Extreme Value Distribution Research Articles

A data-driven spatial approach to characterize the flood hazard.

Model output of localized flood grids are useful in characterizing flood hazards for properties located in the Special Flood Hazard Area (SFHA-areas expected to experience a 1% or greater annual chance of flooding). However, due to the unavailability of higher return-period [i.e., recurrence interval, or the reciprocal of the annual exceedance probability (AEP)] flood grids, the flood risk of properties located outside the SFHA cannot be quantified. Here, we present a method to estimate flood hazards that are located both inside and outside the SFHA using existing AEP surfaces. Flood hazards are characterized by the Gumbel extreme value distribution to project extreme flood event elevations for which an entire area is assumed to be submerged. Spatial interpolation techniques impute flood elevation values and are used to estimate flood hazards for areas outside the SFHA. The proposed method has the potential to improve the assessment of flood risk for properties located both inside and outside the SFHA and therefore to improve the decision-making process regarding flood insurance purchases, mitigation strategies, and long-term planning for enhanced resilience to one of the world's most ubiquitous natural hazards.

Anthropogenic Contributions to the 2021 Pacific Northwest Heatwave

AbstractDaily maximum temperatures during the 2021 heatwave in the Pacific Northwest United States and Canada shattered century old records. Multiple causal factors, including anthropogenic climate change, contributed to these high temperatures, challenging traditional methods of attributing human influence. We demonstrate that the observed 2021 daily maximum temperatures are far above the bounds of Generalized Extreme Value distributions fitted from historical data. Hence, confidence in Granger causal inference statements about the human influence on this heatwave is low. Alternatively, we present a more conditional hindcast attribution study using two regional models. We performed ensembles of simulations of the heatwave to investigate how the event would have changed if it had occurred without anthropogenic climate change and with future warming. We found that global warming caused a ∼0.8°C–1°C increase in heatwave temperatures. Future warming would lead to a ∼5°C increase in heatwave temperature by the end of the 21st century.

Modeling climate extremes using the four-parameter kappa distribution for [formula omitted]-largest order statistics

Accurate estimation of the T-year return levels of climate extremes using statistical distribution is a critical step in the projection of future climate and in engineering design for disaster response. We show how the estimation of such quantities can be improved by fitting the four-parameter kappa distribution for r-largest order statistics (rK4D), which was developed in this study. The rK4D is an extension of the generalized extreme value distribution for r-largest order statistics (rGEVD), similar to the four-parameter kappa distribution (K4D), which is an extension of the generalized extreme value distribution (GEVD). This new distribution (rK4D) can be useful not only for fitting data when three parameters in the GEVD are not sufficient to capture the variability of the extreme observations, but also in reducing the estimation uncertainty by making use of the r-largest extreme observations instead of only the block maxima. We derive a joint probability density function (PDF) of rK4D and the marginal and conditional cumulative distribution functions and PDFs. To estimate the parameters, the maximum likelihood estimation and the maximum penalized likelihood estimation methods were considered. The usefulness and practical effectiveness of the rK4D are illustrated by the Monte Carlo simulation and by an application to the Bangkok extreme rainfall data. A few new distributions for r-largest order statistics are also derived as special cases of the rK4D, such as the r-largest logistic, the r-largest generalized logistic, and the r-largest generalized Gumbel distributions. These distributions for r-largest order statistics would be useful in modeling extreme values for many research areas, including hydrology and climatology.

Probability distributions analysis of travel time variability for the public transit system

Travel time variability (TTV) plays a significant role in analysing the reliability of public transit system. The research works carried out on travel time variability under Indian traffic conditions are very few and these studies did not analyse the performance of travel time distribution in detail, considering different temporal and spatial aggregations. In this study, travel time variability is analysed using travel time distributions considering different temporal and spatial aggregations. The Automatic Vehicle Location (AVL) data of four transit routes of Mysore City, Karnataka, India are used to evaluate travel time distributions with respect to temporal aggregations (peak period, off-peak period, 60 minutes, 30 minutes and 15 minutes) and spatial aggregations (route level and segment level). The performance of travel time distributions is analysed using the Anderson-Darling (AD) test. The segments with signalised intersections and different land-use types are analysed to evaluate the distribution fit for various conditions. The results of both route and segment level analysis report highest accuracy and robustness values for Generalised Extreme Value (GEV) distribution. The distribution is proved to be superior in describing travel time variability of public transit.

