Parameters Of Extreme Value Distribution Research Articles

A P–S–N curve fitting method based on mixed Weibull distribution and expectation-maximization algorithm

In practical engineering, there are many factors affecting the fatigue life of actual components, and a single Weibull distribution is insufficient to accurately describe the impact of various factors on fatigue life. In order to solve the above problems, 316L stainless steel is taken as the experimental object to study the effects of phase difference and stress levels on fatigue life based on the mixed Weibull distribution model. Firstly, the mixed Weibull distribution model is converted into the mixed extreme value distribution model, and used Euler's constant to calculate relevant parameters of extreme value distribution and Expectation-Maximization (EM) algorithm to determine the probability density function of the mixed Weibull distribution. Secondly, the probability density function is integrated to obtain the distribution function and the fatigue life under different failure probabilities is calculated. Finally, the influence of non-proportional additional reinforcement and stress amplitude on fatigue life is studied by taking them as hidden variables of mixed Weibull distribution, which is used to obtain P-S-N curves for various failure probabilities. Through the proposed method, the influence of different factors on fatigue life can be considered from the perspective of statistics, and the accuracy of fatigue life prediction can be promoted.

Vector Generalized Additive Models for Extreme Rainfall Data Analysis: A case study in South Korea

Extreme value theory (EVT) is the branch of statistics concerned with inference about the tails of the distribution function. Accurate estimation of the year return levels of climate extremes is a critical step in the projection of future climate and in engineering design for disaster response. The study of non-stationary behavior in the extremes is important to analyze data in environmental sciences, climate, finance, or sports. The largest order statistics is an extension of the block maxima approach that is used in extreme value modeling. However, in Korea, the scientific literature usually has used the method based on fitting a distribution to an annual maximum. We focus on daily rainfall amounts data sets collected from the Korea Meteorological Administration for Seoul, Busan and Daegu, from 1961 to 2020. First, we fit the GEV df to block maxima data, i.e., annual maximum rainfall. Second, we use the top 10 annual precipitation events. We model, in which both mean and scale parameters of extreme value distribution are modeled as linear or smooth functions of covariate, YEAR, using the vector generalized linear models (VGLMs) and vector generalized additive models (VGAMs).

Coevolution of Extreme Sea Levels and Sea‐Level Rise Under Global Warming

AbstractDesign of coastal defense structures like seawalls and breakwaters can no longer be based on stationarity assumption. In many parts of the world, an anticipated sea‐level rise (SLR) due to climate change will constitute present‐day extreme sea levels inappropriate for future coastal flood risk assessments since it will significantly increase their probability of occurrence. Here, we first show that global annual maxima sea levels (AMSLs) have been increasing in magnitude over the last decades, primarily due to a positive shift in mean sea level (MSL). Then, we apply non‐stationary extreme value theory to model the extremal behavior of sea levels with MSL as a covariate and quantify the evolution of AMSLs in the following decades using revised probabilistic sea‐level rise projections. Our analysis reveals that non‐stationary distributions exhibit distinct differences compared to simply considering stationary conditions with a change in location parameter equal to the amount of MSL rise. With the use of non‐stationary distributions, we show that by the year 2050 many locations will experience their present‐day 100‐yr return level as an event with return period less than 15 and 9 years under the moderate (RCP4.5) and high (RCP8.5) representative concentration pathways. Also, we find that by the end of this century almost all locations examined will encounter their current 100‐yr return level on an annual basis, even if CO2 concentration is kept at moderate levels (RCP4.5). Our assessment accounts for large uncertainty by incorporating ambiguities in both SLR projections and non‐stationary extreme value distribution parameters via a Monte Carlo simulation.

Statistical analysis of progressively first-failure-censored data via beta-binomial removals

<abstract><p>Progressive first-failure censoring has been widely-used in practice when the experimenter desires to remove some groups of test units before the first-failure is observed in all groups. Practically, some test groups may haphazardly quit the experiment at each progressive stage, which cannot be determined in advance. As a result, in this article, we propose a progressively first-failure censored sampling with random removals, which allows the removal of the surviving group(s) during the execution of the life test with uncertain probability, called the beta-binomial probability law. Generalized extreme value lifetime model has been widely-used to analyze a variety of extreme value data, including flood flows, wind speeds, radioactive emissions, and others. So, when the sample observations are gathered using the suggested censoring plan, the Bayes and maximum likelihood approaches are used to estimate the generalized extreme value distribution parameters. Furthermore, Bayes estimates are produced under balanced symmetric and asymmetric loss functions. A hybrid Gibbs within the Metropolis-Hastings method is suggested to gather samples from the joint posterior distribution. The highest posterior density intervals are also provided. To further understand how the suggested inferential approaches actually work in the long run, extensive Monte Carlo simulation experiments are carried out. Two applications of real-world datasets from clinical trials are examined to show the applicability and feasibility of the suggested methodology. The numerical results showed that the proposed sampling mechanism is more flexible to operate a classical (or Bayesian) inferential approach to estimate any lifetime parameter.</p></abstract>

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.

