An evaluation of the self-determined probability-weighted moment method for estimating extreme wind speeds

For the estimation of probability distribution parameters, the method of self-determined probability-weighted moments (SD-PWM) has previously been introduced as a refinement on the original method of probability-weighted moments (PWM). Tables have been created summarizing the solution of the relevant equations for certain probability distributions, but application of these is awkward. In addition, certain associated algorithms are difficult to interpret and contain formulations that do not appear to properly enforce the definitions of self-determined probability-weighted moments. Therefore, new algorithms have been developed to both clarify and simplify the determination of SD-PWM parameter estimates. As an application of the SD-PWM algorithms, the estimation of extreme wind speeds is considered using the Gumbel and generalized extreme value (GEV) distributions. The estimation results are compared to similar results obtained via PWM, the method of moments and the maximum likelihood method. The analyses suggest SD-PWM may be a reasonable tool for analyzing the ability of a particular distribution to describe a sample. Relative to the method of moments and PWM estimates, the SD-PWM estimates compare well based on fits of the cumulative distributions. While the SD-PWM estimates exhibit increased variability relative to the method of moment (MOM) estimates, SD-PWM wind speed estimates are generally conservative relative to the MOM estimates.     

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.     

竖井地基固结概率分析的点估计法

将竖井地基固结度表达式直接作为功能函数,采用点估计法计算固结度的均值、标准差等各阶统计矩,由各阶统计矩可方便计算出竖井地基达到某一固结度的概率。与传统概率分析方法相比,如一次二阶矩法,点估计方法无须事先引入控制指标来建立极限状态方程,计算过程更为简便。实例分析表明,当估计点较多时(如5点),点估计方法具有较高的计算精度,与Monte Carlo法相当,是一种简便而不失精度的概率分析方法,适合工程实际应用。     

Discussion on: moment methods for structural reliability

A few comments on moment method for reliability analysis advocated by Zhao and Ono are presented in the paper. It is demonstrated using a number of numerical examples that the first four moments of a distribution, particularly when they are crudely estimated, do not always allow good estimates of tail probabilities, and that some different equivalent forms of performance function that describe the same structural problem may give totally incorrect values for the reliability when the Zhao and Ono methodology is applied to them. Furthermore, the computation method for the first four moments proposed by the same authors may be incorrect so that the reliability result is unreasonable accordingly.     

Modal identification using spectral moments

The use of spectral moments has been proposed for estimating natural frequencies of vibration and equivalent viscous modal damping ratios from ambient response measurements of dynamically excited structures. The moments tend to be relatively more stable than spectral ordinates and may be derived from raw, unsmoothed periodograms. Central to the analysis are the assumptions of white noise excitation, small damping, and uncoupled modes. Partial spectral moments are recommended for multi-degree-of-freedom dynamic systems, deviations from white noise excitation, and to ignore noisy regions of the spectrum at the frequency extremes.This paper presents an iterative algorithm for implementing the concept which eliminates severe bias errors that are identified in damping estimates obtained with the originally proposed algorithm based on partial spectral moments. An error analysis of the new algorithm, which does not make the small damping assumption, is conducted to evaluate the sensitivity of the parameter estimates to (i) presence of additive noise in the response measurements, and (ii) the non-whiteness of the excitation spectral density function. The results are compared with those from a similar analysis for the well-known maximum response natural frequency and half-power bandwidth damping estimators. Information is provided to help in selecting appropriate integration intervals for computing partial spectral moments.     

Assessment of existing steel structures. A guideline for estimation of the remaining fatigue life

A new methodology, called moment matching, to efficiently estimate repair costs of a building due to future earthquake excitation is presented. As well as excitation uncertainties, other uncertainties considered include those in the structural model and those in the capacity to resist damage and the unit repair costs of structural and non-structural components. Given the first few moments of the basic uncertain variables, moment matching uses specially selected point estimates to propagate the uncertainties in order to more accurately estimate the first few moments of the repair costs. Two buildings are chosen as illustrative examples to demonstrate the use of moment matching: one hypothetical three-degree-of-freedom shear building, and a real seven-storey hotel building. It is shown that the moment matching technique is much more accurate than the First-Order Second-Moment approach when propagating the first two moments, whilst the computational cost is of the same order. The repair cost moments estimated by the moment matching technique are also compared to those obtained by the more computationally demanding Monte Carlo simulation, and it is concluded that as long as the order of the moment matching is sufficient, the comparison is satisfactory. Last, but not least, a procedure for sensitivity analysis is discussed and it is concluded that the most important uncertainties for the real building example are those that correspond to spectral acceleration, component capacity, ground motion details and unit repair costs.     

