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1.
Probability weighted moments (PWM) are widely used in hydrology for estimating parameters of statistical distributions, including the Gumbel distribution. The classical PWM-approach considers the moments βi=E[XFi] with i=0,1 for estimation of the Gumbel scale and location parameters. However, there is no reason why these probability weights (F0 and F1) should provide the most efficient PWM-estimators of Gumbel parameters and quantiles. We explore an extended class of PWMs that does not impose arbitrary restrictions on the values of i. Estimation based on the extended class of PWMs is called the generalized method of probability weighted moments (GPWM) to distinguish it from the classical procedure. In fact, our investigation demonstrates that it may be advantage to use weight functions that are not of the form Fi. We propose an alternative PWM-estimator of the Gumbel distribution that maintains the computational simplicity of the classical PWM method, but provides slightly more accurate quantile estimates in terms of mean square error of estimation. A simple empirical formula for the standard error of the proposed quantile estimator is presented. 相似文献
2.
Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the Gnedenko-Pickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD has been successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h<70 km) in the Harvard catalog over the period 1977–2000 in 12 seismic zones. The GPD is found to approximate the tails of the seismic moment distributions quite well over the lower threshold approximately M 1024 dyne-cm, or somewhat above (i.e., moment-magnitudes larger than m
W
=5.3). We confirm that the b-value is very different (b=2.06 ± 0.30) in mid-ocean ridges compared to other zones (b=1.00 ± 0.04) with a very high statistical confidence and propose a physical mechanism contrasting crack-type rupture with dislocation-type behavior. The GPD can as well be applied in many problems of seismic hazard assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime. 相似文献
3.
A peaks over threshold (POT) method of analysing daily rainfall values is developed using a Poisson process of occurrences and a generalised Pareto distribution (GPD) for the exceedances. The parameters of the GPD are estimated by the method of probability weighted moments (PWM) and a method of combining the individual estimates to define a regional curve is proposed. 相似文献
4.
V. F. Pisarenko M. V. Rodkin T. A. Rukavishnikova 《Izvestiya Physics of the Solid Earth》2017,53(6):805-818
The most general approach to studying the recurrence law in the area of the rare largest events is associated with the use of limit law theorems of the theory of extreme values. In this paper, we use the Generalized Pareto Distribution (GPD). The unknown GPD parameters are typically determined by the method of maximal likelihood (ML). However, the ML estimation is only optimal for the case of fairly large samples (>200–300), whereas in many practical important cases, there are only dozens of large events. It is shown that in the case of a small number of events, the highest accuracy in the case of using the GPD is provided by the method of quantiles (MQs). In order to illustrate the obtained methodical results, we have formed the compiled data sets characterizing the tails of the distributions for typical subduction zones, regions of intracontinental seismicity, and for the zones of midoceanic (MO) ridges. This approach paves the way for designing a new method for seismic risk assessment. Here, instead of the unstable characteristics—the uppermost possible magnitude Mmax—it is recommended to use the quantiles of the distribution of random maxima for a future time interval. The results of calculating such quantiles are presented. 相似文献
5.
The maximum product of spacings (MPS) method is discussed from the standpoint of information theory. MPS parameter and quantile estimates for the generalized Pareto distribution and the two parameter log-logistic distribution are compared with the maximum likelihood(ML) and probability weighted moment (PWM) estimates. 相似文献
6.
Hongjoon Shin Taesoon Kim Sooyoung Kim Jun-Haeng Heo 《Stochastic Environmental Research and Risk Assessment (SERRA)》2010,24(2):183-197
In this study, the parameter estimations for the 3-parameter generalized logistic (GL) distribution are presented based on
the methods of moments (MOM), maximum likelihood (ML), and probability weighted moments (PWM). The asymptotic variances of
the MOM, ML, and PWM quantile estimators for the GL distribution are expressed as functions of the sample size, return period,
and parameters. A Monte Carlo simulation was performed to verify the derived expressions for variances and covariances between
parameters and to evaluate the applicability of the derived asymptotic variances of quantiles for the MOM, ML and PWM methods.
