Generalized Pareto Distribution Research Articles

40 Years Extreme Offshore Waves off the Eastern Continental Shelf of Rio Grande do Norte, Northeast Brazil

Unplanned occupation and extreme metocean events bring disasters to coastal lowlands. Such processes have promoted damages to backshore structures along the Rio Grande do Norte State (RN), Northeast Brazil, harming goods, industries, tourism, traditional populations and sensitive ecosystems. The RN’s coast lacks long-term monitoring data of metocean drivers, which are useful for decision-takers. This study aims to assess extreme ocean waves off RN’s continental shelf. Time series of wind speed, tide and wave parameters have been obtained from AVISO satellite altimetry and ERA5 reanalysis, between 1979 and 2021, in a virtual station at 6.0ºS/34.7ºW. Generalized Pareto Distribution (GPD) fitting of wave significant high (Hs) was performed. Registers of disasters along the coast of study area were collected from local online media. Waves have reached Hs maximum of 3.30 m, average of 0.76 m, coming from east-southeast and southeast. Winter and spring seasons have presented higher waves, related to action of trade winds. Wind speed occurred mainly between 6 and 8 m/s. GPD estimated Hs increasing 2 (summer, winter and spring) and 2.4 times (autumn) above its average, each 2 years. Per decade, Hs may overcome 2.7 times its average in autumn – season that presented the highest proportion of increasing. Previous increasing of wind speed is more important than its scalar value, to form an extreme wave. Hs at the virtual station had its upper peak either at the same day or one-two days before of coastal damage, which means that ocean storms offshore can propagate coastward quickly.

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.

Improved Extreme Rainfall Frequency Analysis Using a Two‐Step Kappa Approach

AbstractAccurate estimation of annual exceedance probabilities (AEPs) of extreme rainfalls through rainfall frequency analysis is a critical step in engineering design for flood mitigation and disaster response. Here we show how the estimation of rainfall frequency curves can be improved by fitting a four‐parameter Kappa distribution to a peaks‐over‐threshold (POT) series. To fit the Kappa distribution to POT data we present a two‐step fitting approach based on maximum likelihood estimation which separately models storm intensity and the arrival frequency. First, a Generalized Pareto distribution (GPA) describing storm intensity is fitted, followed by a Binomial distribution for storm arrivals. We compare the performance of this two‐step Kappa approach to an analogous two‐step Generalized Extreme Value (GEV) approach and to Kappa and GEV distributions fitted to annual maxima series (AMS), using both synthetic and real‐world data representative of Australian climatic conditions. The experiments show that leveraging additional information from the POT series in the two‐step Kappa approach dramatically improves quantile estimation and reduces uncertainty compared to fitting either the GEV or the Kappa distributions to AMS, particularly for rare quantiles. When skew–kurtosis properties of extreme rainfalls are well represented by the Kappa but not the GEV, the two‐step Kappa approach yields unbiased quantiles under extrapolation, while the use of GEV can lead to highly biased estimates. From these results, we believe the two‐step Kappa approach is suitable for both at‐site and regional rainfall frequency analyses as it can accommodate distributions ranging from GPA to GEV without encountering over‐fitting problems generally associated with four‐parameter distributions.

Determination of Current and Future Extreme Sea Levels at the Local Scale in Port-Bouët Bay (Côte d’Ivoire)

The Port-Bouët Bay shoreline is threatened by extreme sea level (ESL) events, which result from the combination of storm tide, wave run-up, and sea level rise (SLR). This study provides comprehensive scenarios of current and future ESLs at the local scale along the bay to understand the evolution of the phenomenon and promote local adaptation. The methodological steps involve first reconstructing historical storm tide and wave run-up data using a hydrodynamic model (D-flow FM) and the empirical model of Stockdon et al. Second, the Generalized Pareto Distribution (GPD) model fitted to the Peaks-Over-Thresholds (POT) method is applied to the data to calculate extreme return levels. Third, we combine the extreme storm tide and wave run-up using the joint probability method to obtain the current ESLs. Finally, the current ESLs are integrated with recent SLR projections to provide future ESL estimates. The results show that the current ESLs are relatively high, with 100-year return levels of 4.37 m ± 0.51, 4.97 m ± 0.57, and 4.48 m ± 0.5 at Vridi, Petit-Bassam, and Sogefiha respectively. By end-century, under the SSP1-2.6, SSP2-4.5, and SSP5-8.5 scenarios, the future SLR is expected to increase the current ESLs by 0.49 m, 0.62 m, and 0.84 m, respectively. This could lead to a more frequent occurrence of the current 100-year return period, happening once every 2 years by 2100, especially under SSP5-8.5. The developed SLR scenarios can be used to assess the potential coastal flood risk in the study area for sustainable and effective coastal management and planning.

