Extreme Value Distribution Research Articles

A generalized extreme value approach for the analysis of stationary climatic covariate in a Mediterranean city

AbstractExtreme value theory (EVT) is used as univariate extreme value analysis (EVA) in order to analyze and model the covariates temperature, relative humidity (RH) and the thermal comfort index (humidex) issued from a dataset of 38 years in Tunis. It is a South Mediterranean area known as a hotspot for climate change. The best approach is to reduce the data considerably by taking annual block maxima from mean monthly data. It will converge to a generalized extreme value distribution in order to estimate the return levels of the studied parameters. The stationarity of the series are checked by augmented Dickey‐Fuller test. The modeling of the three parameters shows a Weibull distribution pattern. The extreme/maximum monthly means temperature of 30.2°C and humidex of 39.4 have a common return level between 300 and 350 years. The highest mean monthly RH of 86.0% is expected to be exceeded every 50 years. For the next 38 years, the maxima monthly mean temperatures are expected to be stable, and the maxima monthly mean RH values, as well as the humidex monthly mean maxima are expected to decrease. The percentile air temperature hot day (TX90p) and night (TN90p) indices show globally linear upward trends and the ones of cold days (TX10p) and cold nights (TN10p) have a downward trend. The diurnal yearly temperature range shows an almost flat trend for its evolution through the years of study.

An objective methodology for waterlogging risk assessment based on the entropy weighting method and machine learning

Waterlogging disasters are one of the most severe and widespread agricultural meteorological disasters. They affect about 15% of land surface globally, causing a significant reduction in crop growth and yields. This paper presents an objective methodology for assessing waterlogging risk, primarily in non-urban, predominantly agricultural areas. The waterlogging risk was assessed by evaluating vulnerability and hazard based on key environmental, anthropogenic, and climatic factors. The weights of factors affecting the waterlogging vulnerability were determined using the entropy weight method (EWM), assuring the objectivity of the overall evaluation results. The obtained waterlogging risk map was validated by comparing it with observed and detected waterlogged sites using Sentinel-2 imagery and Random Forest classification. The key novelties of this study are the use of the entropy weight method to objectively determine the relative importance of factors influencing waterlogging vulnerability, and a two-step validation process which includes field-based comparison and remote sensing validation. The presented methodology was demonstrated in the Vojvodina region, Serbia. The following waterlogging vulnerability factors were used: soil properties, geomorphology, surface depressions, average phreatic water table depth, and land cover. The EWM shows that surface depressions and soil properties have the most significant influence on waterlogging vulnerability. The highest waterlogging hazard classes occur in about 31% of the analyzed territory. The waterlogging hazard was estimated based on water balance for the non-vegetation season and maximum daily precipitation in spring, both modeled using the Generalize Extreme Value distribution function. The highest waterlogging hazard classes occur in about 31% of the analyzed territory. The final risk map shows that the high waterlogging risk occurs in about 11% of the territory. Those are mainly areas in the central, eastern, and southeastern parts of the Vojvodina region, usually along the main watercourses. High agreement between the detected waterlogged areas and the produced waterlogging risk map was achieved, validating the proposed methodology. The presented waterlogging risk assessment methodology is valuable for planning and policy-making for various water management and environmental activities. Although it is demonstrated in Vojvodina, by selecting the appropriate factors of vulnerability and hazard, it can be applied to any other region.

Test of the Equality of Several High-Dimensional Covariance Matrices: A Normal-Reference Approach

As the field of big data continues to evolve, there is an increasing necessity to evaluate the equality of multiple high-dimensional covariance matrices. Many existing methods rely on approximations to the null distribution of the test statistic or its extreme-value distributions under stringent conditions, leading to outcomes that are either overly permissive or excessively cautious. Consequently, these methods often lack robustness when applied to real-world data, as verifying the required assumptions can be arduous. In response to these challenges, we introduce a novel test statistic utilizing the normal-reference approach. We demonstrate that the null distribution of this test statistic shares the same limiting distribution as a chi-square-type mixture under certain regularity conditions, with the latter reliably estimable from data using the three-cumulant matched chi-square-approximation. Additionally, we establish the asymptotic power of our proposed test. Through comprehensive simulation studies and real data analysis, our proposed test demonstrates superior performance in terms of size control compared to several competing methods.