The simulation results generally show good agreement with the analytical results estimated from the asymptotic variances of
parameters and quantiles when the shape parameter (β) of the GL distribution is between −0.10 and 0.10 for the MOM method and between −0.25 and 0.45 for the ML and PWM methods,
respectively. In addition, the actual sample variances and the root mean square error (RMSE) of asymptotic variances of quantiles
for various sample sizes, return periods, and shape parameters were presented. In order to evaluate the applicability of the
estimation methods to real data and to compare the values of estimated parameter, quantiles, and confidence intervals based
on each parameter estimation method, the GL distribution was fitted to the 24-h annual maximum rainfall data at Pohang, Korea. 相似文献
7.
J.-H. Heo J. D. Salas K.-D. Kim 《Stochastic Environmental Research and Risk Assessment (SERRA)》2001,15(4):284-309
Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based
on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances
of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such
variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for
the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation
experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show
that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias
and mean square error. This is specially so for γ≥0.5 even for small sample sizes (e.g. N=10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter
be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of
annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles
based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits
and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low
flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T≥2 while it is about 30% for estimation based on MOM and PWM methods. In addition, the analysis of the rainfall data using
the three-parameter Weibull showed that while ML parameters can be estimated, the corresponding confidence limits and intervals
could not be found because the shape parameter was smaller than 2. 相似文献
8.
基于广义帕累托分布构建地震活动性模型,因其输入参数取值难以避免不确定性,导致依据该模型所得的地震危险性估计结果具有不确定性。鉴于此,本文选取青藏高原东北缘为研究区,提出了基于全域敏感性分析的地震危险性估计的不确定性分析流程和方法。首先,利用地震活动性广义帕累托模型,进行研究区地震危险性估计;然后,选取地震记录的起始时间和震级阈值作为地震活动性模型的输入参数,采用具有全域敏感性分析功能的E-FAST方法,对上述两个参数的不确定性以及两参数之间的相互作用对地震危险性估计不确定性的影响进行定量分析。结果表明:地震危险性估计结果(不同重现期的震级重现水平、震级上限及相应的置信区间)对两个输入参数中的震级阈值更为敏感;不同重现期的地震危险性估计结果对震级阈值的敏感程度不同;对不同的重现期而言,在影响地震危险性估计结果的不确定性上,两个输入参数之间存在非线性效应,且非线性效应程度不同。本文提出的不确定性分析流程和方法,可以推广应用于基于其它类型地震活动性模型的地震危险性估计不确定性分析。 相似文献
9.
ABSTRACT Precipitation prediction is central in hydrology and water resources planning and management. This paper introduces a semi-empirical predictive model to predict monthly precipitation and compares its predictive skill with those of machine learning (ML) methods. The stochastic method presented herein estimates monthly precipitation with one-step-ahead prediction properties. The ML predictive skill of the algorithms is evaluated by predicting monthly precipitation relying on the statistical association between precipitation and environmental and topographic factors. The semi-empirical predictive model features non-negative matrix factorization (NMF) for investigating the influence of multiple predictor variables on precipitation. The semi-empirical predictive model’s parameters are optimized with the hybrid genetic algorithm (GA) and Levenberg-Marquardt algorithm (LM), or GALMA, yielding a validated model with high predictive skill. The methodologies are illustrated with data from Hubei Province, China, which comprise 27 meteorological station datasets from 1988–2017. The empirical results provide valuable insights for developing semi-empirical rainfall prediction models. 相似文献
10.
In earthquake occurrence studies, the so-called q value can be considered both as one of the parameters describing the distribution of interevent times and as an index of
non-extensivity. Using simulated datasets, we compare four kinds of estimators, based on principle of maximum entropy (POME),
method of moments (MOM), maximum likelihood (MLE), and probability weighted moments (PWM) of the parameters (q and τ
0) of the distribution of inter-events times, assumed to be a generalized Pareto distribution (GPD), as defined by Tsallis
(1988) in the frame of non-extensive statistical physics. We then propose to use the unbiased version of PWM estimators to
compute the q value for the distribution of inter-event times in a realistic earthquake catalogue simulated according to the epidemic type
aftershock sequence (ETAS) model. Finally, we use these findings to build a statistical emulator of the q values of ETAS model. We employ treed Gaussian processes to obtain partitions of the parameter space so that the resulting
model respects sharp changes in physical behaviour. The emulator is used to understand the joint effects of input parameters
on the q value, exploring the relationship between ETAS model formulation and distribution of inter-event times. 相似文献
11.
