Saddlepoint Approximation Method in Reliability Analysis:A Review

The escalating need for reliability analysis(RA)and reliability-based design optimization(RBDO)within engineering challenges has prompted the advancement of saddlepoint approximationmethods(SAM)tailored for such problems.This article offers a detailed overview of the general SAM and summarizes the method characteristics first.Subsequently,recent enhancements in the SAM theoretical framework are assessed.Notably,the mean value first-order saddlepoint approximation(MVFOSA)bears resemblance to the conceptual framework of the mean value second-order saddlepoint approximation(MVSOSA);the latter serves as an auxiliary approach to the former.Their distinction is rooted in the varying expansion orders of the performance function as implemented through the Taylor method.Both the saddlepoint approximation and third-moment(SATM)and saddlepoint approximation and fourth-moment(SAFM)strategies model the cumulant generating function(CGF)by leveraging the initial random moments of the function.Although their optimal application domains diverge,each method consistently ensures superior relative precision,enhanced efficiency,and sustained stability.Every method elucidated is exemplified through pertinent RA or RBDO scenarios.By juxtaposing them against alternative strategies,the efficacy of these methods becomes evident.The outcomes proffered are subsequently employed as a foundation for contemplating prospective theoretical and practical research endeavors concerning SAMs.The main purpose and value of this article is to review the SAM and reliability-related issues,which can provide some reference and inspiration for future research scholars in this field.

RBMDO Using Gaussian Mixture Model-Based Second-Order Mean-Value Saddlepoint Approximation

Actual engineering systems will be inevitably affected by uncertain factors.Thus,the Reliability-Based Multidisciplinary Design Optimization(RBMDO)has become a hotspot for recent research and application in complex engineering system design.The Second-Order/First-Order Mean-Value Saddlepoint Approximate(SOMVSA/-FOMVSA)are two popular reliability analysis strategies that are widely used in RBMDO.However,the SOMVSA method can only be used efficiently when the distribution of input variables is Gaussian distribution,which significantly limits its application.In this study,the Gaussian Mixture Model-based Second-Order Mean-Value Saddlepoint Approximation(GMM-SOMVSA)is introduced to tackle above problem.It is integrated with the Collaborative Optimization(CO)method to solve RBMDO problems.Furthermore,the formula and procedure of RBMDO using GMM-SOMVSA-Based CO(GMM-SOMVSA-CO)are proposed.Finally,an engineering example is given to show the application of the GMM-SOMVSA-CO method.

Application of saddlepoint approximation in reliability analysis of dynamic systems

The application of the saddlepoint approximation to reliability analysis of dynamic systems is investigated. The failure event in reliability problems is formulated as the exceedance of a single performance variable over a prescribed threshold level. The saddlepoint approximation technique provides a choice to estimate the cumulative distribution function （CDF） of the performance variable. The failure probability is obtained as the value of the complement CDF at a specified threshold. The method requires computing the saddlepoint from a simple algebraic equation that depends on the cumulant generating function （CGF） of the performance variable. A method for calculating the saddlepoint using random samples of the performance variable is presented. The applicable region of the saddlepoint approximation is discussed in detail. A 10-story shear building model with white noise excitation illustrates the accuracy and efficiency of the proposed methodology.

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization(RBMDO)of engineering structures,the saddlepoint approximation(SA)method can be utilized to enhance the accuracy and efficiency of reliability evaluation.However,the random variables involved in SA should be easy to handle.Additionally,the corresponding saddlepoint equation should not be complicated.Both of them limit the application of SA for engineering problems.The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments.However,the traditional moment matching method is not very accurate generally.In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization,a fourth-moment saddlepoint approximation(FMSA)method is introduced into RBMDO.In FMSA,the approximate cumulative generating functions are constructed based on the first four moments of the limit state function.The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function.Furthermore,the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment,which is based on the performance measure approach strategy.Two engineering examples are introduced to verify the effectiveness of proposed method.