Extreme Value Distributions: An Overview of Estimation and Simulation

The generalized extreme value distribution (GEVD) and various extreme value distributions are commonly applied in air pollution, telecommunications, operational risk management, finance, insurance, material sciences, economics, and hydrology, among many other industries that deal with extreme events. Extreme value distributions (EVDs) typically limit the distribution of maximum and minimum values for many random observations drawn from the same arbitrary distribution. Besides that, it is a crucial method for forecasting future events and emerged as critical method for predicting future events. As a result, prior research is required to select the best estimation method to obtain a reliable value for the parameters of extreme value distributions. This study provides an overview of three-parameter estimation methods based on goodness-of-fit statistics and root mean square error (RMSE). This paper reviewed and compared three estimation methods used to approximate values of parameters for simulated observations taken from the EVD and GEVD. The method of moments (MOMs), maximum likelihood estimator (MLE), and maximum product of spacing (MPS) were the methods investigated in this study. Our findings indicated that the MPS performed better based on the mean square errors (MSEs); meanwhile, the MPS had similar goodness-of-fit statistic values compared to the MLE.

A Novel Tree Ensemble Model to Approximate the Generalized Extreme Value Distribution Parameters of the PM2.5 Maxima in the Mexico City Metropolitan Area

We introduce a novel spatial model based on the distribution of generalized extreme values (GEVs) and tree ensemble models to analyze the maximum concentrations levels of particulate matter with a diameter of less than 2.5 microns (PM2.5) in the Mexico City metropolitan area during the period 2003–2021. Spatial trends were modeled through a decision tree in the context of a non-stationary GEV model. We used a tree ensemble model as a predictor of GEV parameters to approximate nonlinear trends. The decision tree was built by using a greedy stagewise approach, the objective function of which was the log-likelihood. We verified the validity of our model by means of the likelihood and Akaike’s information criterion (AIC). The maps of the generalized extreme value parameters on the spatial plane show the existence of differentiated local trends in the extreme values of PM2.5 in the study area. The results indicated strong evidence of an increase in the west–east direction of the study area. A spatial map of risk with maximum concentration levels of PM2.5 in a period of 25 years was built.

Evgam: An R Package for Generalized Additive Extreme Value Models

This article introduces the R package evgam. The package provides functions for fitting extreme value distributions. These include the generalized extreme value and generalized Pareto distributions. The former can also be fitted through a point process representation. Package evgam supports quantile regression via the asymmetric Laplace distribution, which can be useful for estimating high thresholds, sometimes used to discriminate between extreme and non-extreme values. The main addition of package evgam is to let extreme value distribution parameters have generalized additive model forms, the smoothness of which can be objectively estimated using Laplace's method. Illustrative examples fitting various distributions with various specifications are given. These include daily precipitation accumulations for part of Colorado, US, used to illustrate spatial models, and daily maximum temperatures for Fort Collins, Colorado, US, used to illustrate temporal models.

Baseline Methods for Bayesian Inference in Gumbel Distribution.

Usual estimation methods for the parameters of extreme value distributions only employ a small part of the observation values. When block maxima values are considered, many data are discarded, and therefore a lot of information is wasted. We develop a model to seize the whole data available in an extreme value framework. The key is to take advantage of the existing relation between the baseline parameters and the parameters of the block maxima distribution. We propose two methods to perform Bayesian estimation. Baseline distribution method (BDM) consists in computing estimations for the baseline parameters with all the data, and then making a transformation to compute estimations for the block maxima parameters. Improved baseline method (IBDM) is a refinement of the initial idea, with the aim of assigning more importance to the block maxima data than to the baseline values, performed by applying BDM to develop an improved prior distribution. We compare empirically these new methods with the Standard Bayesian analysis with non-informative prior, considering three baseline distributions that lead to a Gumbel extreme distribution, namely Gumbel, Exponential and Normal, by a broad simulation study.