First-order reliability analysis employing translation systems

A reliability analysis is described in which sample moments are used to transform the design parameters to a space in which they have a multivariate normal distribution. The technique can be rendered quite simple by the use of tabulated data. The analysis then proceeds to generate failure risk estimates based on regions of estimated probability content or on statistical tolerance regions in the normal parameter space. A particularly useful Bayesian analysis based on the multivariate normal distribution is also described.     

随机加权法辅助小样本岩土参数分布拟合

提出小样本条件下基于随机加权法的多项式拟合方法,对岩土参数总体分布密度函数进行推断。应用随机加权法重采样技术对小样本数据信息进行提取,估计总体各阶原点矩。采用勒让德正交多项式拟合推断分布密度函数并采用精度较高的K-S检验法进行检验,确认本方法具有较高的精度。本方法直接根据样本数据进行推断,大大降低了对已有数据及经典分布概型的依赖性,具有明确的数学物理意义,能够满足岩土工程可靠性分析的需要。     

Extreme value theory-based structural health prognosis method using reduced sensor data

This paper presents an extreme value theory (EVT)-based structural health prognosis method that can be used for estimating the quantile values of remaining useful life (RUL) of monitored structure with reduced sensor data. Massive sensor data generated from online structural health monitoring system can be utilised to provide more refined prognosis results such as statistical distribution of RULs. By using the moment estimator from EVT, only a small portion of the full sensor data-set is actually used for estimating the quantile values of the RUL. This can considerably cut the computing time required for structural health prognosis. As a requirement for implementing the EVT-based prognosis, monotonicity relation between damage index (either measured or derived from sensor data) and RUL values has to be satisfied though. Common prognosis problems in civil engineering include fatigue cracking in steel structures and steel rebar corrosion in reinforced concrete structures. To illustrate the EVT-based prognosis, the monotonicity condition is shown for the selected degradation models in two case studies involving fatigue life estimation and pitting corrosion life of steel reinforcing rebar, respectively. The results show that EVT-based structural health prognosis method is computationally efficient without loss of much accuracy.     

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.     

Estimation of failure probabilities for intersections of non-linear limit states

This paper is concerned with estimating the failure probability associated with the intersection of two or more non-linear limit state functions. Previous work has relied on the use of general non-linear minimization routines to establish the intersection point of the limit states and then to use tangents to bound the region for estimating the failure probability. The present paper outlines two approaches based on successive approximation both to find the intersection point and to estimate the intersection probability. It is shown that the results for selected examples converge to the probabilities estimated using Monte Carlo analysis. Comments about extending the results to using second order or asymptotic probability estimates close the paper.     

Minimum cross-entropy method for extreme value estimation using peaks-over-threshold data

Extreme quantile estimation in the peaks-over-threshold (POT) method is based on the Pareto distribution model for peaks of a time series exceeding a high threshold. However, a major practical difficulty associated with this approach is the large and erratic variability of quantile estimates with respect to the threshold value, which is not known a priori. Recognizing the limited applicability of the Pareto model to a narrow and unidentifiable range of peak data, the paper presents a more general non-parametric probabilistic model to improve the statistical accuracy and of POT estimates. The proposed approach relies on a quantitative notion of uncertainty and the minimum cross-entropy principle (Cross-Ent), commonly used in information theory and signal analysis. The model combine a suitable prior distribution with sample estimates of probability-weighted-moments, which exhibit much smaller sampling uncertainty than estimates of ordinary moments. The performance of the Cross-Ent approach is compared with a widely used Pareto model of peak data and an exponential model of transformed data. Examples based on simulated data illustrate that Cross-Ent estimates of 50 and 1000-year quantiles are almost unbiased and insensitive to the threshold value. The analysis of US wind speed data also reveals a remarkably stable trend of Cross-Ent estimates with very limited threshold sensitivity, which is in clear contrast with rapidly fluctuating estimates obtained from the other two methods.     

