Markov chain Monte Carlo in Bayesian models for testing gamma and lognormal S-type process qualities

The process capability index C_pu is widely used to measure S-type process quality. Many researchers have presented adaptive techniques for assessing the true C_pu assuming normality. However, the quality characteristic is often abnormal, and the derived techniques based on the normality assumption could mislead the manager into making uninformed decisions. Therefore, this study provides an alternative method for assessing C_pu of non-normal processes. The Markov chain Monte Carlo, an emerging popular statistical tool, is integrated into Bayesian models to seek the empirical posterior distributions of specific gamma and lognormal parameters. Afterwards, the lower credible interval bound of C_pu can be derived for testing the non-normal process quality. Simulations show that the proposed method is adaptive and has good performance in terms of coverage probability.     

Fatigue crack growth prediction in nuclear piping using Markov chain Monte Carlo simulation

In this paper, we present and demonstrate a methodology to improve probabilistic fatigue crack growth (FCG) predictions by using the concept of Bayesian updating using Markov chain Monte Carlo simulations. The methodology is demonstrated on a cracked pipe undergoing fatigue loading. Initial estimates of the FCG rate are made using the Paris law. The prior probability distributions of the Paris law parameters are taken from the tests on specimen made of the same material as that of pipe. Measured data on crack depth over number of loading cycles are used to update the prior distribution using the Markov chain Monte Carlo. The confidence interval on the predicted FCG rate is also estimated. In actual piping placed in a plant, the measured data can be considered equivalent to the data received from in-service inspection. It is shown that the proposed methodology improves the fatigue life prediction. The number of observations used for updating is found to leave a significant effect on the accuracy of the updated prediction.     

Efficient Technique for Assessing Actual Non‐normal Quality Loss: Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) techniques have been extensively developed and are accepted for solving various real‐world problems. However, process capabilities are rarely analyzed with the means of MCMC. This study integrates the MCMC technique into Bayesian models for assessing the well‐known quality loss index C_pm for gamma and Weibull process distributions. After the MCMC iterations are completed, the quality manager can make reliable decisions via the proposed credible intervals. Furthermore, this study provides performance comparisons of the estimators of C_pm obtained by the MCMC and bootstrap techniques. Simulations show that the MCMC technique performs better than the bootstrap technique in most of the cases that were considered. Copyright © 2016 John Wiley & Sons, Ltd.     

结构物理参数识别的贝叶斯估计马尔可夫蒙特卡罗方法

从结构动力特征方程出发,以结构主模态参数为观测量,推得结构物理参数线性回归模型。对该模型应用贝叶斯估计理论得到物理参数后验联合分布,再结合马尔可夫蒙特卡罗抽样方法给出各个物理参数的边缘概率分布和最优估计值,而提出了基于结构主模态参数的结构物理参数识别贝叶斯估计马尔可夫蒙特卡罗方法。对五层剪切型结构的数值研究表明,此方法能够利用少数主模态参数给出结构质量和刚度参数的概率分布和最优识别值,而且在主模态参数较准确时识别误差很小。     

Weibull extension model: A Bayes study using Markov chain Monte Carlo simulation

Several generalizations of the two-parameter Weibull model have been proposed to model data sets that exhibit complex non-monotone shapes of hazard rate function. The present paper focuses on one such generalization referred to as the Weibull extension model in the literature. Complete Bayesian analysis of the model has been provided using Markov chain Monte Carlo simulation. Finally, a thorough study has been conducted for checking the adequacy of the model for a given data set using some of the graphical and numerical methods based on predictive simulation ideas. A real data set is considered for illustration.     

Robust design using Bayesian Monte Carlo

In this paper, we propose an efficient strategy for robust design based on Bayesian Monte Carlo simulation. Robust design is formulated as a multiobjective problem to allow explicit trade‐off between the mean performance and variability. The proposed method is applied to a compressor blade design in the presence of manufacturing uncertainty. Process capability data are utilized in conjunction with a parametric geometry model for manufacturing uncertainty quantification. High‐fidelity computational fluid dynamics simulations are used to evaluate the aerodynamic performance of the compressor blade. A probabilistic analysis for estimating the effect of manufacturing variations on the aerodynamic performance of the blade is performed and a case for the application of robust design is established. The proposed approach is applied to robust design of compressor blades and a selected design from the final Pareto set is compared with an optimal design obtained by minimizing the nominal performance. The selected robust blade has substantial improvement in robustness against manufacturing variations in comparison with the deterministic optimal blade. Significant savings in computational effort using the proposed method are also illustrated. Copyright © 2007 John Wiley & Sons, Ltd.     

