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.

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.

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.

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.

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.

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.

Inference in the Presence of Likelihood Monotonicity for Polytomous and Logistic Regression

This paper addresses the problem of inference for a multinomial regression model in the presence of likelihood monotonicity. This paper proposes translating the multinomial regression problem into a conditional logistic regression problem, using existing techniques to reduce this conditional logistic regression problem to one with fewer observations and fewer covariates, such that probabilities for the canonical sufficient statistic of interest, conditional on remaining sufficient statistics, are identical, and translating this conditional logistic regression problem back to the multinomial regression setting. This reduced multinomial regression problem does not exhibit monotonicity of its likelihood, and so conventional asymptotic techniques can be used.