Extreme value distribution and reliability of nonlinear stochastic structures

A new approach to evaluate the extreme value distribution （EVD） of the response and reliability of general multi-DOF nonlinear stochastic structures is proposed. The approach is based on the recently developed probability density evolution method, which enables the instantaneous probability density functions of the stochastic responses to be captured. In the proposed method, a virtual stochastic process is first constructed to satisfy the condition that the extreme value of the response equals the value of the constructed process at a certain instant of time. The probability density evolution method is then applied to evaluate the instantaneous probability density function of the response, yielding the EVD. The reliability is therefore available through a simple integration over the safe domain. A numerical algorithm is developed using the Number Theoretical Method to select the discretized representative points. Further, a hyper-ball is imposed to sieve the points from the preceding point set in the hypercube. In the numerical examples, the EVD of random variables is evaluated and compared with the analytical solution. A frame structure is analyzed to capture the EVD of the response and the dynamic reliability. The investigations indicate that the proposed approach provides reasonable accuracy and efficiency.

Applications of Bootstrap in Analyzing General Extreme Value Distributions

The bootstrap method is one of the new ways of studying statistical math which this article uses but is a major tool for studying and evaluating the values of parameters in probability distribution.Our research is concerned overview of the theory of infinite distribution functions.The tool to deal with the problems raised in the paper is the mathematical methods of random analysis(theory of random process and multivariate statistics).In this article,we introduce the new function to find out the bias and standard error with jackknife method for Generalized Extreme Value distributions.

Bivariate Analysis of Pollutants Monthly Maxima in Mexico City Using Extreme Value Distributions and Copula

In the present work, we are interested in studying the joint distributions of pairs of the monthly maxima of the pollutants used by the environmental authorities in Mexico City to classify the air quality in the metropolitan area. In order to obtain the joint distributions a copula will be considered. Since we are analyzing the monthly maxima, the extreme value distributions of Weibull and Fréchet are taken into account. Using these two distributions as marginal distributions in the copula a Bayesian inference was made in order to estimate the parameters of both distributions and also the association parameters appearing in the copula model. The pollutants taken into account are ozone, nitrogen dioxide, sulphur dioxide, carbon monoxide, and particulate matter with diameters smaller than 10 and 2.5 microns obtained from the Mexico City monitoring network. The estimation was performed by taking samples of the parameters generated through a Markov chain Monte Carlo algorithm implemented using the software OpenBugs. Once the algorithm is implemented it is applied to the pairs of pollutants where one of the coordinates of the pair is ozone and the other varies on the set of the remaining pollutants. Depending on the pollutant and the region where they were collected, different results were obtained. Hence, in some cases we have that the best model is that where we have a Fréchet distribution as the marginal distribution for the measurements of both pollutants and in others the most suitable model is the one assuming a Fréchet for ozone and a Weibull for the other pollutant. Results show that, in the present case, the estimated association parameter is a good representation to the correlation parameters between the pair of pollutants analyzed. Additionally, it is a straightforward task to obtain these correlation parameters from the corresponding association parameters.

General Regular Variation of n-th Order and the 2nd Order Edgeworth Expansion of the Extreme Value Distribution (Ⅰ)

In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.

MULTIVARIATE EXTREME VALUE DISTRIBUTION AND ITS FISHER INFORMATION MATRIX

The paper is concerned with the basic properties of multivariate extreme value distribution (in the Logistic model). We obtain the characteristic function and recurrence formula of the density function. The explicit algebraic formula for Fisher information matrix is indicated. A simple and accurate procedure for generating random vector from multivariate extreme value distribution is presented.

Stationary distributions for two-dimensional sticky Brownian motions:Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions,which find applications in many areas including queueing theory and mathematical finance.In this paper,we focus on stationary distributions for sticky Brownian motions.Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions.The kernel method,copula concept and extreme value theory are the main tools used in our analysis.

General Regular Variation of the n-th Order and 2nd Order Edgeworth Expansions of the Extreme Value Distribution （Ⅱ）

In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.

Analyzing the Annual Maximum Magnitude of Earthquakes in Japan by Extreme Value Theory

One of the most important and interesting issues associated with the earthquakes is the long-term trend of the extreme events. Extreme value theory provides methods for analysis of the most extreme parts of data. We estimated the annual maximum magnitude of earthquakes in Japan by extreme value theory using earthquake data between 1900 and 2019. Generalized extreme value (GEV) distribution was applied to fit the extreme indices. The distribution was used to estimate the probability of extreme values in specified time periods. The various diagnostic plots for assessing the accuracy of the GEV model fitted to the magnitude of maximum earthquakes data in Japan gave the validity of the GEV model. The extreme value index, ξ was evaluated as −0.163, with a 95% confidence interval of [−0.260, −0.0174] by the use of profile likelihood. Hence, the annual maximum magnitude of earthquakes has a finite upper limit. We obtained the maximum return level for the return periods of 10, 20, 50, 100 and 500 years along with their respective 95% confidence interval. Further, to get a more accurate confidence interval, we estimated the profile log-likelihood. The return level estimate was obtained as 7.83, 8.60 and 8.99, with a 95% confidence interval of [7.67, 8.06], [8.32, 9.21] and [8.61, 10.0] for the 10-, 100- and 500-year return periods, respectively. Hence, the 2011 off the Pacific coast of Tohoku Earthquake, which was the largest in the observation history of Japan, had a magnitude of 9.0, and it was a phenomenon that occurs once every 500 year.