Linear combinations of order statistics to estimate the quantiles of generalized pareto and extreme values distributions 总被引:2,自引:0,他引:2
G. Salvadori 《Stochastic Environmental Research and Risk Assessment (SERRA)》2003,17(1-2):116-140
Ad hoc techniques for estimating the quantiles of the Generalized Pareto (GP) and the Generalized Extreme Values (GEV) distributions
are introduced. The estimators proposed are based on new estimators of the position and the scale parameters recently introduced
in the Literature. They provide valuable estimates of the quantiles of interest both when the shape parameter is known and
when it is unknown (this latter case being of great relevance in practical applications). In addition, weakly-consistent estimators
are introduced, whose calculation does not require the knowledge of any parameter. The procedures are tested on simulated
data, and comparisons with other techniques are shown.
The research was partially supported by Contract n. ENV4-CT97-0529 within the project “FRAMEWORK” of the European Community – D.G. XII. Grants by “Progetto Giovani Ricercatori” are also acknowledged. 相似文献
12.
Frequency analysis of climate extreme events in Zanjan, Iran 总被引:2,自引:1,他引:1
Saeed Jahanbaksh Asl Ali Mohammad Khorshiddoust Yagob Dinpashoh Fatemeh Sarafrouzeh 《Stochastic Environmental Research and Risk Assessment (SERRA)》2013,27(7):1637-1650
In this study, generalized extreme value distribution (GEV) and generalized Pareto distribution (GPD) were fitted to the maximum and minimum temperature, maximum wind speed, and maximum precipitation series of Zanjan. Maximum (minimum) daily and absolute annual observations of Zanjan station from 1961 to 2011 were used. The parameters of the distributions were estimated using the maximum likelihood estimation method. Quantiles corresponding to 2, 5, 10, 25, 50, and 100 years return periods were calculated. It was found that both candidate distributions fitted to extreme events series, were statistically reasonable. Most of the observations from 1961 to 2011 were found to fall within 1–10 years return period. Low extremal index (θ) values were found for excess maximum and minimum temperatures over a high threshold, indicating the occurrence of consecutively high peaks. For the purpose of filtering the dependent observations to obtain a set of approximately independent threshold excesses, a declustering method was performed, which separated the excesses into clusters, then the de-clustered peaks were fitted to the GPD. In both models, values of the shape parameters of extreme precipitation and extreme wind speed were close to zero. The shape parameter was less negative in the GPD than the GEV. This leads to significantly lower return period estimates for high extremes with the GPD model. 相似文献
13.
Mixed estimation methods for Halphen distributions with applications in extreme hydrologic events 总被引:3,自引:1,他引:2
Fateh Chebana Salaheddine El Adlouni Bernard Bobée 《Stochastic Environmental Research and Risk Assessment (SERRA)》2010,24(3):359-376
The Halphen family of distributions is a flexible and complete system to fit sets of observations independent and identically
distributed. Recently, it is shown that this family of distributions represents a potential alternative to the generalized
extreme value distributions to model extreme hydrological events. The existence of jointly sufficient statistics for parameter
estimation leads to optimality of the method of maximum likelihood (ML). Nevertheless, the ML method requires numerical approximations
leading to less accurate values. However, estimators by the method of moments (MM) are explicit and their computation is fast.