Estimation of the Peak over Threshold-Based Design Rainfall and Its Spatial Variability in the Upper Vistula River Basin, Poland

The proper assessment of design rainfalls with long return periods is very important because they are inputs for many flood studies. In this paper, estimations are performed on daily design rainfall totals from 16 meteorological stations located in the area of the Upper Vistula River Basin (UVB), Poland. The study material consists of a historical series of daily rainfall totals from the period of 1960–2021. The peak over threshold (POT) method is used, and the rainfall depth over threshold is assumed to follow the generalized Pareto distribution (GPD) with parameters estimated from Hill statistics. Alternatively, the competitive method based on annual maxima (AM) is applied. The theoretical distribution of AM is assumed to follow a theoretical distribution function selected by using the Akaike information criterion (AIC) from a family of seven candidate distributions, the parameters of which are estimated by using the maximum likelihood method. The two methods are compared by using the root mean square error (RMSE) and the mean deviation error (MDE) criteria. It is found that the POT-based method with GPD and Hill estimators outperform the AM-based method when considering the highest rainfall events. The confidence intervals of the design rainfalls, derived by using the Monte Carlo simulation method, reflects their large spatial diversity across the UVB. It is shown that the station’s altitude strongly correlates with the threshold, variance, and design rainfall depth of the GPD. This proves the advantage of the GPD with Hill estimates, namely that it can accurately reflect the spatial properties of rainfall and its variability in the UVB. Results can be applied in water-management applications related to floods.

An integrated connectivity diagnostics and dependency analysis framework for supporting water replenishment management

An in-depth investigation on hydrological connectivity and its relationships with water level are fundamental for regional water replenishment management. In this research, an integrated connectivity diagnostics and dependency analysis framework (ICDDAF) was developed for determining the optimal ecological water level in Baiyangdian Wetland, North China. It consisted of two core modules: hydrological connectivity diagnostics (HCD) and hydrological connectivity–water level coexistence analysis (HC–WLCA). The results were as follows. 1) In the HCD module, remote sensing coupling geostatistical connectivity function methods can accurately evaluate the hydrological connectivity considering the directional difference and the validity of distance range. For intra–annual variations, the hydrological connectivity was relatively good in March at the monthly scale, spring at the seasonal scale and the west–east direction, while relatively weak in September and summer, which were related to the periodic characteristics of vegetation growth, overall terrain and flow direction. For inter–annual variations, it was generally weak and showed a downwards, then upwards and finally downwards trend. 2) In the HC–WLCA module, Multivariate Copula Analysis Toolbox was innovatively adopted to describe the coexistence relationships of hydrological connectivity and water level. Beta and generalized Pareto distributions were selected as the optimal marginal distributions for hydrological connectivity and water level indicators respectively, and Frank copula was determined as the optimal joint distribution function; the modelling uncertainty in the coexistence relationships was focused. 3) The water level of 7 m was determined as the optimal ecological water level for maintaining good hydrological connectivity and restoring ecological environment. Considering the coexistence relationships of hydrological connectivity and water level, such an ICDDAF can provide a new perspective for efficient adjustments of water levels based on hydrological connectivity pattern variations under different water replenishment schemes, aiming to achieve ecological restoration of wetlands.

Identification and Characteristics of Historical Extreme High-Temperature Events over the China–Pakistan Economic Corridor

Recently, there has been an increase in the occurrence of extreme high-temperature events across the China–Pakistan Economic Corridor (CPEC). Regional spatiotemporal identification and evaluation of extreme high temperatures are essential for accurate forecasting of future climate changes. When such events generate a meteorological hazard, it is important to understand their temporal and spatial features, return period, and identification criteria. Accurately identifying extreme events can help assess risk and predict their spatial–temporal variation. While past studies have focused on individual sites, extreme heat events generally manifest as spatially and temporally continuous regional events. In this study, we propose an objective identification technique based on gridded data and spatiotemporal continuity to reveal the spatiotemporal characteristics of intensity, frequency, and duration events of extreme heat events in the CPEC from May to October between 1961 and 2015. Furthermore, we estimate the return period of extreme heat in the study region using the generalized Pareto distribution (GPD). Our findings indicate that the historical extreme temperature events (intensity, frequency, and duration) in the CPEC have significantly increased. Areas with a high incidence of extreme heat events are concentrated in eastern Balochistan, northern Sindh, and southeastern Punjab. These findings suggest that disaster prevention and mitigation plans should be targeted towards areas with a high frequency of extreme heat events in the CPEC, allowing policy makers to better prepare for and respond to future events.