Enhanced RTP method for nonlinear statistical prediction of ship structural response

Abstract In this study, the Enhanced RAO-based Translation Process (ERTP) method, which is an enhanced version of the RAO-based Translation Process (RTP) method proposed in the previous study, is newly proposed. The RTP method approximates the Extreme Value Distribution (EVD) in arbitrary irregular waves from the nonlinear response in regular waves based on the idea of the transformation process, in which the nonlinear process is regarded as a transformation of a linear process. However, it was found that the RTP method is inaccurate when the response spectrum has multiple peaks, which is often the case for stresses in ship structures in waves. To solve this problem, the author developed the ERTP method, which defines a nonlinear transformation based on the L p-norm of the response spectrum. Direct Load and Structure Analysis considering the nonlinearity of wave loads was conducted and confirmed that the proposed method can reasonably estimate the EVD even for structural elements with a multi-peaked response spectrum with respect to the wave direction.

Generalised Additive Model-Based Regional Flood Frequency Analysis: Parameter Regression Technique Using Generalised Extreme Value Distribution

This study examines the effectiveness of Generalised Additive Models (GAMs) and log-log linear models for estimating the parameters of the generalised extreme value (GEV) distribution, which are then used to estimate flood quantiles in ungauged catchments. This is known as the parameter regression technique (PRT). Using data from 88 gauged catchments in New South Wales, Australia, flood quantiles were estimated for various annual exceedance probabilities (AEPs) of 50%, 20%, 10%, 5%, 2%, and 1%, corresponding to return periods of 2, 5, 10, 20, 50, and 100 years, denoted by Q2, Q5, Q10, Q20, Q50, and Q100, respectively. These flood quantiles were then used as dependent variables, while several catchment characteristics served as independent variables in the regression. GAMs were employed to capture non-linearities in flood generation processes. This study evaluates different GAMs and log-log linear models, identifying the best ones based on significant predictors and various statistical metrics using a leave-one-out (LOO) validation approach. The results indicate that GAMs provide more accurate and reliable predictions of flood quantiles compared to the log-log linear models, demonstrating better performance in capturing observed values across different quantiles. The absolute median relative error percentage (REr%) ranges from 33% to 39% for the GAMs and from 36% to 45% for the log-log models. GAMs demonstrate better performance compared to the log-log linear models for quantiles Q2, Q5, Q10, Q20, and Q50; however, their performances appear to be similar for Q100.

Clustering of buoyant tracer in quasi-geostrophic coherent structures

We have investigated the dynamics of floating tracer in an idealised turbulent quasi-geostrophic ocean by advecting Lagrangian particles in a high-resolution velocity field enhanced by the potential flow associated with vortex stretching. At first order in the Rossby number expansion, this component of the ageostrophic circulation can be derived through a diagnostic equation in terms of the geostrophic velocities. Borrowing methods from the theory of Lagrangian coherent structures, we identify coherent material loops around strong vortex cores using the Lagrangian averaged vorticity deviation (LAVD). Building on studies of clustering in kinematic, stochastic velocity fields, we utilise methods from statistical topography to show that the coherent vortices dominate the distribution of extreme values of the concentration field. We find that the presence of clusters and voids in a coherent vortex depends on more than just the sense of rotation, but also on the full evolution of the vorticity over its lifecycle. We identify the mechanism behind the cluster formation that respects the symmetries of the quasi-geostrophic equations but can be expected to hold robustly in more complicated regimes, due to the simple physical description. The association of cluster formation with vortex stretching implies that LAVD is a particularly relevant metric for floating tracer dynamics. The detection of intense clustering also has implications for reaction rates between ocean-borne flotsam, meaning that our results are relevant to understanding the general risk of floating microplastics and marine biological populations.

Extreme value theory analysis of high emitter trends across four US cities from 1995 to 2021.