Quantification of vascular pathology in the retina through probability distributions of central artery and vein width equivalent

AbstractPurpose: A semi‐automated method of measuring the Central Retinal Artery Equivalent (CRAE) and Central Retinal Vein Equivalent (CRVE) has been developed. The method was used to calculate parametric probability distribution for CRAE and CRVE in healthy retinas in order to quantify possible vascular pathology degree.Methods: Using the ARIA software (MATLAB), the diameters of the retinal vessels located in a ring with radii 0.5 and 1 diameter of the optic disc were quantified. A measurement of CRAE and CRVE was obtained for two data base of healthy retinas (15 and 52 patients) measured with different retinal cameras. Subsequently, the probability distribution of CRAE and CRVE was calculated, as well as the bootstrap estimators of the mean, standard deviation, skewness, and kurtosis. MATLAB functions were used to select the parametric distribution that best fits the data. The Mahalanobis distance of each retina to healthy distribution was obtained. This classification distance was applied to healthy and glaucoma retinas.Results: The fits to theoretical distributions were good for a Generalized Extreme Value Distribution (gev) characterized by three parameters (scale, mean and standard deviation). They appear to be consistent across different databases. In retinas with glaucoma, the Mahalanobis distance allows the degree of pathology to be quantified. Moreover, the method can be used to classify the healthy retinas.Conclusions: The possible existence of parametric probability distributions for values of vascular calibres opens the possibility of quantifying the degree of vascular pathology.

A New Extension of Generalized Extreme Value Distribution: Extreme Value Analysis and Return Level Estimation of the Rainfall Data

This paper presents an extension of the generalized extreme value (GEV) distribution, based on the T-X family of distributions: Gompertz-generated family of distributions that make the existing distribution more flexible called the Gompertz-general extreme value (Go-GEV) distribution. Some properties of the proposed distribution are introduced, and a new distribution is applied to actual data, namely rainfall in Lopburi Province, by comparing the proposed model with the traditional GEV distribution and estimating the return levels of the rainfall in Lopburi Province. Results showed that the Go-GEV was an alternative flexible distribution for extreme values that fitted with actual data and described the maximum rainfall better than the traditional GEV distribution. The probability density functions of the Go-GEV distribution had various shapes including left-skewed, right-skewed and close to symmetric. Estimation of the return levels of rainfall values in Lopburi Province by the Go-GEV distribution indicated that Buachum Station should be monitored because it had higher precipitation and return levels than Lopburi Station at 2, 5, 10 and 15 years. HIGHLIGHTS The analysis of data with extreme values is complex but ignoring an observation just because it is unacceptable is not best practice. One of the widely used tools in such situations is “Extreme Value Theory” The Gompertz-generalized extreme value (Go-GEV) distribution is a new extension of GEV distribution using deffinition of Gompertz-generalized family of distributions The developed Go-GEV model described the maximum rainfall data better than the GEV distribution because the new extension distribution was more flexible and its probability density function had a more diverse shape, such as both left-skewed, right-skewed and close to symmetrical The developed Go-GEV model is a flexible alternative for applying the model to extreme value data GRAPHICAL ABSTRACT

Spatiotemporal trends in daily and subdaily rainfall extremes and return levels in Turkey

