estimates with different return levels. The results show that the MaxEnt model is more stable and is significantly robust to the variation of sample sizes; and indicates that the MaxEnt model reduces the uncertainty of outcomes and avoids the high risk of bias. The MaxEnt model has a great potential to the applications of the extreme value analysis with small sample size. It offers a wider applicability and helps solve practical problems.

The East Asian Summer Monsoon Response to Global Warming in a High Resolution Coupled Model: Mean and Extremes

Current climate models still have considerable biases in the simulation of the East Asian summer monsoon (EASM), which in turn reduces their reliability of monsoon projections under global warming. We hypothesize that a higher-resolution coupled climate model with atmospheric and oceanic components at horizontal resolutions of 0.25° and 0.1°, respectively, will better capture regional details and extremes of the EASM. Present-day (PD), 2 × CO2 and 4 × CO2 simulations are conducted with the Community Earth System Model (CESM1.2.2) to evaluate PD simulation performance and quantify future changes. Indeed, our PD simulation well reproduces the climatological seasonal mean and intra-seasonal northward advancement of the monsoon rainband, as well as climate extremes. Compared with the PD simulation, the perturbed CO2 experiments show an intensified EASM response to CO2-induced warming. We find that the precipitation increases of the Meiyu-Baiu-Changma band are caused by comparable contributions from the dynamical and thermodynamical components in 2 × CO2, while they are more driven by the thermodynamical component in 4 × CO2 due to stronger upper atmospheric stability. The regional changes in the probability distribution of the temperature show that extreme temperatures warm faster than the most often temperatures, increasing the skewness. Fitting extreme precipitation values with a generalized Pareto distribution model reveals that they increase significantly in 4 × CO2. Changes of temperature extremes scale with the CO2 concentrations over the monsoon domain but not for precipitation extreme changes. The 99th percentile of precipitation over the monsoon region increases at a super Clausius-Clapeyron rate, ~ 8% K–1, which is mainly caused by increased moisture transport through anomalous southerly winds.

Climate change impact on extreme value and its frequency distribution function in a karst basin, Southwest China

Abstract Studying extreme meteorology and its frequency under climate change is helpful to guide flood and drought control. The original achievements and objective of this study are to further contribute to the literature on how to analyze the impact of climate change on extreme rainfall and extreme temperature more reasonably and comprehensively for a karst basin. The Mann–Kendall method, Heuristic segmentation method, cross-wavelet analysis method, generalized extreme value (GEV) model, and generalized Pareto distribution (GPD) model were applied in this paper. The 55-year (1963–2017) extreme rainfall and temperature data recorded in the Chengbi River Basin were applied. The results show that extreme rainfall showed a downward trend (−0.169 and −8.735 mm/10a), while the trends of extreme temperatures were not obvious (Sen's slope estimate is 0). The mutation points range from 1981 to 2002 and the mutation point of extreme rainfall series is earlier than that of extreme temperatures. Compared with the GEV model, the parameters of the GPD model show a smaller variation before and after climate change, and the extreme meteorology values corresponding to the same recurrence period show a decreasing trend after climate change. The performance of GEV and GPD models after climate change is generally more fit than that before climate change.

Global extreme significant wave height within the dominant directional sector

The assessment of global directional extreme significant wave height (SWH) is crucial for the ocean infrastructure, which requires the dominant directional sector to be identified. In this study, 40-year (1979–2018) global SWHs and wave directions obtained from ERA5 are employed as the initial database. To study the extreme wave, storms over the initial threshold are extracted, which show that the non-tropical cyclonic storm is stable with the direction and most directions are within the dominant directional sector. Thus, the directional extrapolation is performed based on the independent non-tropical cyclone sample. To identify the dominant directional sector, an automated method is proposed based on the circular mean sample. Benefiting from this method, global patterns of sector width and representative direction of dominant directional sector can be obtained and classified. To make sufficient use of extreme samples for reasonable extrapolation, the peak over threshold method with the generalized Pareto distribution model is employed to evaluate the global directional extreme SWHs. The spatially comparable patterns accord with the distribution of wind zones and reveal the effect of swells in low latitudes, which can be used to realize design optimization and key protection of ocean infrastructures, especially for infrastructures that are not symmetrical.