Extreme value theory provides a direct means to characterize the distribution of high emitters within a vehicle fleet and calculate statistical confidence intervals for comparisons. Defining a "high emitter" as the maximum emitter in a random sample of N vehicles implies in the limit of large N that high emitters follow an extreme value distribution, comprised of three distinct domains. The analysis of over twenty years of roadside remote sensing emissions measurements in Chicago, Denver, Los Angeles and Tulsa reveals clear differences between gasoline vehicle high emitter distributions across pollutants (hydrocarbons (HC), carbon monoxide (CO) and nitric oxide (NO)), but very similar behavior across the four cities. As Federal emissions regulations became more stringent, HC and CO high emitters evolved from distributions bounded by the stoichiometric limits of HC and CO formation in an engine to progressively longer tailed distributions. In contrast, NO high emitter distributions have changed little over this time period. These distinct behaviors likely reflect differences in the failure mechanisms leading to high HC, CO, versus NO emissions. Whereas average HC, CO and NO fleet emissions fell dramatically over these two plus decades by factors of about 4, 6 and 7, respectively, mean high emitter emissions declined by less than half of these. The paper examines in detail similarities and differences in high emitter distributions versus pollutant, city and vehicle type, how these changed over a period of roughly twenty years and the policy implications that can be drawn.

Electricity Demand Forecasting of Value-at-Risk and Expected Shortfall: The South African Context

The energy landscape, particularly in the electricity sector, is characterised by a complex interplay of various factors that contribute to its inherent volatility. Electricity participants, including investors, regulators, consumers, and policymakers, are constantly seeking methods to better understand and manage the associated risks. The generalised extreme value (GEV) distribution with block minima is applied to model extreme losses on daily electricity demand in South Africa for the period of 1 April 2019 to February 13, 2024. The results of the estimated GEV gave a negative shape parameter implying that both winter and no-winter seasons extremes are correctly model a type III GEV distribution known as the Weibull distribution. When computing the VaR and ES, we found that VaR went as low as 8.74% while for ES had the lowest as 9.57%. Finally, the backtesting procedures further proved that the estimated risk measures are reliable as both the Kupiec and Chrisoffersen tests failed to reject the null hypothesis. In conclusion, the fitted GEV showed some reliance in capturing extremes losses for winter and non-winter returns. Lastly due to reliability of the models, risk analysts together with investors interested in the electricity sector should therefore adopt the procedures used to know the risk of their investment.

Nuclear Data Uncertainty Propagation in the PWR MOX/UO2 Core Transient Benchmark

In light water reactor (LWR) transient analyses using deterministic simulation codes, best-estimate plus uncertainty (BEPU) approaches complement the results obtained from simulation codes with uncertainty quantification. In this study, a stochastic sampling–based uncertainty analysis function was developed under the framework of BEPU, enabling uncertainty quantification of important parameters in LWR steady-state and transient analyses using SIMULATE5 and SIMULATE5-K. In the uncertainty quantification of the dynamic behavior of nuclear reactors, the uncertainty knowledge of thermal-hydraulic parameters was highlighted; however, the nuclear data uncertainty propagation to dynamic behavior was insufficient. To bridge the knowledge gap, the uncertainties of interest parameters in the Organisation for Economic Co-operation and Development/Nuclear Energy Agency and the U.S. Nuclear Regulatory Commission PWR MOX/UO2 Core Transient Benchmark were quantified considering cross-section and kinetics data uncertainties. Consequently, the uncertainties of the maximum fuel enthalpy important for rod ejection transient analyses evaluated on the basis of the generalized extreme value distribution were equally contributed by cross-section and kinetics data uncertainties.

Propagation of Chaos for Point Processes Induced by Particle Systems with Mean-Field Drift Interaction

We study the asymptotics of the point process induced by an interacting particle system with mean-field drift interaction. Under suitable assumptions, we establish propagation of chaos for this point process: It has the same weak limit as the point process induced by i.i.d. copies of the solution of a limiting McKean–Vlasov equation. This weak limit is a Poisson point process whose intensity measure is related to classical extreme value distributions. In particular, this yields the limiting distribution of the normalized upper order statistics.