An extreme wind speed climatology – Atmospheric driver identification using neural networks

Extreme wind speeds are a significant climate risk, potentially endangering human lives, causing damage to infrastructure, affecting maritime and aviation activity, along with the optimal operation of wind energy conversion systems. In this context, accurate knowledge of return levels for various return periods of extreme wind speeds and their atmospheric circulation drivers is essential for effective risk management. In this paper, location-specific extreme wind speed thresholds are identified and return levels of extremes are estimated using the Peaks-Over-Threshold method of the Extreme Value Analysis framework. Furthermore, using an environment-to-circulation approach, the key atmospheric circulation patterns that cause extreme wind speeds are identified. The data used for this analysis are hourly wind speed data, mean sea level pressure and geopotential at 500 hPa from the ERA5 reanalysis dataset, at a horizontal resolution of 0.25° × 0.25°. The thresholds are selected utilizing the Mean Residual Life plots, while the exceedances are modeled with the General Pareto Distribution. The diagnostic metrics exhibit satisfactory goodness-of-fit and the maxima of extreme wind speed return levels are located over marine and coastal areas. The optimal Self-Organizing-Map (2 × 2) is selected using the Davies-Bouldin criterion, and the atmospheric circulation patterns are related to the cyclonic activity in the area. The proposed methodological framework can be applied to other areas, that are endangered by extreme phenomena or in need of accurately assessing the principal drivers of extremes.

Influence of nonstationarity and dependence of extreme wave parameters on the reliability assessment of coastal structures - A case study

The current general consensus is that because of global climate change, coastal infrastructures may be subject to more extreme loads in the future than in the past. There has been an increased recognition that more attention should be paid to incorporating nonstationarity within the design process of coastal structures. In this regard, the present study examines the reliability of a conventional rubble mound breakwater using a variety of probabilistic techniques, incorporating ten nonstationary models to study the impacts of nonstationarity, as well as six copula functions to explore the dependence effects of wave extremes. The Gumbel copula most accurately represents the dependence structure of wave heights and periods in the study area. When the dependence of wave parameters is not correctly modeled, the failure probability of the studied structure is significantly underestimated. Assuming the Gumbel copula for modeling dependence of wave parameters, the results indicate that the maximum failure probability of the structure is underestimated by about 33% when a stationary extreme value distribution is fitted to wave height extremes, compared to a nonstationary Generalized Pareto Distribution (GPD). The insights gained from this study suggest that considering nonstationarity in extreme coastal events has more significant impacts on the reliability of coastal structures when the dependence structure of the marine variables is accurately modeled.

Extreme value modelling of SARS-CoV-2 community transmission using discrete generalized Pareto distributions.

Superspreading has been suggested to be a major driver of overall transmission in the case of SARS-CoV-2. It is, therefore, important to statistically investigate the tail features of superspreading events (SSEs) to better understand virus propagation and control. Our extreme value analysis of different sources of secondary case data indicates that case numbers of SSEs associated with SARS-CoV-2 may be fat-tailed, although substantially less so than predicted recently in the literature, but also less important relative to SSEs associated with SARS-CoV. The results caution against pooling data from both coronaviruses. This could provide policy- and decision-makers with a more reliable assessment of the tail exposure to SARS-CoV-2 contamination. Going further, we consider the broader problem of large community transmission. We study the tail behaviour of SARS-CoV-2 cluster cases documented both in official reports and in the media. Our results suggest that the observed cluster sizes have been fat-tailed in the vast majority of surveyed countries. We also give estimates and confidence intervals of the extreme potential risk for those countries. A key component of our methodology is up-to-date discrete generalized Pareto models which allow for maximum likelihood-based inference of data with a high degree of discreteness.