Even though MM method leads to good results, it is not optimal. In order to combine the advantages of the ML (optimality)
and MM (efficiency and fast computations), two new mixed methods were proposed in this paper. One of the two methods is direct
and the other is iterative, denoted respectively direct mixed method (MMD) and iterative mixed method (MMI). An overall comparison
of the four estimation methods (MM, ML, MMD and MMI) was performed using Monte Carlo simulations regarding the three Halphen
distributions. Generally, the MMI method can be considered for the three Halphen distributions since it is recommended for
a majority of cases encountered in hydrology. The principal idea of the mixed methods MMD and MMI could be generalized for
other distributions with complicated density functions. 相似文献
14.
Non-stationary frequency analysis of extreme precipitation in South Korea using peaks-over-threshold and annual maxima 总被引:2,自引:2,他引:0
Sungwook Wi Juan B. Valdés Scott Steinschneider Tae-Woong Kim 《Stochastic Environmental Research and Risk Assessment (SERRA)》2016,30(2):583-606
The conventional approach to the frequency analysis of extreme precipitation is complicated by non-stationarity resulting from climate variability and change. This study utilized a non-stationary frequency analysis to better understand the time-varying behavior of short-duration (1-, 6-, 12-, and 24-h) precipitation extremes at 65 weather stations scattered across South Korea. Trends in precipitation extremes were diagnosed with respect to both annual maximum precipitation (AMP) and peaks-over-threshold (POT) extremes. Non-stationary generalized extreme value (GEV) and generalized Pareto distribution (GPD) models with model parameters made a linear function of time were applied to AMP and POT respectively. Trends detected using the Mann–Kendall test revealed that the stations showing an increasing trend in AMP extremes were concentrated in the mountainous areas (the northeast and southwest regions) of South Korea. Trend tests on POT extremes provided fairly different results, with a significantly reduced number of stations showing an increasing trend and with some stations showing a decreasing trend. For most of stations showing a statistically significant trend, non-stationary GEV and GPD models significantly outperformed their stationary counterparts, particularly for precipitation extremes with shorter durations. Due to a significant-increasing trend in the POT frequency found at a considerable number of stations (about 10 stations for each rainfall duration), the performance of modeling POT extremes was further improved with a non-homogeneous Poisson model. The large differences in design storm estimates between stationary and non-stationary models (design storm estimates from stationary models were significantly lower than the estimates of non-stationary models) demonstrated the challenges in relying on the stationary assumption when planning the design and management of water facilities. This study also highlighted the need of caution when quantifying design storms from POT and AMP extremes by showing a large discrepancy between the estimates from those two approaches. 相似文献
15.
Youcun Liu Miaojie Lu Xueli Huo Yonghong Hao Hongkai Gao Yan Liu Yonghui Fan Yuhuan Cui Francois Metivier 《水文研究》2016,30(3):424-432
Global climate change models have predicted the intensification of extreme events, and these predictions are already occurring. For disaster management and adaptation of extreme events, it is essential to improve the accuracy of extreme value statistical models. In this study, Bayes' Theorem is introduced to estimate parameters in Generalized Pareto Distribution (GPD), and then the GPD is applied to simulate the distribution of minimum monthly runoff during dry periods in mountain areas of the Ürümqi River, Northwest China. Bayes' Theorem treats parameters as random variables and provides a robust way to convert the prior distribution of parameters into a posterior distribution. Statistical inferences based on posterior distribution can provide a more comprehensive representation of the parameters. An improved Markov Chain Monte Carlo (MCMC) method, which can solve high‐dimensional integral computation in the Bayes equation, is used to generate parameter simulations from the posterior distribution. Model diagnosis plots are made to guarantee the fitted GPD is appropriate. Then based on the GPD with Bayesian parameter estimates, monthly runoff minima corresponding to different return periods can be calculated. The results show that the improved MCMC method is able to make Markov chains converge faster. The monthly runoff minima corresponding to 10a, 25a, 50a and 100a return periods are 0.60 m3/s, 0.44 m3/s, 0.32 m3/s and 0.20 m3/s respectively. The lower boundary of 95% confidence interval of 100a return level is below zero, which implies that the Ürümqi River is likely to cease to flow when 100a return level appears in dry periods. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
16.