Modelling dependence structures of extreme wind speed using bivariate distribution: a Bayesian approach

When investigating extremes of weather variables, it is seldom that a single weather station determines the damage, and extremes may be caused from the combined behaviour of several weather stations. To investigate joint dependence of extreme wind speed, a bivariate generalised extreme value distribution (BGEVD) was considered from frequentist and Bayesian approaches to analyse the extremes of component-wise monthly maximum wind speed at selected weather stations in South Africa. In the frequentist approach, the parameters of extreme value distributions (EVDs) were estimated with maximum likelihood, whereas in the Bayesian approach the Markov Chain Monte Carlo (MCMC) technique was used with the Metropolis–Hastings algorithm. The results showed that when fitted to component-wise maxima of extreme weather variables, the BGEVD provided apparent benefits over the univariate method, which allowed information to be pooled across stations and resulted in improved precision of the estimates for the parameters and return levels of the distributions. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from weather characteristics of four pairs of surrounding weather stations at various distances. The results from the Bayesian analysis showed that posterior inference might be affected by the choice of priors that were used to formulate the informative priors. From the results, it could be inferred that the Bayesian approach provides a satisfactory estimation strategy in terms of precision, compared with the frequentist approach, because it accounts for uncertainty in parameters and return level estimations.

On the Smoothing of the Generalized Extreme Value Distribution Parameters Using Penalized Maximum Likelihood: A Case Study on UVB Radiation Maxima in the Mexico City Metropolitan Area

This paper concerns the use and implementation of penalized maximum likelihood procedures to fitting smoothing functions of the generalized extreme value distribution parameters to analyze spatial extreme values of ultraviolet B (UVB) radiation across the Mexico City metropolitan area in the period 2000–2018. The model was fitted using a flexible semi-parametric approach and the parameters were estimated by the penalized maximum likelihood (PML) method. In order to investigate the performance of the model as well as the estimation method in the analysis of complex nonlinear trends for UVB radiation maxima, a simulation study was conducted. The results of the simulation study showed that penalized maximum likelihood yields better regularization to the model than the maximum likelihood estimates. We estimated return levels of extreme UVB radiation events through a nonstationary extreme value model using measurements of ozone (O3), nitrogen oxides (NOx), particles of 10 μm or less in diameter (PM10), carbon monoxide (CO), relative humidity (RH) and sulfur dioxide (SO2). The deviance statistics indicated that the nonstationary generalized extreme value (GEV) model adjusted was statistically better compared to the stationary model. The estimated smoothing functions of the location parameter of the GEV distribution on the spatial plane for different periods of time reveal the existence of well-defined trends in the maxima. In the temporal plane, a presence of temporal cyclic components oscillating over a weak linear component with a negative slope is noticed, while in the spatial plane, a weak nonlinear local trend is present on a plane with a positive slope towards the west, covering the entire study area. An explicit spatial estimate of the 25-year return period revealed that the more extreme risk levels are located in the western region of the study area.

Modelling dependency effect to extreme value distributions with application to extreme wind speed at Port Elizabeth, South Africa: a frequentist and Bayesian approaches

The dependency effect to extreme value distributions (EVDs) using the frequentist and Bayesian approaches have been used to analyse the extremes of annual and daily maximum wind speed at Port Elizabeth, South Africa. In the frequentist approach, the parameters of EVDs were estimated using maximum likelihood, whereas in the Bayesian approach the Markov Chain Monte Carlo technique with the Metropolis–Hastings algorithm was used. The results show that the EVDs fitted considering the dependency and seasonality effects with in the data series provide apparent benefits in terms of improved precision in estimation of the parameters as well as return levels of the distributions. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from other weather stations. The results from the Bayesian analysis show that posterior inference might be affected by the choice of priors used to formulate the informative priors. The Bayesian approach provides satisfactory estimation strategy in terms of precision compared to the frequentist approach, accounting for uncertainty in parameters and return levels estimation.

Face-Based Multiple User Active Authentication on Mobile Devices

Multiple user active authentications, in contrast with the single user active authentication, require the verification of identity of multiple subjects. Both traditional verification and identification-based solutions fail to address the specific challenges presented in this problem. We introduce Extremal Openset Rejection, a two-fold mechanism with a sparse representation-based identification step and a verification step for this purpose. In the verification step, concentration of the sparsity vector and the overlap between matched and non-matched distributions are considered for decision making. We introduce a semi-parametric model based on Extreme Value Theory for modeling the distributions, and an algorithm to estimate the parameters of extreme value distributions. Effectiveness of the proposed method is demonstrated using three publicly available face-based mobile active authentication data sets.