Incorporating Climatic Extremes Using the GEV Distribution Improves SDM Range Edge Performance

ABSTRACTAimThe changing frequency and intensity of climatic extremes due to climate change can have sudden and adverse impacts on the distribution of species. While species distribution modelling is a vital tool in ecological applications, current approaches fail to fully capture the distribution of climatic extremes, particularly of rare events with the most disruptive potential. Especially at the edges of species' ranges, where conditions are already less favourable, predictions might be inaccurate when these extremes are not well represented.LocationEurope.TaxonTree species.MethodsWe present a novel approach to integrate extreme events into species distribution models based on the generalised extreme value (GEV) distribution. This distribution, following from the extreme value theory has been established as a valuable tool in analysing climatic extremes, both in an ecological context and beyond. The approach relying on the GEV distribution is broadly applicable, readily transferable across species and relies on widely available data. We demonstrate the efficacy of our approach for 28 European tree species, illustrating its superior ability to fully capture the distribution of climatic extremes compared to state‐of‐the‐art methods.ResultsWe found that incorporating parameters on climatic extremes derived from the GEV distribution increased model performance (AICmodel) and characterised range edges more accurately (AUCedge) compared to competing approaches. However, general AUC values were only marginally increased across the species and study period analysed. Overall, the GEV model predicted a narrower niche for the species included in this study.Main ConclusionsIncorporating climatic extremes can impact spatial predictions of species distribution models, especially at range margins. We found that using the GEV distribution to characterise extreme variables in SDMs yields the best performance at these distribution edges. Given the importance of range edges for species conservation, a detailed inclusion of extremes in SDMs employed for those applications will help ensure robust conclusions.

Probability Distribution of Extreme Events in Complex Systems: Application to Climate Data

It has been observed that the statistical structure of certain climate vectors, such as wind speed versus air density and temperature versus humidity, may exhibit more than one mode due to the complexity of climate systems. This study proposes a new bivariate extreme value distribution, called the transformed symmetric logistic extreme value distribution, which can capture the multimodal characteristics of the joint distribution of extreme observations in complex systems. We derive some of its properties, such as marginal distributions, tail indices, conditional distribution, and P(Y<X). The parameters of the new distribution were estimated using the maximum likelihood method. The applicability of the proposed model is illustrated with climate data, including the analysis of the result P(Y<X).

Estimating Extreme Drought Risk Through Classical and Bayesian Paradigms

ABSTRACTDrought poses significant challenges to both the environment and the economy, necessitating proactive mitigation strategies. This study introduces both classical and Bayesian Markov Chain Monte Carlo (MCMC) extreme value probabilistic models for quantifying drought risk. The models utilise the generalised extreme value (GEV) distribution to characterise the distribution of standardised precipitation index (SPI) and non‐stationary standardised precipitation index (NSSPI) variables. Drought risk is probabilistically assessed across five regions in Baluchistan (a drought‐prone area of Pakistan) over two 20‐year periods per region. The study presents a novel approach in probabilistic quantification models, demonstrating slight performance improvement with the Bayesian MCMC paradigm, as evaluated by the continuously ranked probability scoring. Moreover, the application of the presented methodology can be extended to other climatic zones using Bayesian MCMC with informative priors constructed from historical records of the neighbouring regions.

Characteristics of road freezing-thawing environments in China oriented towards construction and maintenance

Abstract Road infrastructure suffers irreversible damage during service due to periodic or seasonal freeze-thaw (F-T) cycles, resulting in significant economic losses to the nation. However, systematic evaluation parameters for the entire road F-T environment are lacking, and there is insufficient understanding and short-term prediction of the underlying F-T environment changes and distribution. In this study, 13 F-T environmental parameters in three categories were summarized by combining meteorological parameters and multi-field evaluation parameters. Taking Shiquan River Town as an example, the statistical probability of road F-T environmental parameters was clarified by using the generalized extreme value (GEV) distribution. The study also clarified the spatiotemporal characteristics of road F-T environment across regions based on key F-T environmental parameters. The frequency of F-T events in China increases from south to north and from east to west, and F-T processes are influenced by regional altitude and latitude. Meanwhile, the temporal variation of the F-T environment in China shows significant seasonality. The F-T environment transitions from short-duration, high-frequency F-T cycles to longer-duration, low-frequency F-T cycles. The initial moisture condition of the road F-T environment in the northeastern regions increases, but the additional moisture during the F-T process decreases. This study provides critical insights into understanding the underlying mechanisms and short-term prediction of the road F-T environment changes, which can be used as a basis for the planning, construction, and maintenance of road infrastructure in China.