Structural damage identification via time domain response and Markov Chain Monte Carlo method

The present work addresses the problem of structural damage identification built on the statistical inversion approach. Here, the damage state of the structure is continuously described by a cohesion parameter, which is spatially discretized by the finite element method. The inverse problem of damage identification is then posed as the determination of the posterior probability densities of the nodal cohesion parameters. The Markov Chain Monte Carlo method, implemented with the Metropolis–Hastings algorithm, is considered in order to approximate the posterior probabilities by drawing samples from the desired joint posterior probability density function. With this approach, prior information on the sought parameters can be used and the uncertainty concerning the known values of the material properties can be quantified in the estimation of the cohesion parameters. The assessment of the proposed approach has been performed by means of numerical simulations on a simply supported Euler–Bernoulli beam. The damage identification and assessment are performed considering time domain response data. Different damage scenarios and noise levels were addressed, demonstrating the feasibility of the proposed approach.     

Finite element model updating using Hamiltonian Monte Carlo techniques

Bayesian techniques have been widely used in finite element model (FEM) updating. The attraction of these techniques is their ability to quantify and characterize the uncertainties associated with dynamic systems. In order to update an FEM, the Bayesian formulation requires the evaluation of the posterior distribution function. For large systems, this function is difficult to solve analytically. In such cases, the use of sampling techniques often provides a good approximation of this posterior distribution function. The hybrid Monte Carlo (HMC) method is a classic sampling method used to approximate high-dimensional complex problems. However, the acceptance rate of HMC is sensitive to the system size, as well as to the time step used to evaluate the molecular dynamics trajectory. The shadow HMC technique (SHMC), which is a modified version of the HMC method, was developed to improve sampling for large system sizes by drawing from a modified shadow Hamiltonian function. However, the SHMC algorithm performance is limited by the use of a non-separable modified Hamiltonian function. Moreover, two additional parameters are required for the sampling procedure, which could be computationally expensive. To overcome these weaknesses, the separable shadow HMC (S2HMC) method has been introduced. This method uses a transformation to a different parameter space to generate samples. In this paper, we analyse the application and performance of these algorithms, including the parameters used in each algorithm, their limitations and the effects on model updating. The accuracy and the efficiency of the algorithms are demonstrated by updating the finite element models of two real mechanical structures. It is observed that the S2HMC algorithm has a number of advantages over the other algorithms; for example, the S2HMC algorithm is able to efficiently sample at larger time steps while using fewer parameters than the other algorithms.     

A note on the Bayesian analysis of experiments with correlated multiple responses using a matrix‐logarithmic covariance model

In the literature, analysis of multiple responses from experiments with replicates has modeled the covariance matrix directly as linear models of the transformed variances and correlations, ie, covariance modeling. This article considers models based on the matrix‐logarithm of the covariance matrix. This so‐called log‐covariance modeling is illustrated with data from actual experiments and compared with the traditional covariance modeling.     

Bayesian system identification of dynamical systems using large sets of training data: A MCMC solution

In the last 20 years the applicability of Bayesian inference to the system identification of structurally dynamical systems has been helped considerably by the emergence of Markov chain Monte Carlo (MCMC) algorithms – stochastic simulation methods which alleviate the need to evaluate the intractable integrals which often arise during Bayesian analysis. In this paper specific attention is given to the situation where, with the aim of performing Bayesian system identification, one is presented with very large sets of training data. Building on previous work by the author, an MCMC algorithm is presented which, through combing Data Annealing with the concept of ‘highly informative training data’, can be used to analyse large sets of data in a computationally cheap manner. The new algorithm is called Smooth Data Annealing.     

Bayesian assurance tests for degradation data

Assurance test plans are chosen to manage consumer's and producer's risks. We develop methods for planning Bayesian assurance tests for degradation data, ie, on the basis of the degradation data collected in the test, a decision is made whether to accept or reject a product. Bayesian assurance tests incorporate prior knowledge in the planning stage and use this information to evaluate posterior consumer's and producer's risks. We consider prior knowledge that takes the form of related degradation data. Assurance test plans are then found that meet the specified requirements for consumer's and producer's risks. We illustrate the planning of such assurance tests with an example involving printhead migration data. We also investigate the impact of measurement error on these assurance test plans.     

基于贝叶斯统计推断的框架结构损伤诊断研究

提出一种基于贝叶斯推理的非线性结构模型修正方法,同时考虑激励的随机性,建立了复合随机振动系统的动力可靠度分析方法。利用实测结构动力响应主分量的瞬时特征参数作为非线性指标构建似然函数,结合拒绝延缓自适应(Delayed Rejection and Adaptive Metropolis, DRAM)算法和高斯过程替代模型实现了非线性结构模型修正及其参数的不确定性量化。根据首次超越破坏准则,利用广义概率密度演化方法,分别对仅考虑激励随机性的确定性模型和同时考虑结构参数与激励不确定性的复合随机振动模型进行动力可靠度分析,并利用蒙特卡洛随机抽样方法验证了计算结果的准确性。研究结果表明：基于振动响应瞬时特征参数的贝叶斯推理方法能够快速、准确地实现结构的非线性模型修正及其参数的不确定性量化。与具有初始设计参数名义值的确定性模型相比,考虑参数不确定性的复合随机模型的动力可靠度总体偏低,因此,在结构安全评估中应考虑非线性模型参数不确定性的影响,使评估结果更加安全、可靠。     