Uncertainty analysis of strong earthquake hazard estimation based on the generalized Pareto distribution model: A case study of the northeastern Tibetan Plateau

A statistical method to analyze the uncertainties of strong earthquake hazard estimation is proposed from Generalized Pareto distribution (GPD) model using the northeastern Tibetan Plateau earthquake catalogue (1885–2017) data. For magnitude threshold of 5.5, the magnitude return levels in 20, 50, 100, 200, and 500 years are 7.19, 7.70, 7.99, 8.22, and 8.45, respectively. The corresponding 95% confidence intervals are [6.77, 7.60], [7.27, 8.12], [7.53, 8.44], [7.71, 8.72], and [7.84, 9.06], respectively. The upper magnitude limit obtained from this GPD model is 9.07 and its 80% confidence interval is [8.16, 9.98]. The sensitivity analysis by the Morris method indicates that the input magnitude threshold has a relatively large influence on the estimation results. Thus, threshold selection is important for the GPD model construction. The sensitivity characteristic ranking of input factors become increasingly stable with the increasing of return period, which implies that GPD model is more suitable for estimating strong earthquakes magnitude return levels and upper magnitude limit. The GPD modeling approach and qualitative uncertainty analysis methods for strong earthquake hazard estimations proposed in this paper can be applied to seismic hazard analysis elsewhere.

Global risks of infectious disease outbreaks and its relation to climate

Infectious disease outbreaks are recurring events which can lead to a large number of fatalities every year. Infectious disease outbreaks occur infrequently and the role of global warming and modes of climate variability for those outbreaks is not clear. Here we use an extreme value statistics approach to examine annual spatially aggregated infectious disease fatality data to compute their probability to occur using generalized Pareto distribution (GPD) models. The GPD provides a good model for modeling the fatality data and reveals that the number of fatalities follows a power-law. We find that the magnitude of Covid-19 is of an expected level given previous fatality data over the period 1900–2020. We also examined whether including co-variates in the GPD models provide better model fits. We find evidence that a pure linear trend is the best co-variate and, thus, has increased the propensity of large outbreaks to occur for most continents and world-wide. This suggests that mainly non-climate factors affect the likelihood of outbreaks. This linear trend function provides a crude representation of socio-economic trends such as improved public health. However, for South America the Atlantic multidecadal oscillation modulates the outbreak propensity as the best co-variate, showing that climate can play some role in infectious disease outbreaks in some regions.

Towards the development of economic damage functions for weather and climate extremes

An important aspect of modeling the coupled human–Earth system is how weather and climate affects the economy. In particular, extreme weather and climate events can cause significant economic damages. Hence, there is a need to better understand the relationship between economic damages and extreme weather and climate events. Here, we analyze economic damage data using regression and probabilistic methods. We provide evidence for upward trends for inflation adjusted damages caused by extreme weather and climate events, while damages normalized by wealth have downward trends. We also find that relative to its wealth, Europe is the most affected continent followed by North America and Asia. We also identify covariates which affect economic damages. Multivariate regression models using either socio-economic or climate covariates explain only small amounts of the damage variation while the probabilistic Generalized Pareto Distribution (GPD) model with covariates fits the damage data well. This suggests that economic damages due to weather and climate extreme events should be modeled using probabilistic methods. Using non-stationary GPD models, we find that mainly population is a significant covariate while also Gross Domestic Product (GDP), global mean surface temperature and modes of climate variability are significant covariates too.

A hierarchical bayesian peak over threshold approach for conflict-based before-after safety evaluation of leading pedestrian intervals

A hierarchical Bayesian peak over threshold (POT) approach is proposed for conflict-based before-after safety evaluation of Leading Pedestrian Intervals (LPI). The approach combines traffic conflicts of different sites and periods to develop a uniform generalized Pareto distribution (GPD) model for the treatment effect estimation. The hierarchical structure has three levels, a data level that consists of modeling the traffic conflict extremes through the POT approach, a latent process level that relates GPD parameters of the data level to certain covariates, and a prior level with prior distributions to characterize the latent process. The approach was applied to a before-after (BA) safety evaluation of leading pedestrian interval (LPI) in Vancouver, BC. Pedestrian-vehicle traffic conflicts were collected from treatment and control sites during the before and after periods using an automated computer vision analysis technique. The treatment effect was measured by the best fitted GPD model with the calculation of the odds ratio (OR). The overall treatment effect varies from 18.1%–20.9% in terms of reduction in the estimated extreme-serious conflicts. The treatment effect indicates a considerable improvement in pedestrian safety after the implementation of the LPI, and the consistent results demonstrate a reliable BA safety evaluation. As such, the proposed approach is recommended as a promising tool for BA safety studies, particularly in cases where collision data is limited.

