Local estimation of failure probability function by weighted approach

In the reliability-based design of engineering systems, it is often required to evaluate the failure probability for different values of distribution parameters involved in the specification of design configuration. The failure probability as a function of the distribution parameters is referred as the ‘failure probability function (FPF)’ in this work. From first principles, this problem requires repeated reliability analyses to estimate the failure probability for different distribution parameter values, which is a computationally expensive task. A “weighted approach” is proposed in this work to locally evaluate the FPF efficiently by means of a single simulation. The basic idea is to rewrite the failure probability estimate for a given set of random samples in simulation as a function of the distribution parameters. It is shown that the FPF can be written as a weighted sum of sample values. The latter must be evaluated by system analysis (the most time-consuming task) but they do not depend on the distribution. Direct Monte Carlo simulation, importance sampling and Subset Simulation are incorporated under the proposed approach. Examples are given to illustrate their application.     

Developing region-specific fragility function for predicting probability of liquefaction induced ground failure

The liquefaction potential index (LPI) has been widely used to develop fragility function for predicting the liquefaction-induced ground failure. As the fragility function tends to vary from one region to another, it is best developed based on region-specific data. When the amount of region-specific data is limited, how to develop the region-specific fragility curve is a challenging problem. In this study, a Hierarchical Bayesian Model (HBM) is suggested for developing region-specific fragility functions based on LPI, which can systematically consider the effects of the amount and characteristics of the local data as well as the data from other regions. The suggested method is illustrated with an example. It is shown that the HBM outperforms the lumped parameter model (LPM) which does not consider the inter-region variability of the fragility curves. When the amount of region-specific data is large, the fragility function developed based on the HBM is very close to that developed based on the independent parameter model (IPM), which constructs a region-specific fragility function utilizing only the region-specific data. When the region-specific data is not enough, the HBM also outperforms the IPM through borrowing information from other regions.     

Local estimation of failure probability function and its confidence interval with maximum entropy principle

An approach is developed to locally estimate the failure probability of a system under various design values. Although it seems to require numerous reliability analysis runs to locally estimate the failure probability function, which is a function of the design variables, the approach only requires a single reliability analysis run. The approach can be regarded as an extension of that proposed by Au [Au SK. Reliability-based design sensitivity by efficient simulation. Computers and Structures 2005;83(14):1048–61], but it proposes a better framework in estimating the failure probability function. The key idea is to implement the maximum entropy principle in estimating the failure probability function. The resulting local failure probability function estimate is more robust; moreover, it is possible to find the confidence interval of the failure probability function as well as estimate the gradient of the logarithm of that function with respect to the design variables. The use of the new approach is demonstrated with several simulated examples. The results show that the new approach can effectively locally estimate the failure probability function and the confidence interval with one single Subset Simulation run. Moreover, the new approach is applicable when the dimension of the uncertainties is high and when the system is highly nonlinear. The approach should be valuable for reliability-based optimization and reliability sensitivity analysis.     

On system failure probability density function

The assessment of the system unreliability is usually accomplished through well-known tools such as block diagram, fault tree, Monte Carlo and others. These methods imply the knowledge of the failure probability density function of each component “k” (pdf p_k). For this reason, possibly, the system failure probability density function (p_sys) has never been explicitly derived.The present paper fills this gap achieving an enlightening formulation which explicitly gives p_sys as the sum of (positive) terms representing the complete set of transitions leading the system from an operating to a failed configuration, due to the failure of “a last” component. As a matter of fact, these are all the independent sequences leading the system to the failure.In our opinion, this formulation is important from both methodological and practical point of views. From the methodological one, a clear insight of the system-vs-components behaviors can be grasped and, in general, the explicit link between p_sys and p_k seems to be a notable result. From a practical point of view, p_sys allows a rigorous derivation of Monte Carlo algorithms and suggests a systematic tool for investigating the system failure sequences.     