Empirical Analysis of Value-at-Risk Estimation Methods Using Extreme Value Theory

This paper investigates methods of value-at-risk (VaR) estimation using extreme value theory (EVT). It compares two different estimation methods, 'two-step subsample bootstrap' based on moment estimation and maximum likelihood estimation (MLE), according to their theoretical bases and computation procedures. Then, the estimation results are analyzed together with those of normal method and empirical method. The empirical research of foreign exchange data shows that the EVT methods have good characters in estimating VaR under extreme conditions and 'two-step subsample bootstrap' method is preferable to MLE.

A New Method for Determining Threshold in Using PGCEVD to Calculate Return Values of Typhoon Wave Height

In using the PGCEVD （Poisson-Gumbel Compound Extreme Value Distribution） model to calculate return values of typhoon wave height, the quantitative selection of the threshold has blocked its application. By analyzing the principle of the threshold selection of PGCEVD model and in combination of the change point statistical methods, this paper proposes a new method for quantitative calculation of the threshold in PGCEVD model. Eleven samples from five engineering points in several coastal waters of Guangdong and Hainan, China, are calculated and analyzed by using PGCEVD model and the traditional Pearson type III distribution （P-III） model, respectively. By comparing the results of the two models, it is shown that the new method of selecting the optimal threshold is feasible. PGCEVD model has more stable results than that of P-III model and can be used for the return wave height in every direction.

Non-conventional modeling of extreme significant wave height through random sets

The analysis and design of offshore structures necessitates the consideration of wave loads. Realistic modeling of wave loads is particularly important to ensure reliable performance of these structures. Among the available methods for the modeling of the extreme significant wave height on a statistical basis, the peak over threshold method has attracted most attention. This method employs Poisson process to character- ize time-varying properties in the parameters of an extreme value distribution. In this paper, the peak over threshold method is reviewed and extended to account for subjectivity in the modeling. The freedom in selecting the threshold and the time span to separate extremes from the original time series data is incorpo- rated as imprecision in the model. This leads to an extension from random variables to random sets in the probabilistic model for the extreme significant wave height. The extended model is also applied to different periods of the sampled data to evaluate the significance of the climatic conditions on the uncertainties of the parameters.

Distributional expansion of maximum from logarithmic general error distribution

Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general error distribution is derived.

Probabilistic Distribution of Ice Forces on a Solitary Pile in the Liaodong Bay

The level ice thickness and compressive strength at the four measuring stations in the Liaodong Bay are inferred according to the hydrologic and meteorologic data there, then the yearly extreme ice forces on a solitary pile are calculated by the use of appropriate formula of ice forces and its probabilistic distribution is determined. Generally, the yearly extreme ice force follows Weibull distribution best as compared with Normal, Lognormal, and Extreme Value I distribution. On the other hand, the short-term distribution of ice forces on a solitary pile is obtained from the model experiment data analysis: It does not refuse Extreme Value I distribution.

A fuzzy quantification approach of uncertainties in an extreme wave height modeling

A non-traditional fuzzy quantification method is presented in the modeling of an extreme significant wave height. First, a set of parametric models are selected to fit time series data for the significant wave height and the extrapolation for extremes are obtained based on high quantile estimations. The quality of these results is compared and discussed. Then, the proposed fuzzy model, which combines Poisson process and gener-alized Pareto distribution （GPD） model, is applied to characterizing the wave extremes in the time series data. The estimations for a long-term return value are considered as time-varying as a threshold is regarded as non-stationary. The estimated intervals coupled with the fuzzy theory are then introduced to construct the probability bounds for the return values. This nontraditional model is analyzed in comparison with the traditional model in the degree of conservatism for the long-term estimate. The impact on the fuzzy bounds of extreme estimations from the non stationary effect in the proposed model is also investigated.