AbstractIn this study, we present the long‐term daily and subdaily station data of annual extreme rainfall (1970–2015) and the trend analyses in rainfall regimes in Turkey. Trends in 5, 10, 15, and 30 min, 1, 2, 6, 12, and 24 hr of extreme rainfall in seven different rainfall regimes are estimated through nonparametric tests. The trends in return levels (2, 20, and 100 years) are defined by an appropriate three‐parameter generalized extreme value distribution and are evaluated in the climatological context of rainfall regimes. Overall, from 5‐min to 2‐hr durations, magnitudes of trends in extreme rainfall constantly increase in all rainfall regimes, which may be attributed to the intensified contribution of convective rainfall as a response to warming. Trend analysis of return levels reveals that, compared with 2‐year return levels, low‐probability high‐impact extreme rainfall events generally have the lowest estimated median trends until 30‐min to 2‐hr range. A shift in the magnitudes of trends occurs generally at 30‐min and 1‐hr durations; trends of rare intense events increase at the expense of less intense extreme rainfall. Thus, the intensification of 1–2‐hr extreme rainfall events can arise from both the increasing trends of more common events in shorter rainfall durations and from the increase in the trends of low‐probability high‐impact extreme rainfall events at this range. Moreover, explicitly in continental rainfall regimes, increases in the magnitudes of the trends in 30‐min to 2‐hr duration range are accompanied by various declines in 6–24‐hr duration range. Although coastal regimes generally have increasing trend values from 5‐min to 24‐hr durations, in northern and southern clusters, changes occur in the variability of extreme rainfall and the trend values of rarer (20‐ and 100‐year return periods) extreme rainfall exhibit increases in 6–12‐hr range.

Smooth Spatial Modeling of Extreme Mediterranean Precipitation

Extreme precipitation events can lead to disastrous floods, which are the most significant natural hazards in the Mediterranean regions. Therefore, a proper characterization of these events is crucial. Extreme events defined as annual maxima can be modeled with the generalized extreme value (GEV) distribution. Owing to spatial heterogeneity, the distribution of extremes is non-stationary in space. To take non-stationarity into account, the parameters of the GEV distribution can be viewed as functions of covariates that convey spatial information. Such functions may be implemented as a generalized linear model (GLM) or with a more flexible non-parametric non-linear model such as an artificial neural network (ANN). In this work, we evaluate several statistical models that combine the GEV distribution with a GLM or with an ANN for a spatial interpolation of the GEV parameters. Key issues are the proper selection of the complexity level of the ANN (i.e., the number of hidden units) and the proper selection of spatial covariates. Three sites are included in our study: a region in the French Mediterranean, the Cap Bon area in northeast Tunisia, and the Merguellil catchment in central Tunisia. The comparative analysis aim at assessing the genericity of state-of-the-art approaches to interpolate the distribution of extreme precipitation events.

Ground-state energy distribution of disordered many-body quantum systems.

Extreme-value distributions are studied in the context of a broad range of problems, from the equilibrium properties of low-temperature disordered systems to the occurrence of natural disasters. Our focus here is on the ground-state energy distribution of disordered many-body quantum systems. We derive an analytical expression that, upon tuning a parameter, reproduces with high accuracy the ground-state energy distribution of the systems that we consider. For some models, it agrees with the Tracy-Widom distribution obtained from Gaussian random matrices. They include transverse Ising models, the Sachdev-Ye model, and a randomized version of the PXP model. For other systems, such as Bose-Hubbard models with random couplings and the disordered spin-1/2 Heisenberg chain used to investigate many-body localization, the shapes are at odds with the Tracy-Widom distribution. Our analytical expression captures all of these distributions, thus playing a role to the lowest energy level similar to that played by the Brody distribution to the bulk of the spectrum.

Wave-function extreme value statistics in Anderson localization

We consider a disordered one-dimensional tight-binding model with power-law decaying hopping amplitudes to disclose wavefunction maximum distributions related to the Anderson localization phenomenon. Deeply in the regime of extended states, the wavefunction intensities follow the Porter-Thomas distribution while their maxima assume the Gumbel distribution. At the critical point, distinct scaling laws govern the regimes of small and large wavefunction intensities with a multifractal singularity spectrum. The distribution of maxima deviates from the usual Gumbel form and some characteristic finite-size scaling exponents are reported. Well within the localization regime, the wavefunction intensity distribution is shown to develop a sequence of pre-power-law, power-law, exponential and anomalous localized regimes. Their values are strongly correlated, which significantly affects the emerging extreme values distribution.