Adaptive directional stratification for controlled estimation of the probability of a rare event

Within the structural reliability context, the aim of this paper is to present a new accelerated Monte-Carlo simulation method, named ADS, Adaptive Directional Stratification, and designed to overcome the following industrial constraints: robustness of the estimation of a low structural failure probability (less than 10⁻³), limited computational resources and complex (albeit often monotonic) physical model. This new stochastic technique is an original variant of adaptive accelerated simulation method, combining stratified sampling and directional simulation and including two steps in the adaptation stage (ADS-2). First, we theoretically study the properties of two possible failure probability estimators and get the asymptotic and non-asymptotic expressions of their variances. Then, we propose some improvements for our new method. To begin with, we focus on the root-finding algorithm required for the directional approach: we present a stop criterion for the dichotomic method and a strategy to reduce the required number of calls to the costly physical model under monotonic hypothesis. Lastly, to overcome the limit involved by the increase of the input dimension, we introduce the ADS-2⁺ method which has the same ground as the ADS-2 method, but additionally uses a statistical test to detect the most significant inputs and carries out the stratification only along them. To conclude, we test the ADS-2 and ADS-2⁺ methods on academic examples in order to compare them with the classical structural reliability methods and to make a numerical sensitivity analysis over some parameters. We also apply the methods to a flood model and a nuclear reactor pressurized vessel model, to practically demonstrate their interest on real industrial examples.     

Formulating informative, data-based priors for failure probability estimation in reliability analysis

Priors play an important role in the use of Bayesian methods in risk analysis, and using all available information to formulate an informative prior can lead to more accurate posterior inferences. This paper examines the practical implications of using five different methods for formulating an informative prior for a failure probability based on past data. These methods are the method of moments, maximum likelihood (ML) estimation, maximum entropy estimation, starting from a non-informative ‘pre-prior’, and fitting a prior based on confidence/credible interval matching. The priors resulting from the use of these different methods are compared qualitatively, and the posteriors are compared quantitatively based on a number of different scenarios of observed data used to update the priors. The results show that the amount of information assumed in the prior makes a critical difference in the accuracy of the posterior inferences. For situations in which the data used to formulate the informative prior is an accurate reflection of the data that is later observed, the ML approach yields the minimum variance posterior. However, the maximum entropy approach is more robust to differences between the data used to formulate the prior and the observed data because it maximizes the uncertainty in the prior subject to the constraints imposed by the past data.     

Error evaluations for the computation of failure probability in static structural reliability problems

The determination of mathematical reliability in static structures is still a motivating field of research. On one hand, the failure probability values are of greater importance in engineering activities; on the other hand, the determination of this probability remains a time consuming computational task when real problems are examined. To obtain a significant value for the failure probability while preserving a reasonable computation cost is therefore an objective to be considered. One of the necessary conditions to reach this target is to design a method to determine the computational error. Knowing the error will then give the capacity to limit the computation time, in particular by avoiding a too accurate probability evaluation. This paper presents a method allowing one to deal with these factors. The RGMR method, presented at the ICASP'7 meeting (July 1995, Paris) was designed to make such an error evaluation possible. Examples of numerical error computations with the RGMR method, particularly when low significant variables are eliminated, are given. These new developments are a step towards the goal mentioned above.     

On the estimation of failure probability having prescribed statistical moments of first passage time

A method of estimating the probability density function and cumulative distribution function when only the ordinary or central moments of the distribution are known is examined. The technique is used in conjunction with previous work which yields the ordinary moments of time to first passage failure to obtain accurate estimates of the failure probability for two representative oscillators. The results are then compared to those obtained by a nearly exact numerical scheme.     

A modified Weibull extension with bathtub-shaped failure rate function

Models with bathtub-shaped failure rate function are useful in reliability analysis, and particularly in reliability related decision making and cost analysis. The traditional Weibull distribution is, however, unable to model the complete lifetime of systems with a bathtub-shaped failure rate function. In this paper, a new model, which is useful for modeling this type of failure rate function, is presented. The model can also be seen as a generalization of the Weibull distribution. Parameter estimation methods are studied for this new distribution. Examples and results of comparison are shown to illustrate the applicability of this new model.     

