岩石力学参数概率分布的信息熵推断

基于信息论的最大熵原理，直接用牛顿迭代法，由样本矩来推断岩石力学参数的概率密度函数，并且用精度较高的K-S检验法，从理论上证明所求的密度函数的正确性。算例表明，该方法避免了复杂的数值计算，可以满足岩石工程随机可靠性分析的要求。直接根据试验样本信息和统计方法推断，而不是先假定成经典的理论概率分布，具有更充分的数学和物理意义。由于概率分布采用指数形式而不是幂函数系多项式，避免了计算中的震荡等不稳定现象。     

岩土参数最优概率分布推断方法及判别准则的研究

 为了获得岩土参数概率分布的最佳推断方法,首先考虑岩土参数均为非负值的特性,提出以“3?”准则为基础并考虑样本数据偏度进行调整的积分区间确定方法;以5组典型岩土参数作为基本信息,利用经典分布拟合法、最大熵法、一般多项式逼近法、正交多项式逼近法和正态信息扩散法分别对其概率分布函数进行了推断,并根据K-S检验法进行检验。通过所得概率分布的检验值、累积概率值和函数曲线的对比,研究上述方法的优劣。结果表明：与经典分布拟合法相比,其他4种方法的检验值普遍较小,均克服了经典分布无法反应样本随机波动性的缺陷,并且满足累积概率值等于1的要求。但最大熵法的检验值存在大于经典分布检验值的情况,一般多项式和正交多项式方法的概率密度函数则在样本数据局部分布区间存在负值情况。正态信息扩散法不存在上述缺陷,而且该方法得到的检验值最小,累积概率值始终为1,并可以随着样本的波动呈现多峰状态,拟合精度最高,是一种比较理想的最佳推断方法。最后给出了岩土参数最优概率分布的判别准则。     

Comparisons of the L- and LH-moments in the selection of the best distribution for regional flood frequency analysis in Lake Urmia Basin

Design flood estimation is an important task that is required in the planning and design of many civil engineering projects. Finding the most suitable distribution to flood samples and selecting the appropriate parameter estimation method are of great importance for flood frequency analysis. In this study, L-moments and LH-moments have been used to characterise the upper part of distributions and larger events in flood data. Three extreme value distributions, that is, generalised extreme value (GEV), generalised logistic (GLO) and generalised Pareto (GPA), through different levels of the LH-moments have been applied to describe the annual maximum flood data obtained from 34 sites in Lake Urmia Basin, northwest Iran. The performances of these distributions have been assessed by evaluating the relative root-mean-square error and the minimum L-kurtosis difference criterion. The result shows that the LH-moments are more efficient for obtaining improved values of flood peaks than the L-moments. For the upper part of distributions, the GEV distribution using the L2-moment is the best, followed by the GLO distribution using the L1-moment and the GPA distribution using the L3-moment. The GLO distribution using the L-moments methods is the best among all distributions for a complete data series.     

POINT-ESTIMATE MOMENT-BASED RELIABILITY ANALYSIS

In moment-based reliability analysis, the probability of failure is calculated from a probability distribution fitted to the first few calculated moments of a limit state function. The calculation of the moments and the distributions fitting in the analysis are described. We suggest using the point estimate method to calculate the moments of the limit state function. A probability distribution from the Gram-Charlier series type A distributions or the Johnson family of distributions is selected to fit the calculated moments. The method, in general, does not require knowledge of the probability distributions of the basic random variables involved in the limit state function but their statistical moments. The method is simple to implement and can be used directly with an available deterministic computer program because it does not require iteration or derivatives of the limit state function.     