Raspberries-LITRP Database: RGB Images Database for the Industrial Applications of Red Raspberries’ Automatic Quality Estimation

This work presents a free new database designed from a real industrial process to recognize, identify, and classify the quality of the red raspberry accurately, automatically, and in real time. Raspberry trays with recently harvested fresh fruit enter the industry’s selection and quality control process to be categorized and subsequently their purchase price is determined. This selection is carried out from a sample of a complete batch to evaluate the quality of the raspberry. This database aims to solve one of the major problems in the industry: evaluating the largest amount of fruit possible and not a single sample. This major dataset enables researchers in various disciplines to develop practical machine-learning (ML) algorithms to improve red raspberry quality in the industry, by identifying different diseases and defects in the fruit, and by overcoming limitations by increasing the performance detection rate accuracy and reducing computation time. This database is made up of two packages and can be downloaded free from the Laboratory of Technological Research in Pattern Recognition repository at the Catholic University of the Maule. The RGB image package contains 286 raw original images with a resolution of 3948 × 2748 pixels from raspberry trays acquired during a typical process in the industry. Furthermore, the labeled images are available with the annotations for two diseases (86 albinism labels and 164 fungus rust labels) and two defects (115 over-ripeness labels, and 244 peduncle labels). The MATLAB code package contains three well-known ML methodological approaches, which can be used to classify and detect the quality of red raspberries. Two are statistical-based learning methods for feature extraction coupled with a conventional artificial neural network (ANN) as a classifier and detector. The first method uses four predictive learning from descriptive statistical measures, such as variance, standard deviation, mean, and median. The second method uses three predictive learning from a statistical model based on the generalized extreme value distribution parameters, such as location, scale, and shape. The third ML approach uses a convolution neural network based on a pre-trained fastest region approach (Faster R-CNN) that extracts its features directly from images to classify and detect fruit quality. The classification performance metric was assessed in terms of true and false positive rates, and accuracy. On average, for all types of raspberries studied, the following accuracies were achieved: Faster R-CNN 91.2%, descriptive statistics 81%, and generalized extreme value 84.5%. These performance metrics were compared to manual data annotations by industry quality control staff, accomplishing the parameters and standards of agribusiness. This work shows promising results, which can shed a new light on fruit quality standards methodologies in the industry.

Record events attribution in climate studies

AbstractWithin the statistical climatology literature, inferring the contributions of potential causes with regard to climate change has become a recurrent research theme during this last decade. In particular, disentangling human induced (anthropogenic) forcings from natural causes represents a nontrivial statistical task, especially when the focal point moves away from mean behaviors and goes towards extreme events with high societal impacts. Most studies found in the field of extreme event attributions (EEA) rely on extreme value theory. Under this theoretical umbrella, it is often assumed that, for a given location, temporal changes in extremes can be detected in both location and scale parameters of an extreme value distribution, while its shape parameter remains unchanged over time. This assumption of constant tail shape parameters between a so‐called factual world (all forcings) and a counterfactual one (without anthropogenic forcing) can be challenged due to the fact that important forcing changes could impact large scale atmospheric and oceanic circulation patterns, and consequently, the latter can reshape the full distribution, including its shape parameter that drives extremal behavior. In this article, we study how allowing different tail shape parameters between the factual and counterfactual worlds can affect the analysis of records. In particular, we extend the work of Naveau et al. in which this case was not treated. We also add properties and theoretical inferential results about records in EEA and propose a procedure for model validation. A simulation study of our approach is detailed. Our method is applied to records of yearly maxima of daily maxima of near‐surface air temperature issued from the numerical climate model CNRM‐CM6‐1 of Météo‐France.

Research on Distribution Model and Detection Spacing of Compaction Degree of Asphalt Pavement Based on the PQI Method