Modeling Malicious Hacking Data Breach Risks

Malicious hacking data breaches cause millions of dollars in financial losses each year, and more companies are seeking cyber insurance coverage. The lack of suitable statistical approaches for scoring breach risks is an obstacle in the insurance industry. We propose a novel frequency–severity model to analyze hacking breach risks at the individual company level, which would be valuable for underwriting purposes. We find that breach frequency can be modeled by a hurdle Poisson model, which is different from the negative binomial model used in the literature. The breach severity shows a heavy tail that can be captured by a nonparametric- generalized Pareto distribution model. We further discover a positive nonlinear dependence between frequency and severity, which our model also accommodates. Both the in-sample and out-of-sample studies show that the proposed frequency–severity model that accommodates nonlinear dependence has satisfactory performance and is superior to the other models, including the independence frequency–severity and Tweedie models.

Hierarchical-Generalized Pareto Model for Estimation of Unhealthy Air Pollution Index

A common way of modeling the exceedance of air pollution index (API) data is to utilize the generalized Pareto distribution (GPD). The marginal GPD model is a good method for describing unhealthy API data. However, for data with multiple locations, integrating the information of GDP models from each location with a hierarchical model (HM) will provide a better result. In this study, a hierarchical-generalized Pareto model (HM-GPD) is used to integrate the information about location and seasonal effects from the marginal GPD models of hourly API exceedance data, along with the information of serial dependence at each location. The accuracy of inferences at a single site and in each season was improved by employing a Gaussian model for the random effects; this model takes advantage of the climatological structure in the data. The temporal dependence was modeled by using first-order Markov chains. This step was performed by operating the posterior draws from HM-GPD to transform the marginal models for each site to unit Frechet by using the Markov chain model likelihood. Overall, the parameters estimated from the HM-GPD are able to provide precise estimation of the return levels for each site.

Study of Directional Declustering for Estimating Extreme Wave Heights in the Yellow Sea

The study of extreme waves is important for the protection of coastal and ocean structures. In this work, a 22-year (1990–2011) wave hindcast in the Yellow Sea is employed to perform the assessment of extreme significant wave heights in this area. To extract the independent sample from this database, the fixed window method is used, which takes the peak significant wave height within five d. With the selected samples, directional declustering is studied to extract the homogenous sample. The results show that most of the independent samples (especially large samples) are observed in the North. In this direction, the peak over threshold (POT) method is used to extract the extreme sample from the homogenous sample, and then the generalized Pareto distribution model is used to extrapolate the extreme significant wave height. In addition to this combination, the annual maxima method with the Gumbel model is also used for estimating extreme values. The comparisons show that the return significant wave heights of the first combination are reliable, resulting from a flexible sampling window in the POT method. With this conclusion, the extreme significant wave height is extrapolated from the Yellow Sea, which can be used to protect the structure in the main directional bin.

Risk assessment of extreme air pollution based on partial duration series: IDF approach

The occurrences of extreme pollution events have serious effects on human health, environmental ecosystems, and the national economy. To gain a better understanding of this issue, risk assessments on the behavior of these events must be effectively designed to anticipate the likelihood of their occurrence. In this study, we propose using the intensity–duration–frequency (IDF) technique to describe the relationship of pollution intensity (i) to its duration (d) and return period (T). As a case study, we used data from the city of Klang, Malaysia. The construction of IDF curves involves a process of determining a partial duration series of an extreme pollution event. Based on PDS data, a generalized Pareto distribution (GPD) is used to represent its probabilistic behaviors. The estimated return period and IDF curves for pollution intensities corresponding to various return periods are determined based on the fitted GPD model. The results reveal that pollution intensities in Klang tend to increase with increases in the length of time between return periods. Although the IDF curves show different magnitudes for different return periods, all the curves show similar increasing trends. In fact, longer return periods are associated with higher estimates of pollution intensity. Based on the study results, we can conclude that the IDF approach provides a good basis for decision-makers to evaluate the expected risk of future extreme pollution events.