An efficient Bayesian modelling of extreme winds in the favour​ of energy generation in Pakistan

Daily and annual maximum wind speed quantiles can be estimated using extreme value theory for any metrological site of interest. These estimates are of vast importance for modelling and predicting maximum wind speed. This paper develops an efficient modelling paradigm of extreme winds by analysing daily and annual maximum wind speed data via frequentist and Bayesian methodologies. For this purpose, the generalized extreme value (GEV) model is used for yearly maxima, and the generalized Pareto distribution (GPD) is used for daily exceedance over a high threshold. In frequentist inference, the parameters of both models are estimated using the maximum likelihood and linear moments method. In contrast, the Bayesian Markov Chain Monte Carlo procedure with the Metropolis–Hasting algorithm is used. In addition, the informative priors for both models are constructed empirically using historical records of wind speed data from five other weather stations of Pakistan and one belonging to India. The results show that the Bayesian modelling provides apparent benefits in terms of improved accuracy in the estimation of the parameters as well as return levels of both distributions. Furthermore, the Bayesian analysis expresses that posterior inference might be affected by the choice of priors used to formulate the informative priors. Overall, based on assessment measures, the GPD fitted through Bayesian informative priors provides an efficient estimation strategy in terms of precision than other frameworks when uncertainty in parameters and return levels are taken into account. Our methodology can be implemented easily to other regions by considering the prior information from the border area stations of other countries (e.g., China, Afghanistan, India, and Iran). Moreover, the return level estimates of the GPD based on informative Bayesian priors are very beneficial in policymaking and wind energy generation engineering for the Thatta region of the country.

Uncertainty analysis of extreme mooring loads associated with environmental contours and peak tension distributions

This paper aims to assess the uncertainty on the extreme mooring loads of floating system considering short-term variability. Two environmental contour approaches based on the inverse First and Second Order Reliability Methods are employed to identify critical sea states that may give rise to extreme loads. The uncertainty related to the construction of environmental contours is addressed including significant differences due to marginal distribution fitting, parameter estimation methods and joint models. Three measured datasets are analysed using a known conditional joint distribution and proposed mixed copula model. 3-h time domain numerical simulation for each sea state is conducted and the characteristic extreme responses of mooring lines subjected to design loads are assessed. The uncertainties due to various statistical models including the average conditional exceedance rate method as well as global maxima, peak-over-threshold method combined with Gumbel distribution, Generalized Extreme Value distribution, Generalized Pareto distribution and 3-parameter Weibull distribution are investigated and quantified. It is observed that marginal distributions, joint models and parameters estimation methods have apparent effect on design loads estimation, and the extreme tensions of the semi-submersible platform shows significant difference using various probabilistic models. The results indicate that those epistemic uncertainties should be account for in the reliability analysis or safety factor calibration for mooring systems.

Improved inference for the generalized Pareto distribution under linear, power and exponential normalization

Improved inference for the generalized Pareto distribution under linear, power and exponential normalization

Best Fit versus Default Distribution and the Impact on the Reliability over the Design Lifetime of Hydraulic Structures

In the present study, Flood Frequency Analysis (FFA) is performed on the daily inflows of a reservoir dam taken as a case study. The Peaks-Over-Threshold (POT) approach was adopted. A comparison between the default generalized Pareto distribution and the best distribution fitted to the data has been carried out. After the risk analysis, the reliability of the structure decreases to 25.60% for the chosen threshold values if the best distribution is adopted instead of the default fit.

CharTM: The dynamic stability characterization for memory based on tail distribution modeling

Dynamic stability has become a major constraint to static random access memory (SRAM) circuits design. In this paper, the relationship between the read/write delay and corresponding yield will be given by our CharTM, an unified memory yield analysis method by modeling the tail of performance distribution. In charTM, we design an efficient weighted regression-based high-dimensional meta-model to collect the tail samples without running simulations and construct the tail distribution by generalized Pareto distribution(GPD) with an adaptive tail fraction selection strategy to improve the accuracy in the high-sigma region. The effectiveness of CharTM is verified on bit cell, read, and write critical paths of an SRAM. The results show that the efficiency of CharTM exceeds that of a commercial yield analysis tool by 5.3–16.5 times on our validation cases with the same level of accuracy, and exceeds that of state-of-the-art methods by 13–82 times with much higher accuracy, especially on read and write paths.

An Assessment of Uncertainties in Flood Frequency Estimation Using Bootstrapping and Monte Carlo Simulation

Reducing uncertainty in design flood estimates is an essential part of flood risk planning and management. This study presents results from flood frequency estimates and associated uncertainties for five commonly used probability distribution functions, extreme value type 1 (EV1), generalized extreme value (GEV), generalized pareto distribution (GPD), log normal (LN) and log Pearson type 3 (LP3). The study was conducted using Monte Carlo simulation (MCS) and bootstrapping (BS) methods for the 10 river catchments in eastern Australia. The parameters were estimated by applying the method of moments (for LP3, LN, and EV1) and L-moments (for GEV and GPD). Three-parameter distributions (e.g., LP3, GEV, and GPD) demonstrate a consistent estimation of confidence interval (CI), whereas two-parameter distributions show biased estimation. The results of this study also highlight the difficulty in flood frequency analysis, e.g., different probability distributions perform quite differently even in a smaller geographical area.

