Unbiased estimation of probability weighted moments and partial probability weighted moments from systematic and historical flood information and their application to estimating the GEV distribution

Unbiased estimators of probability weighted moments (PWM) and partial probability weighted moments (PPWM) from systematic and historical flood information are derived. Applications are made to estimating parameters and quantiles of the generalized extreme value (GEV) distribution. The effect of lower bound censoring, which might be deliberately introduced in practice, is also considered.     

On the use of partial probability weighted moments in the analysis of hydrological extremes

The use of partial probability weighted moments (PPWM) for estimating hydrological extremes is compared to that of probability weighted moments (PWM). Firstly, estimates from at‐site data are considered. Two Monte Carlo analyses, conducted using continuous and empirical parent distributions (of peak discharge and daily rainfall annual maxima) and applying four different distributions (Gumbel, Fréchet, GEV and generalized Pareto), show that the estimates obtained from PPWMs are better than those obtained from PWMs if the parent distribution is unknown, as happens in practice. Secondly, the use of partial L‐moments (obtained from PPWMs) as diagnostic tools is considered. The theoretical partial L‐diagrams are compared with the experimental data. Five different distributions (exponential, Pareto, Gumbel, GEV and generalized Pareto) and 297 samples of peak discharge annual maxima are considered. Finally, the use of PPWMs with regional data is investigated. Three different kinds of regional analyses are considered. The first kind is the regression of quantile estimates on basin area. The study is conducted applying the GEV distribution to peak discharge annual maxima. The regressions obtained with PPWMs are slightly better than those obtained with PWMs. The second kind of regional analysis is the parametric one, of which four different models are considered. The congruence between local and regional estimates is examined, using peak discharge annual maxima. The congruence degree is sometimes higher for PPWMs, sometimes for PWMs. The third kind of regional analysis uses the index flood method. The study, conducted applying the GEV distribution to synthetic data from a lognormal joint distribution, shows that better estimates are obtained sometimes from PPWMs, sometimes from PWMs. All the results seem to indicate that using PPWMs can constitute a valid tool, provided that the influence of ouliers, of course higher with censored samples, is kept under control. Copyright © 2007 John Wiley & Sons, Ltd.     

Further research on application of probability weighted moments in estimating parameters of the Pearson type three distribution

This paper contains further research results on the recently developed method for estimating parameters of P-III distribution by using probability weighted moments (PWM). The computing procedure can be largely simplified and the PWM method is appropriate and functional for various conditions of the skewness coefficient. Statistical experiments made based on the new procedure have shown that PWM estimators are almost unbiased and compare favorably with those by the conventional moment method.     

Expressions relating probability weighted moments to parameters of several distributions inexpressible in inverse form

The probability weighted moment (PWM) method can generally be used in estimating parameters of a distribution whose inverse form cannot be expressed explicitly. For several distributions, such as normal, log-normal and Pearson Type Three distributions, the expressions relating PWM to parameters have the same forms. Such expressions may be readily employed in practice for estimating the parameters.     

Unbiased estimation of flood risk with the GEV distribution

Conventional flood frequency analysis is concerned with providing an unbiased estimate of the magnitude of the design flow exceeded with the probabilityp, but sampling uncertainties imply that such estimates will, on average, be exceeded more frequently. An alternative approach is therefore, to derive an estimator which gives an unbiased estimate of flow risk: the difference between the two magnitudes reflects uncertainties in parameter estimation. An empirical procedure has been developed to estimate the mean true exceedance probabilities of conventional estimates made using a GEV distribution fitted by probability weighted moments, and adjustment factors have been determined to enable the estimation of flood magnitudes exceeded with, on average, the desired probability.     