Probability of infancy problems for space launch vehicles

This paper addresses the treatment of ‘infancy problems’ in the reliability analysis of space launch systems. To that effect, we analyze the probability of failure of launch vehicles in their first five launches. We present methods and results based on a combination of Bayesian probability and frequentist statistics designed to estimate the system's reliability before the realization of a large number of launches. We show that while both approaches are beneficial, the Bayesian method is particularly useful when the experience base is small (i.e. for a new rocket). We define reliability as the probability of success based on a binary failure/no failure event. We conclude that the mean failure rates appear to be higher in the first and second flights (≈1/3 and 1/4, respectively) than in subsequent ones (third, fourth and fifth), and Bayesian methods do suggest that there is indeed some difference in launch risk over the first five launches. Yet, based on a classical frequentist analysis, we find that for these first few flights, the differences in the mean failure rates over successive launches or over successive generations of vehicles, are not statistically significant (i.e. do not meet a 95% confidence level). This is true because the frequentist analysis is based on a fixed confidence level (here: 95%), whereas the Bayesian one allows more flexibility in the conclusions based on a full probability density distribution of the failure rate and therefore, permits better interpretation of the information contained in a small sample. The approach also gives more insight into the considerable uncertainty in failure rate estimates based on small sample sizes.     

New super-kurtic probability density function for use in computer simulation

The philosophy of computer simulation and its application to hydrological processes is described in this paper. The structure of natural hydrologic time processes is indicated and the techniques to filter out white noise is explained. The limitations of the well-known probability density functions (PDF) such as the Gaussian, Pearson’s and Johnson’s etc. in hydrologic applications are set forth. A new super-kurtic PDF developed by the author specially for hydrological processes is introduced and a numerical example is given.     

Exponential age sensitive method for failure rank estimation

This paper presents a new method for estimating the modified failure ranks used for probability plotting and least squares estimation from randomly censored failure samples. The proposed method uses a Bayesian smoothed piecewise hazard rate approximation to derive modified failure ranks that are sensitive to the age at censoring. Using a simulation study, the new estimator is shown to be more robust than the known failure ranks estimators, particularly for heavy censoring and small failure samples. Practical use includes field failure data analysis, for which the survival and failure probability predictions are more realistic, i.e. less pessimistic, when the new estimator is used compared with predictions obtained from using the traditional Johnson method for failure rank estimators. Copyright © 2000 John Wiley & Sons, Ltd.     

基于贝叶斯理论及MCMC-MH算法推演地基土材料阻尼比的概率分布模型

浅层土壤的材料阻尼比参数分布可以通过多通道表面波(MASW)分析法提取的表面波衰减曲线来反演识别,但是衰减曲线对于较大深度和小空间尺度土壤性质的变化不敏感,反演的土壤材料阻尼比分布是非唯一的和不确定的.基于该研究建立了土体材料阻尼比随深度变化的先验概率分布模型,利用Nataf变换和Karhunen-Loeve将其分解为...     

Radial basis function neural networks solution for stationary probability density function of nonlinear stochastic systems

A radial basis function neural networks (RBF-NN) solution of the reduced Fokker–Planck-Kolmogorov (FPK) equation is proposed in this paper. The activation functions consist of normalized Gaussian probability density functions (PDFs). The use of normalized Gaussian PDFs leads to a simple constraint on the coefficients for normalization of the RBF-NN solution, which as a constraint is imposed with the help of the method of Lagrange multiplier. The relationship between the proposed RBF-NN PDF solution and the generalized cell mapping with short-time Gaussian approximation is discussed, which provides a justification for Gaussian PDFs with varying means and variances in the state space. The optimal number of neurons or activation functions, which leads to the smallest error, is investigated. Four examples are presented to show the effectiveness of the proposed solution method. The results indicate that the proposed solution method is a very efficient and accurate way to compute the stationary PDF of nonlinear stochastic systems. It is also found that the distribution of the optimal coefficients as a function of the mean of Gaussian activation functions is similar to the steady-state PDF solution. Finally, we should point out that an important advantage of the RBF-NN method over methods such as finite element and finite difference is its ability to obtain solutions of the FPK equation for multi-degree-of-freedom stochastic systems.     