The Multivariate Saddlepoint Approximation to the Distribution of Estimators： A General Approach

We develop the theory of multivariate saddlepoint approximations. Our treatment differs from the one in Barndorff-Nielsen and Cox （1979, equation （4.7）） in two aspects： 1） our results are satisfied for random vectors that are not necessarily sums of independent and identically distributed random vectors, and 2） we consider that the sample is taken from any distribution, not necessarily a member of the exponential family of densities. We also show the relationship with the corresponding multivariate Edgeworth approximations whose general treatment was developed by Durbin in 1980, emphasizing that the basic assumptions that support the validity of both approaches are essentially similar.

基于自适应窗和形状自适应小波变换的SAR图像相干斑抑制

结合SAR图像空域的先验知识和小波域系数的特性,提出了一种新的SAR图像相干斑抑制算法.使用最近提出的局部多项式近似-置信区间交叉(local polynomial approximation-intersection of confidence intervals(LPA-ICI))构造自适应窗,寻找到与SAR图像中每个像素点相对应的同质区域,在每个同质区域内利用本文给出的快速形状自适应小波变换进行硬阈值收缩抑斑,最后根据本文提出的稀疏加权方法融合多个估计样本获得最终抑斑图像.实验结果表明本文提出的算法有着很好的抑斑性能,尤其是在去除重构图像中的"振铃"效应以及有效保留原始SAR图像中的点目标方面性能更突出.

基于改进DPC近似的多子阵SAS距离多普勒算法

传统修正的偏置相位中心近似方法往往忽略距离历程近似误差的方位空变性,使得基于该方法的成像算法重建的图像出现近距离散焦和方位向偏移现象。为解决该问题,该文对修正的偏置相位中心近似方法进行改进,将双根号形式距离历程表示为类收发合置项、距离空变项和方位空变项之和的形式,有效提高了距离历程的近似精度;在利用驻定相位原理求取方位向驻定相位点的过程中,通过充分考虑距离历程方位空变性的影响,推导了更为精确的方位向驻定相位点,有效提高了点目标响应二维谱的准确性。在此基础上,提出一种适用于高分辨多子阵合成孔径声呐成像的距离多普勒算法,仿真试验和实测数据成像结果验证了该算法的有效性。

基于BP神经网络的非线性函数逼近及SAS实现

BP神经网络作为目前应用最广泛的神经网络模型之一,其主要的特点在于非线性函数逼近的能力。提出基于BP神经网络的非线性函数逼近方法,开创性地使用功能强大的统计分析软件SAS编写BP算法,实现非线性函数的逼近,并对训练过程中的权值变化以及训练前后的函数曲线逼近效果进行对比分析。

污染型类高斯噪声中信号的Robust-SA检测

采用了Ｒｏｂｂｉｎｓ－Ｍｏｒｒｏ随机逼近算法，得到了Ｒｏｂｕｓｔ－ＳＡ检测器。它的主要特点是具有Ｍ－检测器的优点，且便于实时处理和计算机编程。理论分析与仿真例子都表明：此检测器具有良好的顽健性，能很好的检测信号。

Saddlepoint approximation based structural reliability analysis with non-normal random variables

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

Saddlepoint approximation based line sampling method for uncertainty propagation in fuzzy and random reliability analysis

For structural system with random basic variables as well as fuzzy basic variables,uncertain propagation from two kinds of basic variables to the response of the structure is investigated.A novel algorithm for obtaining membership function of fuzzy reliability is presented with saddlepoint approximation(SA)based line sampling method.In the presented method,the value domain of the fuzzy basic variables under the given membership level is firstly obtained according to their membership functions.In the value domain of the fuzzy basic variables corresponding to the given membership level,bounds of reliability of the structure response satisfying safety requirement are obtained by employing the SA based line sampling method in the reduced space of the random variables.In this way the uncertainty of the basic variables is propagated to the safety measurement of the structure,and the fuzzy membership function of the reliability is obtained.Compared to the direct Monte Carlo method for propagating the uncertainties of the fuzzy and random basic variables,the presented method can considerably improve computational efficiency with acceptable precision.The presented method has wider applicability compared to the transformation method,because it doesn't limit the distribution of the variable and the explicit expression of performance function, and no approximation is made for the performance function during the computing process.Additionally,the presented method can easily treat the performance function with cross items of the fuzzy variable and the random variable,which isn't suitably approximated by the existing transformation methods.Several examples are provided to illustrate the advantages of the presented method.