Determining the probability distribution of rework costs in construction and engineering projects

Rework arises due to design errors, changes and omissions during design and has been found to contribute to 52% of a project's cost overrun. The statistical characteristics of rework costs experienced from contract award in 276 construction and engineering projects were analysed. The skewness and kurtosis values of rework costs are computed to determine if the empirical distribution of the data follows a normal distribution. The empirical distributions for rework costs are found to be non-Gaussian. Theoretical probability distributions are fitted to the rework data. The Kolmogorov–Smirnov and Anderson–Darling non-parametric tests are used to determine the ‘Goodness of Fit’ of the selected probability distributions. A Generalised Pareto probability function is found to provide the best overall distribution fit for rework costs. The Generalised Pareto distribution is used to calculate the probability of rework being experienced for the selected sample. Projects with a contract value <A$1million had higher rework probabilities than that of >A$101 million. Larger projects may be better managed and longer completion times provide an opportunity to make adjustments to facilitate cost control. The anticipation that rework will occur using the probabilities that are derived can enable a quantitative risk assessment to be undertaken prior to the commencement of construction.     

Determination of groundwater sampling frequencies through Bayesian decision theory

Bayesian decision theory is applied to determine an optimum sampling frequency for chlorides in a public water supply well. The well is located approximately one mile downstream of a landfill with an established contaminant plume and has exhibited increasing chloride concentrations. A digital computer program is utilized to solve the optimization problem using Bayes' theorem for continuous probability distributions. The sampling frequency obtained by this method is compared with the frequency used by the authority operating the public water supply well and to the frequencies estimated by other methods that have been recommended for determining sampling frequency. The sampling frequencies obtained using Bayesian decision theory are close to those obtained with other mathematical methods. The results are also similar to the sampling frequency chosen by the operating authority when monitoring the well.     

Estimation of the maximum entropy quantile function using fractional probability weighted moments

In a previous study, the conventional or integral-order probability weighted moments (IPWM) and the principle of maximum entropy were combined to derive an analytical form of the quantile function of a random variable [Pandey MD. Direct estimation of quantile functions using the maximum entropy principle. Struct Safety 2000;22(1):61–79]. This method is extended and improved in the present paper by utilizing the concept of fractional probability weighted moments (FPWMs). A general estimation method is proposed in which the Monte Carlo simulations and optimization algorithms are combined to estimate fractionals of FPWM that would lead to the best-fit quantile function. The numerical examples presented in the paper illustrate that the accuracy of the proposed FPWM based quantile function is superior to that estimated from the use of conventional IPWMs.     

Evaluation of the FEH Rainfall-Runoff Method for Catchments in the UK

The Flood Estimation Handbook, which was published in 1999, has substantially replaced the Flood Studies Report for design flood estimation in the UK. Although there are substantial differences between the handbook and the report, the philosophy of the rainfall-runoff method has been retained in the handbook. However, there are differences in the equations that have been used to estimate parameters of the rainfall-runoff method.
This paper provides a comprehensive evaluation of the performance of the Flood Estimation Handbook rainfall-runoff method for flood-frequency estimation, using data from eighty-eight catchments in the UK. The performance is evaluated by reference to the flood quantiles obtained from single-site frequency analysis, and comparison is made with the performance of the Flood Studies Report methodology. In general terms, the handbook over-estimates flood quantiles, with respect to (a) those estimated from observed records and (b) the report's quantiles. The paper explores the causes of these over-estimations and seeks linkages with particular catchment characteristics.     

Direct estimation of quantile functions using the maximum entropy principle

The paper presents a distribution free method for estimating the quantile function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints specified in terms of the probability-weighted moments estimated from observed data. Traditionally, MaxEnt is used for estimating the probability density function under specified moment constraints. The density function is then integrated to obtain the cumulative distribution function, which needs to be inverted to obtain a quantile corresponding to some specified probability. For correct modelling of the distribution tail, higher order moments must be considered in the analysis. However, the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain. The difficulty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt Principle in extreme quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting the PWM as the moment of quantile function, the paper derives an analytical form of quantile function using MaxEnt principle. Monte Carlo simulations are performed to assess the accuracy of MaxEnt quantile estimates computed from small samples.     

PIPE SYSTEM PRESSURE PROBABILITIES WITH FLUCTUATING DEMANDS

A method for calculating the probability of very low pressures at nodes within a pressurised pipe network subject to a set of independent nodal demands with known probability distributions is presented. The method is applicable to tree networks, and has been verified by checking against a Monte-Carlo simulation.

The method is introduced and discussed in the context of low pressure gas distribution systems and water supply networks. Examples taken from the gas and water industries show the method to be computationally rapid and accurate.     

