Extreme quantile estimation using order statistics with minimum cross-entropy principle

The paper presents a general approach to the estimation of the quantile function of a non-negative random variable using the principle of minimum cross-entropy (CrossEnt) subject to constraints specified in terms of expectations of order statistics estimated from observed data.Traditionally CrossEnt is used for estimating the probability density function under specified moment constraints. In such analyses, consideration of higher order moments is important for accurate modelling of the distribution tail. Since the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain, the use of CrossEnt quantile estimates in extreme value analysis is fairly limited.The present paper is an attempt to overcome this problem via 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 a PWM as the moment of quantile function, the paper derives an analytical form of quantile function using the CrossEnt principle. Monte Carlo simulations are performed to assess the accuracy of CrossEnt quantile estimates obtained from small samples.     

Using partial probability weighted moments and partial maximum entropy to estimate quantiles from censored samples

The maximum entropy principle constrained by probability weighted moments is an useful technique for unbiasedly and efficiently estimating the quantile function of a random variable from a sample of complete observations. However, censored or incomplete data are often encountered in engineering reliability and lifetime distribution analysis. This paper presents a new distribution free method for the estimation of the quantile function of a non-negative random variable using a censored sample of data, which is based on the principle of partial maximum entropy (MaxEnt) in which partial probability weighted moments (PPWMs) are used as constraints. Numerical results and practical examples presented in the paper confirm the accuracy and efficiency of the proposed partial MaxEnt quantile function estimation method for censored samples.     

基于Legendre正交多项式逼近法的结构可靠性分析

提出了结构可靠性分析的Legendre正交多项式逼近法。主要是基于数值逼近原理,以Legendre正交函数族做基,利用功能函数的高阶矩信息,通过计算功能函数概率密度函数的逼近表达式,然后根据工程结构可靠性的一般表达式来计算结构的失效概率,进行可靠性分析。通过数值检验,证明该方法可以很好地逼近各种经典理论分布曲线(正态分布、指数分布等6种经典分布)。文后给出了结构构件失效概率的实例计算,并和其它几种常用方法进行对比,进一步表明了Legendre正交多项式数值逼近法在结构可靠性分析理论上的正确性和实用性。     

Reliability Analysis of Zero‐Failure Data with Poor Information

For costly and dangerous experiments, growing attention has been paid to the problem of the reliability analysis of zero‐failure data, with many new findings in world countries, especially in China. The existing reliability theory relies on the known lifetime distribution, such as the Weibull distribution and the gamma distribution. Thus, it is ineffective if the lifetime probability distribution is unknown. For this end, this article proposes the grey bootstrap method in the information poor theory for the reliability analysis of zero‐failure data under the condition of a known or unknown probability distribution of lifetime. The grey bootstrap method is able to generate many simulated zero‐failure data with the help of few zero‐failure data and to estimate the lifetime probability distribution by means of an empirical failure probability function defined in this article. The experimental investigation presents that the grey bootstrap method is effective in the reliability analysis only with the few zero‐failure data and without any prior information of the lifetime probability distribution. Copyright © 2011 John Wiley & Sons, Ltd.     

First-passage failure of quasi linear systems subject to multi-time-delayed feedback control and wide-band random excitation

A procedure for studying the first-passage failure of quasi-linear systems subject to multi-time-delayed feedback control and wide-band random excitation is proposed. The stochastic averaging method for quasi-integrable Hamiltonian systems is first introduced. The backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are then established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed.     