Saddlepoint approximations for studentized compound Poisson sums with no moment conditions in audit sampling

Saddlepoint approximations for the studentized compound Poisson sums with no moment conditions in audit sampling are derived. This result not only provides a very accurate approximation for studentized compound Poisson sums, but also can be applied much more widely in statistical inference of the error amount in an audit population of accounts to check the validity of financial statements of a firm. Some numerical illustrations and comparison with the normal approximation method are presented.

Detecting Change-Point via Saddlepoint Approximations

It’s well-known that change-point problem is an important part of model statistical analysis. Most of the existing methods are not robust to criteria of the evaluation of change-point problem.In this article, we consider "mean-shift" problem in change-point studies. A quantile test of single quantile is proposed based on saddlepoint approximation method. In order to utilize the information at different quantile of the sequence, we further construct a "composite quantile test" to calculate the probability of every location of the sequence to be a change-point. The location of change-point can be pinpointed rather than estimated within a interval. The proposed tests make no assumptions about the functional forms of the sequence distribution and work sensitively on both large and small size samples,the case of change-point in the tails, and multiple change-points situation. The good performances of the tests are confirmed by simulations and real data analysis. The saddlepoint approximation based distribution of the test statistic that is developed in the paper is of independent interest and appealing.This finding may be of independent interest to the readers in this research area.

SADDLEPOINT APPROXIMATIONS TO THETWO-SAMPLE PERMUTATION TESTS

A saddlepoint approximation for a two-sample permutation test was obtained by Robinson[7].Although the approximation is very accurate, the formula is very complicated and difficult toapply. In this papert we shall revisit the same problem from a different angle. We shall first turnthe problem into a conditional probability and then apply a Lugannani-Rice type formula to it,which was developed by Skovagard[8] for the mean of i.i.d. samples and by Jing and Robinson[5]for smooth function of vector means. Both the Lugannani-Rice type formula and Robinson'sformula achieve the same relative error of order O(n-3/2), but the former is very compact andmuch easier to use in practice. Some numerical results will be presented to compare the twoformulas.

多子阵合成孔径声纳距离-多普勒成像算法

基于相位中心近似(PCA)的传统多子阵合成孔径声纳(SAS)成像方法忽略了近似误差的方位空变性,使得分布式目标的聚焦结果发生畸变。为解决这个问题,该文从收发阵元空间分置采样与相位中心近似采样的几何模型出发推导了一种考虑近似误差方位空变性的双程斜距历程,并将多子阵合成孔径声纳2维频域系统函数分解为收发分置畸变项和类收发合置项。在此基础上,采用复数相乘、插值实现收发分置畸变项的补偿,并利用距离-多普勒算法进行成像处理。相对传统方法,该文方法在整个测绘带内的近似误差更小,不会带来方位向上的位置偏移,能得到与真实目标位置一致的成像结果。

基于鞍点逼近的机械零部件可靠性及其灵敏度分析

将鞍点逼近理论与灵敏度分析方法相结合,讨论具有随机参数的机械零部件可靠性及其灵敏度问题。在基本随机参数概率分布已知的前提下,应用鞍点逼近技术获得了外载荷作用下随机结构响应的概率密度函数和分布函数。在实例计算中,对螺栓、汽车前轴进行数值分析,将所提方法得出的数值分析结果与Monte-Carlo方法计算的结果进行对比验证,得出由鞍点逼近技术计算概率密度函数和分布函数具有计算结果准确度高、计算速率较快、尾部分布较好的优点。应用灵敏度分析方法,利用鞍点逼近技术计算的随机结构响应的概率密度函数,得到机械零部件随机参数可靠性灵敏度的变化规律,研究设计参数的改变对螺栓、汽车前轴可靠性的影响,为机械零部件的可靠性设计提供理论依据。