Estimation of the annual maximum distribution from samples of maxima in separate seasons

Quantile estimates of the annual maximum distribution can be obtained by fitting theoretical distributions to the maxima in separate seasons, e.g. to the monthly maxima. In this paper, asymptotic expressions for the bias and the variance of such estimates are derived for the case that the seasonal maxima follow a Gumbel distribution. Results from these expressions are presented for a situation with no seasonal variation and for maximum precipitation depths at Uccle/Ukkel (Belgium). It is shown that the bias is often negligible and that the variance reduction by using seasonal maxima instead of just the annual maxima strongly depends on the seasonal variation in the data. A comparison is made between the asymptotic standard error of quantile estimates from monthlymaxima with those from a partial duration series. Much attention is paid to the effect of model misspecification on the resulting quantile estimates of the annual maximum distribution. The use of seasonal maxima should be viewed with caution when the upper tail of this distribution is of interest.     

The generalized method of moments as applied to the generalized gamma distribution

The generalized gamma (GG) distribution has a density function that can take on many possible forms commonly encountered in hydrologic applications. This fact has led many authors to study the properties of the distribution and to propose various estimation techniques (method of moments, mixed moments, maximum likelihood etc.). We discuss some of the most important properties of this flexible distribution and present a flexible method of parameter estimation, called the generalized method of moments (GMM) which combines any three moments of the GG distribution. The main advantage of this general method is that it has many of the previously proposed methods of estimation as special cases. We also give a general formula for the variance of theT-year eventX _T obtained by the GMM along with a general formula for the parameter estimates and also for the covariances and correlation coefficients between any pair of such estimates. By applying the GMM and carefully choosing the order of the moments that are used in the estimation one can significantly reduce the variance ofT-year events for the range of return periods that are of interest.     

Estimation of the generalized logistic distribution of extreme events using partial L-moments

Abstract

Statistical analysis of extreme events is often carried out to predict large return period events. In this paper, the use of partial L-moments (PL-moments) for estimating hydrological extremes from censored data is compared to that of simple L-moments. Expressions of parameter estimation are derived to fit the generalized logistic (GLO) distribution based on the PL-moments approach. Monte Carlo analysis is used to examine the sampling properties of PL-moments in fitting the GLO distribution to both GLO and non-GLO samples. Finally, both PL-moments and L-moments are used to fit the GLO distribution to 37 annual maximum rainfall series of raingauge station Kampung Lui (3118102) in Selangor, Malaysia, and it is found that analysis of censored rainfall samples of PL-moments would improve the estimation of large return period events.

Editor D. Koutsoyiannis; Associate editor K. Hamed

Citation Zakaria, Z.A., Shabri, A. and Ahmad, U.N., 2012. Estimation of the generalized logistic distribution of extreme events using partial L-moments. Hydrological Sciences Journal, 57 (3), 424–432.     

Estimation of confidence interval for hydrological design value for some continuous distributions under complete and censored samples

The quantile of a probability distribution, known as return period or hydrological design value of a hydrological variable is the value corresponding to fixed non-exceedence probability and is very important notion in hydrology. In hydraulic engineering design and water resources management, confidence interval (CI) estimation for a population quantile is of primary interest and among other applications, is used to assess the pollution level of a contaminant in water, air etc. The accuracy on such estimation directly influences the engineering investments and safety. The two parameter Weibull, Pareto, Lognormal, Inverse Gaussian, Gamma are some commonly used probability models in such applications. In spite of its practical importance, the problem of CI estimation of a quantile of these widely applicable distributions has been less attended in the literature. In this paper, a new method is proposed to obtain a CI for a quantile of any distribution for which [or the probability distribution of any one-to-one function of the underlying random variable (RV)] generalized pivotal quantities (GPQs) exist for its parameters. The proposed method is elucidated by constructing CIs for quantiles of Weibull, Pareto, Lognormal, Extreme value distribution of type-I for minimum, Exponential and Normal distributions for complete as well as type II singly right censored samples. The empirical performance evaluation of the proposed method evinced that the proposed method has exact well concentrated coverage probabilities near the nominal level even for small uncensored samples as small as 5 and for censored samples as long as the proportion of censored observations is up to 0.70. The existing methods for Weibull distribution have poor or dispersed coverage probabilities with respect to the nominal level for complete samples. Applications of the proposed method in ground water monitoring and in the assessment of air pollution are illustrated for practitioners.     