Estimation of minimum cross-entropy quantile function using fractional probability weighted moments

The principle of minimum cross-entropy provides a systematic approach to derive the posterior distribution of a random variable given a prior and additional information in terms of its product moments. This approach can be extended to derive directly the quantile function by using probability weighted moments (PWMs) as constraints in the cross-entropy minimization approach, as shown in a previous study [Pandey MD. Extreme quantile estimation using order statistics with minimum cross-entropy principle. Probabilistic Engineering Mechanics 2001;16(1):31–42]. The objective of the present paper is to extend and improve the previous method by incorporating the use of the fractional probability weighted moments (FPWMs) in the place of conventional integer-order PWMs. A new and general estimation method is proposed in which the Monte Carlo simulations and optimization algorithms are combined to estimate FPWMs that would subsequently lead to the best-fit quantile function. The numerical examples presented in the paper show a substantial improvement in accuracy by the use of the proposed method over the conventional approach.     

Second-order statistics of integrated intensities and detected photons,the exact analysis I. Theory

Abstract

We express the exact probability density distribution function as the product of a gamma distribution and a series of associated Laguerre polynomials, with the expansion coefficients determined by moments of the integrated intensity. Orthogonal polynomials with respect to the exact probability distribution function are then expanded in similar fashion. These polynomials are then used to construct an expansion of the joint probability distribution function in the second-order photoelectron statistics. Since the polynomials are identical with the corresponding Laguerre polynomials when the exact probability distribution function is the gamma distribution, the new polynomials are generalized versions of the associated Laguerre polynomials. The joint photoelectron statistics may be studied with these new polynomials.     

Sample regeneration algorithm for structural failure probability function estimation

An efficient strategy to approximate the failure probability function in structural reliability problems is proposed. The failure probability function (FPF) is defined as the failure probability of the structure expressed as a function of the design parameters, which in this study are considered to be distribution parameters of random variables representing uncertain model quantities. The task of determining the FPF is commonly numerically demanding since repeated reliability analyses are required. The proposed strategy is based on the concept of augmented reliability analysis, which only requires a single run of a simulation-based reliability method. This paper introduces a new sample regeneration algorithm that allows to generate the required failure samples of design parameters without any additional evaluation of the structural response. In this way, efficiency is further improved while ensuring high accuracy in the estimation of the FPF. To illustrate the efficiency and effectiveness of the method, case studies involving a turbine disk and an aircraft inner flap are included in this study.     

A system reliability estimation method by the fourth moment saddle point approximation and copula functions

Despite many advances in the field of computational system reliability analysis, estimating the joint probability distribution of correlated non-normal state variables on the basis of incomplete statistical data brings great challenges for engineers. To avoid multidimensional integration, system reliability estimation usually requires the calculation of marginal failure probability and joint failure probability. The current article proposed an integrated approach for estimating system reliability on the basis of the high moment method, saddle point approximation, and copulas. First, the statistic moment estimation based on the stochastic perturbation theory is presented. Thereafter, by constructing CGF (concise cumulant generating function) for the state variable with its first four statistical moments, a fourth moment saddle point approximation method is established for the component reliability estimation. Second, the copula theory is briefly introduced and extensively utilized two-dimensional copulas are presented. The best fit copula for estimating the probability of system failure is selected according to the AIC (Akaike Information Criterion). Finally, the derived method is applied to three numerical examples for the sake of a comprehensive validation.     

The omega probability distribution and its applications in reliability theory

A new three‐parameter probability distribution called the omega probability distribution is introduced, and its connection with the Weibull distribution is discussed. We show that the asymptotic omega distribution is just the Weibull distribution and point out that the mathematical properties of the novel distribution allow us to model bathtub‐shaped hazard functions in two ways. On the one hand, we demonstrate that the curve of the omega hazard function with special parameter settings is bathtub shaped and so it can be utilized to describe a complete bathtub‐shaped hazard curve. On the other hand, the omega probability distribution can be applied in the same way as the Weibull probability distribution to model each phase of a bathtub‐shaped hazard function. Here, we also propose two approaches for practical statistical estimation of distribution parameters. From a practical perspective, there are two notable properties of the novel distribution, namely, its simplicity and flexibility. Also, both the cumulative distribution function and the hazard function are composed of power functions, which on the basis of the results from analyses of real failure data, can be applied quite effectively in modeling bathtub‐shaped hazard curves.     