Influence of changes in extreme daily rainfall distribution on the stability of residual soil slopes

Many landslides triggered by intense rainfall have occurred in moun-tainous areas in Thailand,causing major economic losses and infra-structure damage.Extreme daily rainfall is a significant trigger for hillslope instability.Increases in extreme daily rainfall intensity due to climate change may be one of the key factors responsible for the increased landslides.Thus,in this context,changes in the intensity of extreme daily rainfall in Chiang Mai Province in North Thailand and their effects on hillslope stability are analyzed.Extreme rainfall is modeled using a generalized extreme value distribution and esti-mated for various return periods.A numerical analysis of seepage and an infinite slope stability model are combined to understand the hillslope response under extreme rainfall conditions.The analysis period is divided into two periods of 34 years:1952 to 1985 and 1986 to 2019.According to the analysis results,the distribution of extreme daily rainfall changes in terms of location.The average annual daily maximum rainfall increased by approximately 11.13%.The maximum decrease in the safety factor is approximately 4.5%;therefore,these changes in extreme daily rainfall should be consid-ered in future landslide prevention policies.

二项-二维对数正态分布及其在极端海况预测中的应用

Extreme value analysis is an indispensable method to predict the probability of marine disasters and calculate the design conditions of marine engineering.The rationality of extreme value analysis can be easily affected by the lack of sample data.The peaks over threshold(POT)method and compound extreme value distribution(CEVD)theory are effective methods to expand samples,but they still rely on long-term sea state data.To construct a probabilistic model using shortterm sea state data instead of the traditional annual maximum series(AMS),the binomial-bivariate log-normal CEVD(BBLCED)model is established in this thesis.The model not only considers the frequency of the extreme sea state,but it also reflects the correlation between different sea state elements(wave height and wave period)and reduces the requirement for the length of the data series.The model is applied to the calculation of design wave elements in a certain area of the Yellow Sea.The results indicate that the BBLCED model has good stability and fitting effect,which is close to the probability prediction results obtained from the long-term data,and reasonably reflects the probability distribution characteristics of the extreme sea state.The model can provide a reliable basis for coastal engineering design under the condition of a lack of marine data.Hence,it is suitable for extreme value prediction and calculation in the field of disaster prevention and reduction.

THE STATISTICAL DISTRIBUTION OF EXTREME VALUES OF WAVE-CURRENT FORCES ACTING ON SLENDER CIRCULAR CYLINDER IN IRREGULAR WAVES

Based on model test,the statistical distribution of extreme values of wave- current in-line forces acting on vertical circular cylinder is analyzed in this paper.It is shown that the results calculated by the simplified method,proposed by authors,agree well with the test data;Weibull distribution is also adoptable in the region of high KC number, and the shape parameter a and scale parameter β are related well with KC number respectively.

Disaster prevention design criteria for the estuarine cities:New Orleans and Shanghai The lesson from Hurricane Katrina

The accurate prediction of the typhoon （hurricane） induced extreme sea environments is very important for the coastal structure design in areas influenced by typhoon （hurricane）. In 2005 Hurricane Katrina brought a severe catastrophe in New Orleans by combined effects of hurricane induced extreme sea environments and upper flood of the Mississippi River. Like the New Orleans City, Shanghai is located at the estuarine area of the Changjiang River and the combined effect of typhoon induced extreme sea en- vironments, flood peak runoff from the Changjiang River coupled with the spring tide is the dominate factor for disaster prevention design criteria. The Poisson-nested logistic trivariate compound extreme value distribution （PNLTCEYD） is a new type of joint probability model which is proposed by compounding a discrete distribution （typhoon occurring frequency） into a continuous multivariate joint distribution （ typhoon induced extreme events）. The new model gives more reasonable predicted results for New Orleans and Shanghai disaster prevention design criteria.

Spatial Modeling of the Highest Daily Maximum Temperature in Korea via Max-stable Processes

This paper examines the annual highest daily maximum temperature （DMT） in Korea by using data from 56 weather stations and employing spatial extreme modeling. Our approach is based on max-stable processes （MSP） with Schlather＇s characterization. We divide the country into four regions for a better model fit and identify the best model for each region. We show that regional MSP modeling is more suitable than MSP modeling for the entire region and the pointwise generalized extreme value distribution approach. The advantage of spatial extreme modeling is that more precise and robust return levels and some indices of the highest temperatures can be obtained for observation stations and for locations with no observed data, and so help to determine the effects and assessment of vulnerability as well as to downscale extreme events.

Design Code Calibration of Offshore Platform Against Typhoon/Hurricane Attacks

Hurricanes Katrina and Rita resulted in the largest number of platforms destroyed and damaged in the history of Gulf of Mexico operations. With the trend of global warming, sea level rising and the frequency and intensity of typhoon increase. How to determine a reasonable deck elevation against the largest hurricane waves has become a key issue in offshore platforms design and construction for the unification of economy and safety. In this paper, the multivariate compound extreme value distribution （MCEVD） model is used to predict the deck elevation with different combination of tide, surge height, and crest height. Compared with practice recommended by American Petroleum Institute （API）, the prediction by MCEVD has probabilistic meaning and universality.