Impulse response of the kinematic diffusion model as a probability distribution of hydrologic samples with zero values

It is hypothesized that the unit impulse response of a linearized kinematic diffusion (KD) model is a probability distribution suitable for frequency analysis of hydrologic samples with zero values. Such samples may include data on monthly precipitation in dry seasons, annual low flow, and annual maximum peak discharge observed in arid and semiarid regions. The hypothesized probability distribution has two parameters, which are estimated using the methods of moments (MOM) and maximum likelihood (MLM). Also estimated are errors in quantiles for MOM and MLM. The distribution shows an equivalency of MOM and MLM with respect to the mean value—an important property for ML-estimation in the case of the unknown true distribution function. The hypothesized KD distribution is tested on 44 discharge data series and compared with the Muskingum-like (M-like) probability distribution function. A comparison of empirical distribution with KD and M-like distributions shows that MOM better reproduces the upper tail of the distribution, while MLM is more robust for higher sample values and more conditioned on the value of the probability of the zero value event. The KD-model is suitable for frequency analysis of short samples with zero values and it is more universal than the M-like model as its modal value cannot be only equaled to zero value but also to any positive value.     

Method of moments of the Halphen distribution parameters

Halphen laws have been proposed as a complete system of distributions with sufficient statistics that lead to estimation with minimum variance. The Halphen system provides a flexibility to fit a large variety of data sets from natural events. In this paper we present the method of moments (MM) to estimate the Halphen type B and IB distribution parameters. Their computation is very fast when compared to those given by the maximum likelihood method (ML). Furthermore, this estimation method is very easy to implement since the formulae are explicit. Some simulations show the equivalence of both methods when estimating the quantiles for finite sample size.     

一种基于经验分布的大地震复发概率计算方法

以历史重演原则和构造类比原则为基础,提出了一种基于经验分布的大地震复发概率计算方法,该方法不作任何复发概率分布的强假定,直接通过对大量地震序列数据的蒙特卡罗随机抽样来模拟未来大地震的复发规律,进而统计得到未来一段时间内的大地震发生概率,并以鲜水河断裂带炉霍段和道孚段为实例,利用本文给出的复发概率计算方法得出炉霍段和道孚段未来50年大地震发生概率分别为0.15和0.31。      

Improved fuzzy weighted optimum curve-fitting method for estimating the parameters of a Pearson Type-III distribution

ABSTRACT

The objective of the curve-fitting method is to determine the optimal distribution by parameter estimation. The selection of the parameter estimation methods and the determination of the parameter estimation results may vary according to the different aims of the curve fitting, as well as the different accuracies and positions of the points. To solve the problem, the fuzzy weighted optimum curve-fitting method (FWOCM) was used to deal with the characters. The deficiencies of the original FWOCM were analysed, and it was found that the membership function and nomograph were unable to effectively deal with the curve fitting. An improved method and its indexes were evaluated, using effectiveness and unbiasedness as the assessment criteria, while scoring and percentage methods were chosen to comprehensively assess the statistical results. Compared with FWOCM, the results showed greater effectiveness and unbiasedness in the improved method.     

Can we determine the transverse macrodispersivity by using the method of moments?