It Is Normal: The Probability Distribution of Temperature Extremes

The probability of heat extremes is often estimated using the non-stationary generalized extreme value distribution (GEVD) applied to time series of annual maximum temperature. Here, this practice was assessed using a global sample of temperature time series, from reanalysis (both at the grid point and the region scale) as well as station observations. This assessment used forecast negative log-likelihood as the main performance measure, which is particularly sensitive to the most extreme heat waves. It was found that the computationally simpler normal distribution outperforms the GEVD in providing probabilistic year-ahead forecasts of temperature extremes. Given these findings, it is suggested to consider alternatives to the GEVD for assessing the risk of extreme heat.

Assessing the Effect of Bias Correction Methods on the Development of Intensity–Duration–Frequency Curves Based on Projections from the CORDEX Central America GCM-RCM Multimodel-Ensemble

This work aims to examine the effect of bias correction (BC) methods on the development of Intensity–Duration–Frequency (IDF) curves under climate change at multiple temporal scales. Daily outputs from a 9-member CORDEX-CA GCM-RCM multi-model ensemble (MME) under RCP 8.5 were used to represent future precipitation. Two stationary BC methods, empirical quantile mapping (EQM) and gamma-pareto quantile mapping (GPM), along with three non-stationary BC methods, detrended quantile mapping (DQM), quantile delta mapping (QDM), and robust quantile mapping (RQM), were selected to adjust daily biases between MME members and observations from the SJO weather station located in Costa Rica. The equidistant quantile-matching (EDQM) temporal disaggregation method was applied to obtain future sub-daily annual maximum precipitation series (AMPs) based on daily projections from the bias-corrected ensemble members. Both historical and future IDF curves were developed based on 5 min temporal resolution AMP series using the Generalized Extreme Value (GEV) distribution. The results indicate that projected future precipitation intensities (2020–2100) vary significantly from historical IDF curves (1970–2020), depending on individual GCM-RCMs, BC methods, durations, and return periods. Regardless of stationarity, the ensemble spread increases steadily with the return period, as uncertainties are further amplified with increasing return periods. Stationary BC methods show a wide variety of trends depending on individual GCM-RCM models, many of which are unrealistic and physically improbable. In contrast, non-stationary BC methods generally show a tendency towards higher precipitation intensities as the return period increases for individual GCM-RCMs, despite differences in the magnitude of changes. Precipitation intensities based on ensemble means are found to increase with the change factor (CF), ranging between 2 and 25% depending on the temporal scale, return period, and non-stationary BC method, with moderately smaller increases for short-durations and long-durations, and slightly higher for mid-durations. In summary, it can be concluded that stationary BC methods underperform compared to non-stationary BC methods. DQM and RQM are the most suitable BC methods for generating future IDF curves, recommending the use of ensemble means over ensemble medians or individual GCM-RCM outcomes.

Multicomponent Stress–Strength Reliability with Extreme Value Distribution Margins: Its Theory and Application to Hydrological Data

This paper focuses on the estimation of multicomponent stress–strength models, an important concept in reliability analyses used to determine the probability that a system will function successfully under varying stress conditions. Understanding and accurately estimating these probabilities is essential in fields such as engineering and risk management, where the reliability of components under extreme conditions can have significant consequences. This is the case in applications where one seeks to model extreme hydrological events. Specifically, this study examines cases where the random variables X (representing strength) and Y (representing stress) follow extreme value distributions. New analytical expressions are derived for multicomponent stress–strength reliability (MSSR) when different classes of extreme distributions are considered, using the extreme value H-function. These results are applied to three l-max stable laws and six p-max stable laws, providing a robust theoretical framework for multicomponent stress–strength analyses under extreme conditions. To demonstrate the practical relevance of the proposed models, a real dataset is analyzed, focusing on the monthly water capacity of the Shasta Reservoir in California (USA) during August and December from 1980 to 2015. This application showcases the effectiveness of the derived expressions in modeling real-world data.