高斯型点扩展函数估计的最近邻算法

本文针对计算光学切片中的最近邻算法提出了一种改进算法.通过小波变换计算出高斯点扩展函数的方差值,再根据相邻图像成像及高斯函数特性,得出所需的高斯型层间点扩展函数.同时,文章还给出了两种高斯型层间点扩展函数方差的获得方式及获得过程,对最近邻算法中的加权因子的取值范围做出了讨论,对传统的最近邻算法做出了改进.实验表明,本算法能够更有效地复原符合最近邻要求的切片图像.在点扩展函数未知的情况下,复原效果要优于传统方法.     

A new artificial neural network-based response surface method for structural reliability analysis

This paper presents a new artificial neural network-(ANN)based response surface method in conjunction with the uniform design method for predicting failure probability of structures. The method involves the selection of training datasets for establishing an ANN model by the uniform design method, approximation of the limit state function by the trained ANN model and estimation of the failure probability using first-order reliability method (FORM). In the proposed method, the use of the uniform design method can improve the quality of the selected training datasets, leading to a better performance of the ANN model. As a result, the ANN dramatically reduces the number of required trained datasets, and shows a good ability to approximate the limit state function and then provides a less rigorous formulation in the context of FORM. Results of three numerical examples involving both structural and non-structural problems indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure. Compared with the conventional ANN-based response surface method, the proposed method is much more economical to achieve reasonable accuracy when dealing with problems where closed-form failure functions are not available or the estimated failure probability is extremely small. Finally, several important parameters in the proposed method are discussed.     

Reliability analysis for wind turbines with incomplete failure data collected from after the date of initial installation

Reliability has an impact on wind energy project costs and benefits. Both life test data and field failure data can be used for reliability analysis. In wind energy industry, wind farm operators have greater interest in recording wind turbine operating data. However, field failure data may be tainted or incomplete, and therefore it needs a more general mathematical model and algorithms to solve the model. The aim of this paper is to provide a solution to this problem. A three-parameter Weibull failure rate function is discussed for wind turbines and the parameters are estimated by maximum likelihood and least squares. Two populations of German and Danish wind turbines are analyzed. The traditional Weibull failure rate function is also employed for comparison. Analysis shows that the three-parameter Weibull function can obtain more accuracy on reliability growth of wind turbines. This work will be helpful in the understanding of the reliability growth of wind energy systems as wind energy technologies evolving. The proposed three-parameter Weibull function is also applicable to the life test of the components that have been used for a period of time, not only in wind energy but also in other industries.     

An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability

This paper presents a simulation technique for reliability analysis of linear dynamical systems. It is based on simple additive rules of probability (in contrast to other probabilistic approaches such as importance sampling). It is shown that the proposed appoach is identical to a newly developed approach, Importance Sampling using Elementary Events (ISEE) [Au SK, Beck JL. First excursion probabilities for linear sytems by very efficient importance sampling. Probabl Eng Mech 2001;16(3):193–208]. A simple formula for the coefficient of variation of the estimator of the failure probability using the samples is also given. A 10-story building model with nonstationary excitation is utilized to demonstrate the accuracy and efficiency of the proposed method.     

A convenient approach for estimating time-dependent structural reliability in the load space

A procedure, formulated in the space of the load processes, is described for estimating the reliability of structures subject to multi-parameter time-varying loading. For most realistic reliability problems the load process space is of low order. As a result, the required multidimensional integration is significantly simplified. The proposed approach also has well defined steps. As a result, there is increased transparency and reduced problems of integration instability and non-convergence. Both loads and resistances are described in terms of random variable parameters and time dependent structural resistances can be considered. Numerical examples are given to illustrate the proposed method. Example applications are given for a fixed base rigid-plastic portal frame subjected to time dependent loads and resistances. Linear and non-linear limit state equations and Normal and non-Normal distribution of the random variables are considered and compared, in some cases, to the results evaluated using Monte Carlo simulation.     

分析基于失效概率的矩独立重要性测度的单层密度权重法

为了有效地分析结构系统输入变量不确定性对结构系统失效概率的影响程度,研究了对基于失效概率的矩独立重要性测度的计算,并基于单层Monte Carlo计算法和密度权重计算法,建立了准确、高效求解失效概率矩独立重要性测度的单层密度权重法。工程算例表明,与已有的计算方法相比,该方法仅需较少的模型计算量就能得到足够准确的结果,极大地提高了计算效率,且具有很好的工程适用性。