基于降维积分的结构系统可靠性分析研究

复杂结构系统一般具有多个失效模式.传统系统可靠性分析模型是在假设各失效模式相互独立的条件下建立的.而在工程实际问题中,由于结构系统的组成单元之间紧密联系,系统的失效模式大多是相互耦合的.简单地在失效模式相互独立的假设条件下进行系统可靠性分析与评价常常会导致过大的误差,甚至得出错误的结论.提出一种相关失效模式结构系统可靠性分析方法.利用降维法和Gauss--Hermite数值积分技术计算随机参数结构系统极限状态函数的统计矩,采用极限状态函数的前四阶累积量拟合其累积量生成函数,通过鞍点逼近方法拟合结构系统极限状态函数的概率密度函数和累积分布函数,进而获取结构系统的可靠度(或失效概率).数值算例表明该方法具有较高的计算精度和效率,通用性强.

渐开线圆柱齿轮系统运动精度可靠性分析

为确定齿轮系统在随机因素影响下满足精度要求的概率,基于齿轮运动精度理论与鞍点逼近技术,提出一种渐开线圆柱齿轮系统运动精度可靠性分析方法,用以确定齿轮系统在随机因素影响下满足精度要求的概率。分析了齿轮加工误差、轴的径向跳动以及装配偏心对齿轮系统运动误差的影响,推导了齿轮系统运动误差的均值、方差、三阶和四阶原点矩的函数表达式,利用鞍点逼近方法拟合了齿轮系统运动误差的概率密度函数与概率分布函数。结合国家标准建立了齿轮系统运动精度可靠性分析模型。数值算例表明,该方法具有较高的求解精度和效率。

基于鞍点估计及其改进法的可靠性灵敏度分析

鞍点估计可以直接逼近非正态变量空间中线性功能函数概率分布,进而得出功能函数的失效概率.在此基础上进行了基于鞍点估计的可靠性灵敏度分析.对于非线性功能函数,尤其是强非线性功能函数,基于鞍点估计进行可靠性及灵敏度分析时存在较大的误差,为此建立了基于鞍点估计的改进方法——鞍点线抽样方法的可靠性灵敏度分析.在标准化的变量空间中利用线抽样方法的样本点将系统失效概率转化为一系列线性响应功能函数失效概率的平均值,从而可靠性灵敏度转化为一系列线性响应功能函数的失效概率对随机变量分布参数偏导数的平均值,再采用鞍点概率估计方法直接估计非正态变量标准化空间中这一系列线性响应功能函数的失效概率及可靠性灵敏度.通过比较两种方法的基本思想、实现过程和算例结果可以发现:(1)第1种方法只适用于线性程度较好的功能函数的情况,其误差主要来源于非线性极限状态函数的线性化;(2)改进方法给出的是失效概率及失效概率对随机变量分布参数偏导数的估计值,这些估计值随样本点数的增加而趋于真值,并且该方法可以考虑功能函数的非线性对失效概率的影响,因此具有广泛的适用范围.

多失效模式下基于鞍点逼近和Nataf变换的基本变量总效应分析

针对工程实际中经常出现的多失效模式可靠性模型,建立多失效模式情况下的基本随机变量总效应分析的鞍点逼近方法。在所提模型中,利用鞍点逼近方法和Nataf变换结合来近似多失效模式下响应功能函数的分布函数,并以此来求解基本随机变量总效应分析中所需的多失效模式响应功能函数的无条件联合概率密度和条件联合概率密度函数,进而求得基本变量对多失效模式响应功能函数概率分布影响的总效应。给出所提方法的实现步骤和原理,并通过算例验证所提方法的合理性和可行性。