Estimating the frequency distributions of PM10 and PM2.5 by the statistics of wind speed at Sha-Lu,Taiwan

The frequency distribution of air pollutant concentration varies with the meteorological conditions and pollutant emission level. There exists a simple relation between the frequency distribution of wind speed and frequency distribution of air pollutant concentration. The concentration of air pollutant, C, at cumulative probability, p, is inversely proportional to the wind speeds, u, at probability of (100 - p) when the distributional types and shape factors of both data are the same. The relationship is shown as K=Cp u(100 - p), where K is constant. In this study, three theoretical distributions (log-normal, Weibull and type V Pearson distributions) are selected to fit the measured data of PM10, PM2.5 and wind speed. The frequency distributions of air pollutants can be estimated from the simple relationship of air pollutant concentration and wind speed. The results show that the log-normal distribution is the best one to represent the data of PM10, PM2.5 and wind speed. The K values of PM10 and PM2.5 are nearly constant from the 30-80th percentiles. It was also found that the distributions of PM10 and PM2.5 can be successfully estimated from the distribution of wind speed. The Kolmogorov-Smirnov (K-S) test shows that there is no significant discrepancy between the estimated and measured distribution of PM10 and PM2.5 at the 95% confidence level. Therefore, the distribution of air pollutants is easily estimated when the wind speed data are known.     

A Microcomputer-Based Geographic Information System for Hydrologic Simulation

Abstract: A microcomputer-based geographic information system (GIS) supporting hydrologic simulation is described. The hydrologically-oriented GIs can automatically provide parameters required for prediction of flood hydroghraphs and physically-based synthetic flood frequency curves for ungaged watersheds. The hydrologic model uses the Soil Conservation Service (SCS) runoff curve number (CN) method and a synthetic unit hydrograph to represent the physical transformation of rainfall into runoff The standard SCS method was modified to account for randomness of key parameters whose probability distributions can be derived from regional rainfall and runoff data. The GIs, and hydrologic theory and techniques, including the modified SCS method, derivation of a regional dimensionless unit hydrograph, and Monte Carlo simulation procedure, are described. Results of hydrologic analysis and simulation by the system implemented for an area of 11000 km² in the Alberta foothills are presented.     

基于Beta分布的改进的一次可靠度法研究

为了提高一次可靠度方法的计算精度和计算效率,文章利用Beta分布拟合了功能函数中的随机变量的概率分布,利用VBA语言给出了将Beta分布和对数正态分布转化为标准正态分布的方法,采用蒙特卡洛模拟法比较了采用不同概率分布的一次可靠度计算精度。算例分析表明:当功能函数中随机变量服从对数正态分布时,失效概率被高估;当随机变量服从正态分布时失效概率被低估。利用本文提出的Beta分布拟合随机变量的概率不仅大大提高了计算精度,而且在保证迭代收敛的前提下提高了计算效率。     

Estimation of complicated distributions using B-spline functions

The distributions of some of random variables are quite complicated and difficult to determine using ordinary statistical models. A method is presented in this paper which gives satisfactory estimation of the complicated distribution of a continuous random variable. There are two key steps in the method: one being that the probability density function (p.d.f.) of a random variable is approximated by a linear combination of B-splines and the other being that the best model is determined by entropy analysis. Extensive numerical experiments have made it clear that the proposed method is useful to determine the p.d.f. directly from a set of sample points without using any prior knowledge of the distribution form.     

准脆性材料的破坏概率与强度尺寸效应

基于最弱链模型和缺陷的Poisson分布假设,综合体积和材质因素,建立准脆性材料破坏概率和强度尺寸效应的统计模型和一般表达式。由此可知物体的破坏概率随体积单调增加,强度随体积单调减小,在通常有限的尺度范围内,当体积很大时,强度变化很小可视作常值;无缺陷材料则没有尺寸效应。通过分别采用Weibull分布和对数正态分布,得到材料破坏概率和强度尺寸效应的具体形式并进行对比分析,最后利用典型岩石强度试验结果验证其适用性。分析结果发现,这2种分布都能描述材料的尺寸效应,而且体积相同时Weibull分布给出的强度值较低。当试样体积与参考体积接近时,2种分布给出的强度结果差别很小,而对数正态分布在整个体积范围都能给出更接近工程实际的强度值。在准脆性材料强度尺寸效应的统计途径研究中,采用对数正态分布可作为Weibull尺寸效应理论的简单改进。     