The pavement quality indicator (PQI) is a non-destructive piece of equipment for detecting the compaction degree of asphalt pavement, which can avoid primary damage to the pavement compared with the traditional core-drilling method. In this paper, the PQI method was applied to evaluate the compaction quality of asphalt pavement through data collection, calibration and statistical analysis, and the probability-distribution characteristics of compaction degree were also explored, by fitting the data with probability-distribution models. Furthermore, the optimal detection-spacing was determined by comparing the statistical results of compaction degree measured at different detection-spacings. Test results showed that the calibrated PQI data was close to the actual data of the core sample, and their error rate was within 1%. The compaction degree of the test road was mostly located between 92% and 99%, and the variable coefficient was entirely below 2%, demonstrating that the pavement-compaction quality was satisfactory and uniform. The normal distribution model, lognormal distribution model and extreme-value distribution model had relatively high accuracy in fitting the compaction-degree frequency data, while the sine-wave distribution model was low in fitting accuracy. By comparing the predicted value with the actual value of compaction degree, the normal distribution model was determined as the most appropriate model for describing the frequency distribution of compaction degree. In addition, the detection spacing was selected as 50 m, considering the reliability, accuracy and efficiency. The research results provide technical support for the compaction quality-control of asphalt pavement in a non-destructive, timely, accurate and multi-point manner, ultimately contributing to the excellent service performance and service life of asphalt pavement.

Characterizing breaking wave slamming loads on bridge piers with probabilistic models of slamming maxima, rise and decay times

The piers of sea-crossing bridges always suffer from considerable breaking wave loads. But the uncertainties and variability of wave slamming maxima and time history have yet been understood quantitatively. This paper conducted a series of experiments in a wave flume to investigate the slamming load of breaking waves on bridge piers with different cross-sections. Four piers with three kinds of cross-sections under three breaking wave conditions were employed in the experimental program. The time history of breaking wave slamming load was characterized by three parameters, i.e., slamming maxima, rise and decay times. To account for the statistical variability of breaking wave load, a pair-copula decomposed trivariate joint probability model was developed to determine the joint distribution of slamming maxima, rise and decay times. The effects of pier geometry on the probabilistic modeling of slamming maxima, rise and decay times are analyzed based on the experimental results. The slamming maxima, rise and decay times follow the generalized extreme value (GEV), log-logistic, and gamma distribution, respectively. Their marginal and joint probabilities are sensitive to the pier shape. This study provides an effective probability method to characterize breaking wave slamming loads based on the experimental results.

Importance of non-stationary analysis for assessing extreme sea levels under sea level rise

Abstract. Increased coastal flooding caused by extreme sea levels (ESLs) is one of the major hazards related to sea level rise. Estimates of return levels obtained under the framework provided by extreme-event theory might be biased under climatic non-stationarity. Additional uncertainty is related to the choice of the model. In this work, we fit several extreme-value models to two long-term sea level records from Venice (96 years) and Marseille (65 years): a generalized extreme-value (GEV) distribution, a generalized Pareto distribution (GPD), a point process (PP), the joint probability method (JPM), and the revised joint probability method (RJPM) under different detrending strategies. We model non-stationarity with a linear dependence of the model's parameters on the mean sea level. Our results show that non-stationary GEV and PP models fit the data better than stationary models. The non-stationary PP model is also able to reproduce the rate of extremes occurrence fairly well. Estimates of the return levels for non-stationary and detrended models are consistently more conservative than estimates from stationary, non-detrended models. Different models were selected as being more conservative or having lower uncertainties for the two datasets. Even though the best model is case-specific, we show that non-stationary extremes analyses can provide more robust estimates of return levels to be used in coastal protection planning.