Nurulkamal Masseran Muhammad Aslam Mohd Safari 《Stochastic Environmental Research and Risk Assessment (SERRA)》2020,34(3):545-559
The occurrences of extreme pollution events have serious effects on human health, environmental ecosystems, and the national economy. To gain a better understanding of this issue, risk assessments on the behavior of these events must be effectively designed to anticipate the likelihood of their occurrence. In this study, we propose using the intensity–duration–frequency (IDF) technique to describe the relationship of pollution intensity (i) to its duration (d) and return period (T). As a case study, we used data from the city of Klang, Malaysia. The construction of IDF curves involves a process of determining a partial duration series of an extreme pollution event. Based on PDS data, a generalized Pareto distribution (GPD) is used to represent its probabilistic behaviors. The estimated return period and IDF curves for pollution intensities corresponding to various return periods are determined based on the fitted GPD model. The results reveal that pollution intensities in Klang tend to increase with increases in the length of time between return periods. Although the IDF curves show different magnitudes for different return periods, all the curves show similar increasing trends. In fact, longer return periods are associated with higher estimates of pollution intensity. Based on the study results, we can conclude that the IDF approach provides a good basis for decision-makers to evaluate the expected risk of future extreme pollution events. 相似文献
17.
An extreme value analysis of the flow of Burbage Brook is carried out by modelling peaks over a high threshold. The aims are to illustrate recently developed statistical techniques and to report on interesting features of the flow of the brook over a 58-year period. Peak flows are found to show marked seasonal variation and a downward trend. Then-year return level is estimated for various values ofn, and the reliability of the estimates is assessed. 相似文献
18.
J. H. Barrett 《Stochastic Hydrology and Hydraulics》1992,6(3):151-165
An extreme value analysis of the flow of Burbage Brook is carried out by modelling peaks over a high threshold. The aims are to illustrate recently developed statistical techniques and to report on interesting features of the flow of the brook over a 58-year period. Peak flows are found to show marked seasonal variation and a downward trend. Then-year return level is estimated for various values ofn, and the reliability of the estimates is assessed. 相似文献
19.
B. Bobee L. Perreault F. Ashkar 《Stochastic Environmental Research and Risk Assessment (SERRA)》1993,7(1):41-65
We refocus attention on moment ratio diagrams and their uses in hydrology with four major objectives: (1) to summarize the information available in the literature about possible uses of the traditional moment ratio diagram introduced by Karl Pearson, which uses the coefficient of skewness and of kurtosis to compare the shapes of various distributions commonly used in hydrology; (2) to complete this traditional MRD by integrating into it the regions occupied by the log-Pearson Type III and generalized gamma distributions which are more and more used in hydrology; (3) to present another MRD which uses ratios of moments of orders –1 (harmonic mean), quasi zero (geometric mean) and 1 (arithmetic mean); (4) to stress the need to consider the different MRD's (along with the more recently introduced L-moment ratio diagrams) as complementary tools for choosing between distributions fitted to hydrologic data. Finally, using Monte Carlo simulation we compare the two types of diagrams as tools to identify and discriminate between different distributions. 相似文献
20.
The work presents statistical methods for estimating the distribution parameters of rare, strong earthquakes. Using the two
main theorems of extreme value theory (EVT), the distribution of T-maximum (the maximum magnitude over the time period T). Two methods for estimating the parameters of this distribution are proposed using the Generalized Pareto Distribution (GPD)
and the General Extreme Value Distribution (GEV). In addition, the that allow the determination of the distribution of the
T-maximum for an arbitrary value of T are proposed. The approach being used clarifies the nature of the instability of the widely accepted M max parameter. In the work, instead of unstable values of the M max parameter, the robust parameter Q
T
(q), the q level quantile for the distribution of the T-maximum, is proposed to be used. The described method has been applied to the Harvard Catalogue of Seismic Moments of 1977–2006
and to the Magnitude Catalogue for Fennoscandia in 1900–2005. Moreover, the estimates of parameters of the corresponding GPD
and GEV distributions, in particular, the most interesting shape parameter and the values of the M
max and Q
T
(q) parameters are given. 相似文献