Percentile estimation for water quality data

A survey of methods used by the Regional Water Authorities found that three techniques are commonly employed for percentile estimation: (1) assuming a parametric probability distribution; (2) using a non-parametric percentile estimation technique, and (3) statistically testing the fit of an assumed distribution.In this study, mathematical probability distribution fitting techniques were applied to 334 water quality datasets for a variety of rivers and effluents distributed throughout England and Wales. The results indicated that the “best” estimate of the 95 percentile from a dataset should be based upon probability distribution fitting. An untested assumption made about the underlying distribution of the data may produce an erroneous result in the subsequent water quality consent determination. The three parametric distributions selected (normal, lognormal and Pearson Type 3) could be fitted to half of the datasets studied. Results suggested that the Pearson Type 1 distribution may also be suitable for fitting to water quality data. The arbitrary use of non-parametric techniques may fail to make the most effective use of the information contained in the data.     

Finite element reliability and sensitivity analysis of structures using the multiplicative dimensional reduction method

Finite element reliability analysis (FERA) has been used to evaluate the reliability of structures. In FERA, approximate methods are commonly used to estimate the mean and variance of the structural response, while its probability distribution is primarily derived based on the Monte Carlo simulation (MCS) method. This paper advances FERA by combining it with the multiplicative dimensional reduction method (M-DRM). The proposed M-DRM allows fairly accurate estimation of the statistical moments, as well as the probability distribution of the structural response. The distribution of the response is obtained using fractional moments, which are calculated from the M-DRM, along with the maximum entropy principle. The variance of the response, based on global sensitivity measures, is obtained as a by-product of the analysis. The proposed approach is integrated with the OpenSees software and is illustrated through examples of nonlinear finite element analyses of reinforced concrete and steel frames. The paper shows that the proposed approach is an accurate and efficient alternative for FERA.     

'Rev Thomas Bayes: Get Ready to Wiggle' — Bayesian Modelling,Radiocarbon Wiggle-Matching,and the North Wing of Baguley Hall

Abstract

The principles of the Bayesian approach for modelling chronologies has been outlined in Bayliss (this volume). This paper applies these methods to a 'real world' problem of dating a phase of construction where the potential for treering dating is limited. The accuracy and precision that these new techniques can deliver is demonstrated. Bayesian modelling of radiocarbon dates and tree-ring dating independently estimate construction of the north wing in the third quarter of the fifteenth century.     

Developing indicators to detect changes in the seasonality,frequency and duration of medium and high river flows

Indicators specifically designed to detect changes in the seasonality, frequency and duration of medium and high river flows have been developed to monitor river flow regimes. Four primary methods to display and interpret the indicator results have been developed to meet the needs of different target audiences. The variability in annual indicator values is accommodated by comparing the probability distributions of the annual values rather than seeking to identify trends in annual values themselves. Changes in the distribution of indicator values are consistent with those anticipated to arise as a result of climate change. This finding is confirmed by calculating the indicators from example flow data for catchments across England and Wales, which has been perturbed to reflect climate change scenario model outputs.     

Bayesian analysis of the T-year events for flood data fitted by a three-parameter lognormal distribution

The three-parameter lognormal distribution is widely used in hydrology to fit maximum annual flood events. The method most commonly used to estimate the parameters and 7”-year flood event is the method of maximum likelihood. In this paper, Bayes estimates of the parameters and of the 7”-yearflood and their posterior variances of estimates are obtained by using Lindley's Bayesian approximation procedure. These estimates are then compared to estimates obtained by the method of maximum likelihood. Twelve flood series from all over Canada that are shown to be well-fitted by the three-parameter lognormal distribution are used as numerical examples. In all cases the posterior variances of the Bayes estimates of the 7”-year flood events are less than the corresponding variances of their maximum likelihood counterparts.