A Review of Extreme Air Temperature Analysis in Croatia

A historical review of extreme air temperature analysis in Croatia is presented. Two capital works on the subject were published in the 1970s by the Croatian Meteorological and Hydrological Service (DHMZ) and Faculty of Science University of Zagreb (PMF-Zagreb), respectively. The first is a monography on extreme value theory or extreme value analysis (EVA) with an application on more than a century-long time series of annual minima air temperature for Zagreb Grič weather station (Croatia) for the period 1862–1969. It is just a case study, with a lot of instructions regarding how to estimate the generalized extreme value (GEV) distribution parameters. The second is a master’s thesis with an application of the EVA on maxima air temperature time series for 41 weather stations from Croatia for the period 1950–1969. The shortness of the time series of the presented data caused instability in the estimation of GEV distribution parameters in transition areas from continental to maritime climate, but in general, the results are acceptable after a reduction of the 1950–1969 time series data on a ‘normal climate period’ 1910–1969. Both works were pioneering for that time in the South-Eastern Europe scale. A routine application of GEV distribution on the extreme air temperature (both minimum and maximum) for ten representative weather stations from Croatia is represented in Climate atlas of Croatia for the period 1961–1990, published by DHMZ in 2008. Theoretically estimated results fit well with empirical data. A review of long-term “warm” and “cold” indices of extreme air temperature for 41 weather stations from Croatia for the period 1951–2010 is represented in the Sixth National Communication Report of Croatia under the UNFCCC published by the Ministry for Environment and Nature Protection of Croatia (MZOIP) in 2014, showing a positive trend of “warm” and a negative trend of “cold” indices during the period 1951–2010 which tackled the non-stationarity of extreme air temperature time series. That topic of non-stationarity is more extensively considered using the results of a series of scientific papers published in the international journals which conducted a study of extreme air temperature of the wider Western Europe territory, including Croatia and other countries close to Croatia. Some authors of these papers stated that the GEV distribution parameters have to be considered as a function of time rather than fixed in time using covariates like North Atlantic Oscillation (NAO), coherent atmospheric blocking regions, linear trends in data caused by global warming and others covariates. The EVA results, connected with the global climate warming, could contribute to the national Natural Disaster Risk Reduction (NDRR) efforts.

Modelling of critical defect distributions for estimating the size effect of selective laser melted 316L stainless steel

The paper presents modelling the effect of specimen size on the fatigue behaviour of SLM 316L stainless steel. The failure mechanism assumes an important role in fatigue crack initiated from random material defects. The finite element calculations showed an increase in the non-local stress distribution of the Crossland criterion for varying defect location and size. A procedure for generating critical defects in a highly stressed volume is proposed to simulate the distributions of extreme values of equivalent stress and material sensitivity coefficient for size effect. The predicted fatigue strength for selected defect size distribution parameters correlates well with experimental data.

An Integrated Estimating Approach for Design Wind Speed under Extreme Wind Climate in the Yangtze River Inland Waterway

Developing the engineering design standard of wind speed is a key aspect of the climate research in the Yangtze River Inland Waterway (YRIW), which is highly sensitive to extreme weather and climate processes. An engineering design wind speed projection model was established to evaluate the distribution of extreme wind speeds in the YRIW region at spatiotemporal scales from 1979 to 2100, integrating the Weibull distribution and generalized extreme value (GEV) distribution characteristics. We also used high-precision climate model products and integrated analysis methods to predict the evolution of engineering design wind speeds in the study area in the future. The results show that: (1) The maximum wind speed in the study area shows a decline—recovery trend in the historical period in general and a weak increase in Wuhan and Shanghai. (2) The maximum wind speed does not follow the Weibull distribution, and the extracted extreme wind speed types include type I, II, and III GEV distributions. (3) The updated inland port project design wind speed can meet the climatic and topographic characteristics of the YRIW. (4) The model of CNRM-CM6-1-HR product accurately captures the spatial and temporal characteristics of the maximum wind speed. (5) In the future, the design wind speed shows a slight decrease in Shanghai, Jiujiang and Yueyang. These findings provide a scientific theoretical reference and engineering reference for the development of design wind speeds for inland port projects at various cross-sections in the YRIW.

On a new extreme value distribution: characterization, parametric quantile regression, and application to extreme air pollution events

On a new extreme value distribution: characterization, parametric quantile regression, and application to extreme air pollution events

Goodness-of-Fit of Wind Speed for Probability Distribution in Central Western Brazil

The adjustment of statistical models to wind speed data makes it possible to know in detail the wind energy potential of a given location, representing a relevant data in the selection of the location for new installations of wind farms. The objective of this study was to evaluate the fit of average wind speed data for Brasilia, Campo Grande, Cuiaba and Goiania in central western Brazil, to the probability distributions functions (PDF) of: GEV, GUM and LN. The statistical criteria, KS, R2, χ2, RMSE, AIC and BIC were considered as judgment criteria to assess the adequacy of PDF. As the main result, the GEV distribution was the one that presented the best result of adjustments to the functions , mainly in the potential of wind energy use and extreme winds that cause felling of trees in the urban environment. These distributions can be used as an alternative distribution that adequately describes the wind speed data in the region. The weakest configurations were obtained by the GUM and LN distributions