建筑结构荷载随机参数的概率加权矩法计算

用概率加权矩理论来计算建筑结构荷载随机模型的参数，推导了极值Ⅰ型分布与样本概率加权矩的关系，用概率加权矩法计算其分布参数，并与其他方法的结果进行了比较，说明了概率加权矩法的正确性和实用性。     

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。     

Bayesian estimation of complicated distributions

In a previous paper (Zong Z, Lam KY. Estimation of complicated disributions using B-spline functions. Structural safety 1998; 20(4): 323–32), we used a linear combination of B-spline functions to approximate complicated distributions. The method works well for large samples. In this paper, we extend the method to small samples. We still use a linear combination of B-spline functions to approximate a complicated probability density function (p.d.f). Strongly influenced by statistical fluctuations, the combination coefficients (unknown parameters) estimated from a small sample are highly irregular. Useful information is, however, still contained in these irregularities, and likelihood function is used to pool the information. We then introduce smoothness restriction, based on which the so-called smooth prior distribution is constructed. By combining the sample information (likelihood function) and the smoothness information (smooth prior distribution) in the Bayes' theorem, the influence of statistical fluctuations is effectively removed, and greatly improved estimation, which is close to the true distribution, can be obtained by maximizing the posterior probability. Moreover, an entropy analysis is employed to find the most suitable prior distribution in an “objective” way. Numerical experiments have shown that the proposed method is useful to identify an appropriate p.d.f. for a continuous random variable directly from a sample without using any prior knowledge of the distribution form. Especially, the method applies to large or small samples.     

Probability distributions of extreme pressure coefficients

Several thousand extreme pressure coefficients from repeated time-history samples, from a wall tap and a roof tap on a model of the Texas Tech University Building in a simulated atmospheric boundary layer, were used to better determine the appropriate probability distributions for the data. Both the Type I (Gumbel) Extreme Value Distribution, and the Generalized Extreme Value (GEV) distribution, with variable shape factor, k, were used to fit the data; in the latter case the method of probability weighted moments was used. The GEV with small positive shape factor was found to fit the data well. The distributions obtained indicated upper limits of peak pressures about 35% higher than the highest values actually measured.     

The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach

The paper evaluates the effectiveness of the method of L-moments for estimating parameters of the Pareto distribution model of peaks over a sufficiently high threshold, and compares its performance against a widely used method of de Haan (de Haan L. Extreme value statistics. In: Galambos J, Lechner J, Simin E, editor. Extreme value theory and applications, vol. 1. 1994. p. 93–122). Monte Carlo simulations and actual wind speed data collected at various stations in the United States have been utilized in this study. In the de Haan method, the first two moments of peaks of log-transformed data are used for the parameter estimation, whereas the L-moment method utilizes linear combinations of expectations of order statistics of peaks in the original data. Despite the procedural differences, the paper shows that the de Haan and two L-moments based estimates of the tail shape parameter are in fairly close agreement in simulated data as well as in the US wind speed data. Furthermore, higher order estimates of the shape parameter are obtained using the L-skewness of peaks data. Such estimates appear to provide a more stable upper bound, which can be useful in identifying meaningful values of design quantiles.     

Shear and bending flexibility in closed-form moment solutions for continuous beams and bridge structures

Shear deformations can affect final member-end-moments for statically indeterminate continuous beams and frame structures, though for typical civil engineering structures their effect is small and moments can be based on flexural deformations only. When a member is deep relative to span length, however, shear deformations should be considered in the analysis. This can be included in the stiffness method and in a modified form of moment distribution where the carry-over factor is less than one-half due to the added flexibility from shear. In a prior paper the first author presented a new approach for solving statically indeterminate beams and bridge frames, with final end moments given in closed-form expressions. The advantages of this new approach are that no simultaneous equations are required as in the stiffness method, moments are not distributed back and forth as in moment distribution, and manual calculations may be used which give exact results for as many spans as desired. While only flexural deformations were considered in the original paper, this paper presents a closed-form approach that has been modified to include shear deformations. Final expressions are given for continuous beams and bridge frames, providing exact member-end-moments that match results from the stiffness method when shear deformations are included in the analysis.     

基于支持向量机的边坡可靠性分析

将支持向量机与一阶二次矩方法结合,提出了边坡可靠性分析的支持向量机方法。利用极限平衡分析构造学习样本,通过支持向量机学习,建立安全系数与随机变量之间映射关系的支持向量机表达,进而实现边坡极限状态函数及其偏导数的显式表达,从而计算边坡的可靠性指标。该方法避免了传统可靠性分析的缺点。利用一个算例进行了分析,结果表明:该方法计算效率高,结果可靠,对含有大量随机变量的复杂岩土工程可靠性分析具有很大的潜力,具有广泛的应用前景和工程价值。     

变刚度框架-剪力墙结构的力法解及其简化

以剪力墙弯矩为未知数 ,以框架楼层的剪切刚度为参变数 ,建立了一组基本方程式。此方程组适用于楼层刚度变化不规则的框架 剪力墙结构分析。基于基本方程组的特性 ,还提出了分段计算的方法 ,以便很快求得任一区段内的楼层内力和相对位移。