Variations in parameters of extreme value distributions of water level along the eastern Baltic Sea coast

Time series of water level at several sites of the eastern Baltic Sea contain statistically almost impossible outliers. Their presence may considerably modify the appearance of the relevant extreme value distributions. We analyse systematically the variation in the parameters of the Generalised Extreme Value (GEV), Gumbel and Weibull distributions for extreme water levels along a 600 km coastal stretch in the north-eastern Baltic Sea. The parameters of the distributions are evaluated using a) three different methods, b) the output of two different ocean models and c) two versions of block maxima (for calendar years and for stormy seasons), and compared with similar parameters retrieved for measured water level time series at five sites. The location and scale parameters of GEV and Gumbel distributions vary smoothly by about ±30% from their average values. The variation is modest along the open shores of the Baltic Sea proper. All these parameters increase almost linearly along the southern coast of the Gulf of Finland. The absolute values of the shape parameter of the GEV distribution are mostly <0.2. Different methods for evaluation lead to values of the same magnitude but with a different sign on the open shores of the Baltic Proper. This feature indicates that extreme water levels may follow radically different extreme value distributions in different locations of semi-enclosed seas.

A Skewed and Heavy-Tailed Latent Random Field Model for Spatial Extremes

ABSTRACTThis article develops Bayesian inference of spatial models with a flexible skew latent structure. Using the multivariate skew-normal distribution of Sahu et al., a valid random field model with stochastic skewing structure is proposed to take into account non-Gaussian features. The skewed spatial model is further improved via scale mixing to accommodate more extreme observations. Finally, the skewed and heavy-tailed random field model is used to describe the parameters of extreme value distributions. Bayesian prediction is done with a well-known Gibbs sampling algorithm, including slice sampling and adaptive simulation techniques. The model performance—as far as the identifiability of the parameters is concerned—is assessed by a simulation study and an analysis of extreme wind speeds across Iran. We conclude that our model provides more satisfactory results according to Bayesian model selection and predictive-based criteria. R code to implement the methods used is available as online supplementary material.

Parameter Estimation Method of Extreme Value Distribution under Periodic Inspection Based on Interval Statistic

In order to solve the reliability estimation with periodic inspection test data of products, the parameter estimation method of extreme value distribution based on interval statistic is given in this paper. The mean and variance formulas of interval statistic for the periodic inspection test data are given, the best linear unbiased estimation of extreme value distribution parameters under periodic inspection is also established, and the confidence limits of extreme value distribution percentile for the periodic inspection test data are obtained based on it.

Confidence intervals for parameters of IWD based on MLE and bootstrap

In this paper, we will study the joint confidence regions for the parameters of inverse Weibull distribution (IWD) in the point of view of record values. Based on this new censoring scheme, the approximate confidence intervals and percentile bootstrap confidence intervals as well as approximate joint confidence region for the parameters of IWD, are developed. One of the applications of the joint confidence regions of the parameters is to find confidence bounds for the functions of the parameters. Joint confidence regions for the parameters of extreme value distribution are also discussed. In this way we will discuss some numerical examples with real data set and simulated data, to illustrate the proposed method. A simulation study is performed to compare the proposed joint confidence regions.

Analysis of extreme waves with seasonal variation in the gulf of Mexico using a time-dependent GEV model

This paper considers a time-dependent GEV model to assess the extreme value of significant wave. The model incorporates the monthly seasonality of maximum values in a block of time of recorded events. The extreme value distribution parameters are fitted considering the seasonal behavior of the significant wave height using harmonic functions representing annual and semiannual cycles. The model is applied to the records in 10 buoys in the Gulf of Mexico. The obtained results are compared with some of the available results in the literature.

Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. [2006] have found analytical results.

Assigning spectrum-specific P-values to protein identifications by mass spectrometry

Although many methods and statistical approaches have been developed for protein identification by mass spectrometry, the problem of accurate assessment of statistical significance of protein identifications remains an open question. The main issues are as follows: (i) statistical significance of inferring peptide from experimental mass spectra must be platform independent and spectrum specific and (ii) individual spectrum matches at the peptide level must be combined into a single statistical measure at the protein level. We present a method and software to assign statistical significance to protein identifications from search engines for mass spectrometric data. The approach is based on asymptotic theory of order statistics. The parameters of the asymptotic distributions of identification scores are estimated for each spectrum individually. The method relies on new unbiased estimators for parameters of extreme value distribution. The estimated parameters are used to assign a spectrum-specific P-value to each peptide-spectrum match. The protein-level confidence measure combines P-values of peptide-to-spectrum matches. We extensively tested the method using triplicate mouse and yeast high-throughput proteomic experiments. The proposed statistical approach improves the sensitivity of protein identifications without compromising specificity. While the method was primarily designed to work with Mascot, it is platform-independent and is applicable to any search engine which outputs a single score for a peptide-spectrum match. We demonstrate this by testing the method in conjunction with X!Tandem. The software is available for download at ftp://genetics.bwh.harvard.edu/SSPV/. ssunyaev@rics.bwh.harvard.edu Supplementary data are available at Bioinformatics online.