Spatial distribution of the level of return of extreme precipitation events in Northeast Brazil

AbstractThis study aimed to estimate levels of return of extreme daily precipitation events, associating them with natural disasters in Northeast Brazil (NEB), a region characterized by different climatic conditions and low rates of social and economic development. For this, generalized Pareto distribution (GPD) models were adjusted to the daily extreme precipitation data estimated by the Tropical Rainfall Measuring Mission (TRMM) 3B42 product of the multisatellite precipitation analysis for a period of 16 years (2000–2015). In addition, the estimates of the GPD model were compared using two data sources, TRMM and pluviometer. The investigation showed that the results of the GPD model estimated by means of the extreme data from the rain gauge and the TRMM were statistically the same, with 95% confidence. Thus, using the data referring to the 2,082 grid points of the TRMM, it was possible to map the spatial distribution of the estimates of the levels of return of extreme precipitation to the return periods of 2, 5 and 10 years, per seasonal period. In general, the results indicated that the intensity of expected extreme precipitation depends on the seasonal period and the place of occurrence of precipitation. The eastern NEB stood out as the region where the highest intensities of extreme precipitation are expected. Extreme precipitation values of up to 178 mm are expected in 2 years. The areas where natural disasters occurred in the years 2016, 2017 and 2018 are similar to those in which the highest rainfall intensities are expected. The results of this study can allow the evaluation of the spatial distribution of risks related to extreme precipitation events, and therefore, support the planning of regional public policies and environmental management for the prevention of natural disasters in NEB.

The coupling method in extreme value theory

A coupling method is developed for univariate extreme value theory, providing an alternative to the use of the tail empirical/quantile processes. We emphasize the Peak-over-Threshold approach that approximates the distribution above high threshold by the Generalized Pareto Distribution (GPD) and compare the empirical distribution of exceedances to the empirical distribution associated with the limit GPD model. Sharp bounds for their Wasserstein distance in the second order Wasserstein space are provided. As an application, we recover standard results on the asymptotic behavior of the Hill estimator, the Weissman extreme quantile estimator or the probability weighted moment estimators, shedding some new light on the theory.

Bayesian hierarchical modeling of traffic conflict extremes for crash estimation: A non-stationary peak over threshold approach

This study presents a Bayesian hierarchical model to estimate crashes from traffic conflict extremes in a non-stationary context. The model combines a peak over threshold approach with non-stationary thresholds in terms of regression quantiles and covariate-dependent parameters of the generalized Pareto distribution. The developed model was applied to estimate rear-end crashes from traffic conflicts of the same type collected from four signalized intersections. The conflicts were measured by the modified time to collision (MTTC) and traffic volume, shock wave area, average shock wave speed, and platoon ratio of each signal cycle were employed as covariates. Thresholds corresponding to quantiles ranging from 80% to 95% were tested and the threshold stability plot indicated the 90% quantile was reasonable. Threshold excesses were then declustered at the signal cycle level, and the remained ones were used to develop the Bayesian hierarchical generalized Pareto distribution models (BHM_GPD). The model estimation results show that accounting for non-stationarity significantly improves the model fit. As well, the best fitted model generated accurate crash estimates with relatively narrow confidence intervals. The developed BHM_GPD model was also compared to the Bayesian hierarchical generalized extreme value model (BHM_GEV). The results show that the two models generate comparable crash estimates in terms of accuracy, but the crash estimates from the BHM_GPD model are generally more precise than those of BHM_GEV model. It is also found that although the peak over threshold approach combined with declustering reduces the number of extreme samples, it ensures the use of actual extremes. Moreover, the limited sample size issue is overcome by the proposed Bayesian hierarchical framework, which allows sharing information from different sites and accounting for unobserved heterogeneity. The findings also imply that the BHM_GEV model is preferred when traffic conflicts are relatively evenly distributed over blocks; otherwise the BHM_GPD model should be a better choice.

Tree-based Threshold Model for Non-stationary Extremes with Application to the Air Pollution Index Data

Air pollution index (API) is a common tool used to describe the air quality in the environment. High level of API indicates the greater level of air pollution which will gives bad impact on human health. Statistical model for high level of API is important for the purpose of forecasting the level of API so that the public can be warned. In this study, extremes of API are modelled using Generalized Pareto Distribution (GPD). Since the values of API are determined by the value of five pollutants namely sulphur dioxide, nitrogen dioxide, carbon monoxide, ozone and suspended particulate matter, data on API exhibit non-stationarity. Standard method for modelling the non-stationary extremes using GPD is by fixing the high constant threshold and incorporating the covariate model in the GPD parameters for data above the threshold to account for the non-stationarity. However, high constant threshold value might be high enough on certain covariate for GPD approximation to be a valid model for extreme values, but not on the other covariates which leads to the violation of the asymptotic basis of GPD model. New method for the threshold selection in non-stationary extremes modelling using regression tree is proposed to the API data. Regression tree is used to partition the API data into a stationary group with similar covariate condition. Then, a high threshold value can be applied within a group. Study shows that model for extremes of API using tree-based threshold gives a good fit and provides an alternative to the model based on standard method.