Fitting Probability Distributions and Statistical Trend Analysis of Rainfall of Agro-climatic Zone of West Bengal

This research aimed to identify the most appropriate probability distribution for modeling average monthly rainfall in the agro-climatic zones of West Bengal and to detect any trends in this data. The study utilized historical rainfall data spanning 51 years (1970-2020) obtained from the IMD in Pune. To determine the best-fitting distribution and assess trends, 23 different probability distributions were employed, with the Mann-Kendall test and Sen’s slope estimator used for trend analysis. Goodness-of-fit tests, including the Kolmogorov-Smirnov, Anderson-Darling, and Chi-square tests, were employed to determine the most suitable distribution. The findings indicated that the Generalized Extreme Value, Gamma, and Lognormal (3-parameter) distributions were the best fits for two specific districts. The monthly rainfall distributions can be effectively used for predicting future monthly rainfall events in the region. The Mann-Kendall test revealed an increasing trend in rainfall for Kalimpong and Nadia Districts and a decreasing trend for Malda District.

Assessing the Suitability of Probability Distribution Models for Streamflow Analysis in Peninsular Malaysia

Streamflow analysis is indeed a need for Malaysia which always threaten by extreme streamflow events including flooding. However, the determination of the most suitable probability distribution for streamflow analysis has been the major concern and received considerable critical attention recently. The purpose of this study is to analyse the performances of several distributions in fitting the streamflow data collected form 11 streamflow stations in Peninsular Malaysia. and select the best performed distribution eventually. Normal, Generalized Extreme Value (GEV), three-parameter Gamma (G3), two-parameter Gamma (G2), three-parameter Weibull (W3), two-parameter Weibull (W2), three-parameter Log-Normal (LN3) and two-parameter Log-Normal (LN2) were used in this study to fit the maximum, minimum, and mean streamflow with monthly and seasonal time scales by using Easyfit software. The performances of distributions were evaluated through constructing Probability-Probability (P-P) Plot, Cumulative Distribution Function (CDF) and conducting goodness-of-fit tests including Chi-square () Test, Anderson-Darling (AD) Test and Kolmogorov-Smirnov (KS) Test. All the results were tabulated for comparison and the most suitable probability distribution was selected. Overall, the results indicated that the GEV distribution was the best fit distribution for most of the stations and most of the data series or time scales, while the LN3 distribution appeared to be the second-best distribution. These findings add to a growing body of literature on the selection of probability distribution especially for the streamflow analysis in Peninsular Malaysia. Besides, these findings provide significant information and practical knowledge in supporting the future development of streamflow-related plans or management including the flood and drought mitigation plans.

Fisher-Tippet Law Truncated Form for Loading Modeling of Machinery Structures

Introduction. Statistical data are used as a basis for assessing the reliability of engineering structures. However, incomplete data or inaccurate modeling of random variables may lead to an overestimation of reliability indicators. In practice, laws with infinitely decreasing or increasing distribution functions of an exponential family are usually used to model random variables characterizing the bearing capacity, load, and resource of engineering structures. To improve the accuracy of modeling of random variables, truncated forms of distribution laws are often used. These forms allow us to consider the random variable within a specified interval, excluding impossible values. Several studies have suggested using the Fisher-Tippett law with three parameters for modeling random variables related to the loading of engineering structures. The advantage of this law is that it limits the range of the random variable on the right side, but the left side of the distribution function decreases indefinitely, which is not ideal for load characteristics. To improve the accuracy of predicting random variables that characterize the load, it would be helpful to have a left-sided restriction using the Fisher-Tippett law. Currently, there are no descriptions of truncated forms of the distribution law in scientific literature. This article will explore the justification and development of a three-parameter truncated form of the Fisher-Tippett law and its use in calculation methods. The goal is to create a left-sided truncated version of the Fisher-Tippett distribution with three parameters to model random variables within a specific range.Materials and Methods. The article provides a detailed description of the history of the Fisher-Tippet law, including its three-parameter form, and justifies the need for obtaining its truncated form.Results. As a result of the research, a truncated form of the Fisher-Tippet three-parameter law in differential and integral forms was obtained and substantiated. The findings included graphs and calculations that demonstrated the normalization of a random variable within a given range.Discussion and Conclusion. The conclusion was drawn about the advantages and disadvantages of the truncated form of the Fisher-Tippet law. The possibility of its practical application in the schematization of random loading processes under operating conditions and testing of machine elements and structures to assess fatigue life and determine fatigue resistance characteristics was established. The direction of further research is related to the practical use of the truncated form, particularly with the need to develop a method for evaluating the parameters of the truncated distribution and verifying the consistency of the proposed model