One of the main assumptions that renders the stochastic theories applicable to real aquifers is the ergodic hypothesis, i.e. the possibility to exchange ensemble and spatial averages of a variable of interest. The principal aim of this paper is to elucidate the conditions that allow for an exchange between ensemble and spatially averaged second moments of concentration field (S_ij); the fulfillment of the ergodic condition underlies the applicability of the dispersion coefficients or other related quantities obtained by the stochastic theories to actual aquifers. The fulfillment of the ergodic hypothesis is assessed here by analyzing the diminishing of the variance of S_ij as the initial size of the plume V₀ grows, i.e. the tendency of S_ij toward its expected value 〈S_ij〉. It is shown that it is not always possible to assume ergodicity for solute plumes in heterogeneous aquifers. For the typical plume configurations encountered in applications, transverse and vertical spreading are the most problematic in this respect. In particular, satisfying the ergodic hypothesis depends largely on the initial plume configuration and size, on one hand, and the direction of the moment of interest, on the other. Numerical simulations based on the analytic element method elucidate the results. The relevance of the results is mostly felt for the inference of macrodispersive parameters, which are often derived through S_ij. The work indicates that S_ij may be a distorted and inadequate measure of the plume spread. This should serve as a warning against application of results based on ensemble averages to real-life plumes, particularly when estimating macrodispersion coefficient from field experiment.     

Geostatistical modeling of clay spatial distribution in siliciclastic rock samples using the plurigaussian simulation method

In order to implement secondary and enhanced oil recovery processes in complex terrigenous formations as is usual in turbidite deposits, a precise knowledge of the spatial distribution of shale grains is a crucial element for the fluid flow prediction. The reason of this is that the interaction of water with shale grains can significantly modify their size and/or shape, which in turn would cause porous space sealing with the subsequent impact in the flow. In this work, a methodology for stochastic simulations of spatial grains distributions obtained from scanning electron microscopy images of siliciclastic rock samples is proposed. The aim of the methodology is to obtain stochastic models would let us investigate the shale grain behavior under various physico-chemical interactions and flux regimes, which in turn, will help us get effective petrophysical properties (porosity and permeability) at core scale. For stochastic spatial grains simulations a plurigaussian method is applied, which is based on the truncation of several standard Gaussian random functions. This approach is very flexible, since it allows to simultaneously manage the proportions of each grain category in a very general manner and to rigorously handle their spatial dependency relationships in the case of two or more grain categories. The obtained results show that the stochastically simulated porous media using the plurigaussian method adequately reproduces the proportions, basic statistics and sizes of the pore structures present in the studied reference images.     

Effective probability distribution approximation for the reconstruction of missing data

Stochastic Environmental Research and Risk Assessment - Spatially distributed processes can be modeled as random fields. The complex spatial dependence is then incorporated in the joint probability...     

利用 Yabuki＆Matsu'ura 反演方法计算2011年日本东北地区太平洋海域Mw9.0级地震同震滑动分布

Yabuki & Matsu'ura反演方法是利用ABIC最佳模型参数选取方法和平滑的滑动分布作为约束条件,由形变观测数据计算发震断层滑动分布.本文基于日本列岛同震GPS观测数据和发震断层曲面构造模型,利用Yabuki&Matsu'ura反演方法计算2011年日本东北地区太平洋海域Mw9.0级地震的发震断层同震滑动分布.反演结果表明,断层面上的最大滑动量为35 m,较大滑动分布在浅于30 km的震源中心上部,最大破裂集中在20 km深度的地方,其地震矩约为3.63×1022N·m,对应的矩震级为Mw9.0.模拟结果显示Yabuki&Matsu'ura反演方法更适用于倾角低于40°的断层模型反演.最后,本文基于上述方法获得的发震断层滑动模型,利用地球体位错理论正演计算该地震在中国及其邻区产生的远场形变,正演计算结果基本可以解释由中国GPS陆态网络观测到的同震形变.     

利用贝叶斯极值分布函数估算河套断陷带的发震概率

正Campbell(1982,1983)将贝叶斯概率理论和极值概率模型相结合,发展出一种估算地震发生概率的贝叶斯极值分布模型。在此模型中,地震活动的先验估计值是基于地震矩、滑动速率、地震复发率和震级等数据计算得到的,而后将估计值用于贝叶斯理论的后验估算,或者用于研究区的历史地震活动性的评估方面(李拴虎等,2016)。