Extreme value analysis for waves in the Persian Gulf: Skill assessment of different methods for a fetch-limited basin

Estimating the design wave height with specific return period using extreme value analysis (EVA) is of significant importance in coastal and marine engineering. The purpose of this study is to compare different methods such as the Maximum Likelihood, the L-Moment and the least square methods to estimate the parameters of Generalized Pareto Distribution (GPD), Gumbel, and Weibull distributions in order to determine 100 years return period wave height at 16 selected points within the Persian Gulf. The Peaks Over Threshold (POT) method was employed for all tests. The hourly wave data were available for the total period of 30 years from 01/01/1990 to 31/12/2019 using reanalyzed ECMWF-ERA5 dataset as well as hindcast results from a calibrated local WAVEWATCH-III model. In order to find the threshold cutoff value, one of the pivotal parameters in EVA, adaptive approaches were employed which determined the threshold value based on the distribution function and the goodness of fit for extreme data at specific site. It was shown that subjective constant values used in most of previous studies in this water body were much less than what should be used to have the best fit. Moreover, the determined threshold value was not constant at different points and varied depending on the type of distribution employed. The results showed superiority of the Gumbel distribution in most of the points in the Persian Gulf. Sensitivity Analysis to the threshold values showed that using a low threshold value would lead to increasing the return wave height, which was significant in the case of Pareto and Gumbel distributions. Sensitivity analysis to input data confirmed that ECMWF-ERA5 underestimated the wave height and consequently, the underestimation of designed wave height at some locations by more than 40%. Also, the results from Annual Maxima Method (AMM) was similar to POT method but with ±20% differences in values.

Assessing the importance of the choice threshold in quantifying market risk under the POT approach (EVT)

From a theoretical point of view, the selection of thresholds is a critical issue in the framework of the Peaks Over Threshold (POT) approach, which is why in the last decade numerous methodologies have been proposed for its selection. In this paper, we address this subject from an empirical point of view by assessing to what extent the selection of the threshold is decisive in quantifying the market risk. For measuring market risk, we use the Value at Risk (VaR) and the Expected Shortfall (ES) measures. The results obtained indicate that there is a large set of thresholds that provide similar Generalized Pareto Distribution (GPD) quantiles estimators and as a consequence similar market risk measures. Just only, for large thresholds, those corresponding to the 98th and 99th percentile of the GPD some differences are found. It means that the choice of threshold in the framework of the POT method may not be relevant in quantifying market risk when we use the VaR and ES measures for this task.

Compilation and Extrapolation of Load Spectrum of Tractor Ground Vibration Load Based on CEEMDAN-POT Model

Effective indoor testing on a four-poster rig of agricultural machinery should mimic the real-world environment while the machinery is in farming and transportation. The emerging load spectrum extrapolation and compilation are capable of speeding up the use of the indoor vibration testing of tractors. However, most ground load spectrum studies consider a particular type of ground rather than all the rough farmland conditions that tractors experience daily. Such a problem inevitably raises doubts about validating the load spectrum-based fatigue durability and reliability testing of tractors. For a remedy, this article proposes a CEEMDAN-POT (Peak Over Threshold) model to comprehensively build a full life-cycle ground load spectrum of tractor vibration with six ground conditions and different field operations. The study acquires the real four vertical acceleration signals from the front and rear axles of the tractor to collect representative vibration load data. Furthermore, the study refines the measured load data by proposing a CEEMDAN-Wavelet threshold method, which is proved to be effective for the load signal decomposition and denoising. Lastly, the study presents a time domain extrapolation method, integrating the sample principle of the POT model, the proper parameter estimation with the Generalized Pareto Distribution (GPD) function, and the POT super-threshold model. The statistical analysis implies that the produced load spectrum fully preserves the statistical features observed in the original one. After extrapolation, the overall distribution of the rainflow matrix becomes more consistent while the mean and amplitude of the spectrum data increase. This study unifies a load spectrum of tractors operating and transporting on various farm ground conditions, providing the real load data of the indoor four-poster rig test.

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.