Novel computational methods for high‐dimensional stochastic sensitivity analysis

This paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high‐dimensional complex systems subject to random input. The first method represents a novel integration of the polynomial dimensional decomposition (PDD) of a multivariate stochastic response function and score functions. Applied to the statistical moments, the method provides mean‐square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD‐saddlepoint approximation (SPA) or PDD‐SPA method, entailing SPA and score functions; and the PDD‐Monte Carlo simulation (MCS) or PDD‐MCS method, utilizing the embedded MCS of the PDD approximation and score functions. For all three methods developed, the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Numerical examples, including a 100‐dimensional mathematical problem, indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities, but also computationally efficient solutions. A practical example involving robust design optimization of a three‐hole bracket illustrates the usefulness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

On the existence of maximum entropy distributions with four and more assigned moments

It is known that the probability distribution satisfy the Maximum Entropy Principle (MEP) if the available data consist in four moments of probability density function. Two problems are typically associated with use of MEP: the definition of the range of acceptable values for the moments M_i; the evaluation of the coefficients a_j. Both problems have already been accurately resolved by analytical procedures when the first two moments of the distribution are known.

In this work, the analytical solution in the case of four known moments is provided and a criterion for confronting the general case (whatever the number of known moments) is expounded. The first four moments are expressed in nondimensional form through the expectation and the coefficients of variation, skewness and kurtosis. The range of their acceptable values is obtained from the analytical properties of the differential equations which govern the problem and from the Schwarz inequality.     

Universal properties of kernel functions for probabilistic sensitivity analysis

Development of probabilistic sensitivities is frequently considered an essential component of a probabilistic analysis and often critical towards understanding the physical mechanisms underlying failure and modifying the design to mitigate and manage risk. One useful sensitivity is the partial derivative of the probability-of-failure and/or the system response with respect to the parameters of the independent input random variables. Calculation of these partial derivatives has been established in terms of an expected value operation (sometimes called the score function or likelihood ratio method). The partial derivatives can be computed with typically insignificant additional computational cost given the failure samples and kernel functions — which are the partial derivatives of the log of the probability density function (PDF) with respect to the parameters of the distribution. The formulation is general such that any sampling method can be used for the computation such as Monte Carlo, importance sampling, Latin hypercube, etc. In this paper, useful universal properties of the kernel functions that must be satisfied for all two parameter independent distributions are derived. These properties are then used to develop distribution-free analytical expressions of the partial derivatives of the response moments (mean and standard deviation) with respect to the PDF parameters for linear and quadratic response functions. These universal properties can be used to facilitate development and verification of the required kernel functions and to develop an improved understanding of the model for design considerations.     

Probabilistic crack trajectory analysis by a dimension reduction method

This paper proposes a novel analysis method of stochastic crack trajectory based on a dimension reduction approach. The developed method allows efficiently estimating the statistical moments, probability density function and cumulative distribution function of the crack trajectory for cracked elastic structures considering the randomness of the loads, material properties and crack geometries. First, the traditional dimension reduction method is extended to calculate the first four moments of the crack trajectory, in which the responses are eigenvectors rather than scalars. Then the probability density function and cumulative distribution function of the crack trajectory can be obtained using the maximum entropy principle constrained by the calculated moments. Finally, the simulation of the crack propagation paths is realized by using the scaled boundary finite element method. The proposed method is well validated by four numerical examples performed on varied cracked structures. It is demonstrated that this method outperforms the Monte Carlo simulation in terms of computational efficiency, and in the meanwhile, it has an acceptable computational accuracy.     

Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations

Reliability–sensitivity, which is considered as an essential component in engineering design under uncertainty, is often of critical importance toward understanding the physical systems underlying failure and modifying the design to mitigate and manage risk. This paper presents a new computational tool for predicting reliability (failure probability) and reliability–sensitivity of mechanical or structural systems subject to random uncertainties in loads, material properties, and geometry. The dimension reduction method is applied to compute response moments and their sensitivities with respect to the distribution parameters (e.g., shape and scale parameters, mean, and standard deviation) of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate failure probability, probability density functions, and cumulative distribution functions. The rigorous analytic derivation of the parameter sensitivities of the failure probability with respect to the distribution parameters of basic random variables is derived. Results of six numerical examples involving hypothetical mathematical functions and solid mechanics problems indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the failure probability and reliability–sensitivity. Copyright © 2012 John Wiley & Sons, Ltd.     

含非闭合隶属函数模糊变量的结构失效概率分布研究

结合工程实际,提出了非闭合隶属函数的截断可能性分布模型,并对模糊强度和模糊应力进行截断处理,给出了结构模糊随机失效概率随截断参数的分布,并给出了结构模糊随机失效概率分布的数值计算方法。所提出的方法不仅可以考虑基本变量的随机模糊性,而且可以考虑安全和失效状态的随机模糊性。关于强度和应力两个基本变量的情况易于推广应用到多个变量的情况,以解决多变量体系中含有非闭合隶属函数模糊变量的安全分析问题。     

变保费率复合Poisson-Geometric过程风险模型的Gerber-Shiu折现惩罚函数

本文研究了当保费率随时间变化时的复合Poisson-Geometric过程的风险模型.通过无穷小方法,得到了该模型的Gerber-Shiu折现惩罚函数所满足的更新方程.在此基础上,推导出破产概率,破产前瞬时盈余,以及破产时刻赤字分布满足的更新方程.特别地,当个体索赔服从指数分布时,通过求解微分方程,得到了该模型的破产概率的显式表达式和所满足的不等式.最后通过数值模拟和算例分析,提出了保险公司的赔付政策和保费政策对自身风险的影响.     

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.     

A semi-analytical simulation method for reliability assessments of structural systems

A semi-analytical simulation method is proposed in this paper to assess system reliability of structures. Monte Carlo simulation with variance-reduction techniques, systematic and antithetic sampling, is employed to obtain the samples of the structural resistance in this method. Variance-reduction techniques make it possible to sufficiently simulate the structural resistance with less runs of structural analysis. When resistance samples and its moments determined, exponential polynomial method (EPM) is used to fit the probability density function of the structural resistance. EPM can provide the approximate distribution and statistical characteristic of the structural resistance and then the first-order second-moment method can be carried out to calculate the structural failure probability. Numerical examples are provided for a structural component and two ductile frames, which illustrate the method proposed facilitates the evaluation of system reliability in assessments of structural safety.     

非线性流滞阻尼器耗能结构随机地震响应和首超时间分析

对非线性流滞阻尼器耗能结构在Kanai-Tajimi谱地震激励下的随机响应及其随机失效时间和动力可靠性进行了系统研究。首先建立了结构的非线性运动方程;然后,基于随机平均法,将结构响应幅值近似为一维markov扩散过程,获得了扩散过程漂移系数和扩散系数的解析表达式;其次,利用扩散过程与FPK方程的对应关系,获得了幅值平稳概率密度函数和幅值任意阶矩的解析表达式;再次,利用幅值与结构位移和速度的相互转化关系,获得了结构位移与速度的平稳联合概率密度函数和位移、速度方差以及位移期望穿越率的解析表达式;最后,利用扩散过程的后向Kolmogrov方程,基于首超失效模型,建立了结构动力可靠性函数方程和结构随机失效时间统计矩方程,并利用一维扩散过程的边界分类性质,将统计矩方程的奇异定性边界条件转化为等价的定量边界条件,进而获得了失效时间任意阶统计矩的解析解,并利用此矩,对结构动力可靠性和失效时间概率分布函数进行了近似分析,给出了算例,从而建立了结构非线性随机地震响应及其随机失效时间和动力可靠性的分析方法。