利用贝叶斯推理估计二维含源对流扩散方程参数

为了克服观测数据的不确定性给参数反演带来的困难,利用贝叶斯推理建立了二维含源对流扩散方程参数估计的数学模型。通过贝叶斯定理,获得了模型参数的后验分布,从而获得反问题的解。对于多参数反演问题,基于数值解计算得到的参数后验分布很难直观地表现出来,采用马尔科夫链蒙特卡罗方法对参数的后验分布进行采样,获得了扩散系数和降解系数的估计值。研究了观测点位置对计算结果的影响;同时研究了似然函数的形式对估计结果的影响,结果表明在异常值可能出现时采用Laplace分布型的似然函数可以获得稳健估计。对不同观测点数目下的估计值进行了对比,认为对于二维稳态对流扩散方程的双参数估计问题,至少需要两个观测点才有可能得到合理的解。     

Combining stochastics and analytics for a fast Monte Carlo decay chain generator

Various Monte Carlo programs, developed either by small groups or widely available, have been used simulate decays of radioactive chains, from the original parent nucleus to the final stable isotopes. These chains include uranium, thorium, radon, and others, and generally have long-lived parent nuclei. Generating decays within these chains requires a certain amount of computing overhead related to simulating unnecessary decays, time-ordering the final results in post-processing, or both. We present a combination analytic/stochastic algorithm for creating a time-ordered set of decays with position and time correlations, and starting with an arbitrary source age. Thus the simulation costs are greatly reduced, while at the same time avoiding chronological post-processing. We discuss optimization methods within the approach to minimize calculation time, and extension of the algorithm to include various source types.     

Using Markov Chain Monte Carlo methods to solve full Bayesian modeling of PWR vessel flaw distributions

We present a hierarchical Bayesian method for estimating the density and size distribution of subclad-flaws in French Pressurized Water Reactor (PWR) vessels. This model takes into account in-service inspection (ISI) data, a flaw size-dependent probability of detection (different functions are considered) with a threshold of detection, and a flaw sizing error distribution (different distributions are considered). The resulting model is identified through a Markov Chain Monte Carlo (MCMC) algorithm. The article includes discussion for choosing the prior distribution parameters and an illustrative application is presented highlighting the model's ability to provide good parameter estimates even when a small number of flaws are observed.     

A study using Monte Carlo Simulation for failure probability calculation in Reliability-Based Optimization

In this study, a Reliability-Based Optimization (RBO) methodology that uses Monte Carlo Simulation techniques, is presented. Typically, the First Order Reliability Method (FORM) is used in RBO for failure probability calculation and this is accurate enough for most practical cases. However, for highly nonlinear problems it can provide extremely inaccurate results and may lead to unreliable designs. Monte Carlo Simulation (MCS) is usually more accurate than FORM but very computationally intensive. In the RBO methodology presented in this paper, limit state approximations are used in conjunction with MCS techniques in an approximate MCS-based RBO that facilitates the efficient calculation of the probabilities of failure. A FORM-based RBO is first performed to obtain the initial limit state approximations. A Symmetric Rank-1 (SR1) variable metric algorithm is used to construct and update the quadratic limit state approximations. The approximate MCS-based RBO uses a conditional-expectation-based MCS, that was chosen over indicator-based MCS because of the smoothness of the probability of failure estimates and the availability of analytic sensitivities. The RBO methodology was implemented for an analytic test problem and a higher-dimensional, control-augmented-structure test problem. The results indicate that the SR1 algorithm provides accurate limit state approximations (and therefore accurate estimates of the probabilities of failure) for these test problems. It was also observed that the RBO methodology required two orders of magnitude fewer analysis calls than an approach that used exact limit state evaluations for both test problems.     

A Bayesian Semiparametric Analysis of the Reliability and Maintenance of Machine Tools

A Bayesian semiparametric proportional hazards model is presented to describe the failure behavior of machine tools. The semiparametric setup is introduced using a mixture of Dirichlet processes prior. A Bayesian analysis is performed on real machine tool failure data using the semiparametric setup, and development of optimal replacement strategies are discussed. The results of the semiparametric analysis and the replacement policies are compared with those under a parametric model.     

Bayesian Computations for a Class of Reliability Growth Models

In this article, we consider the development and analysis of both attribute- and variable-data reliability growth models from a Bayesian perspective. We begin with an overview of a Bayesian attribute-data reliability growth model and illustrate how this model can be extended to cover the variable-data growth models as well. Bayesian analysis of these models requires inference over ordered regions, and even though closed-form results for posterior quantities can be obtained in the attribute-data case, variable-data models prove difficult. In general, when the number of test stages gets large, computations become burdensome and, more importantly, the results may become inaccurate due to computational difficulties. We illustrate how the difficulties in the posterior and predictive analyses can be overcome using Markov-chain Monte Carlo methods. We illustrate the implementation of the proposed models by using examples from both attribute and variable reliability growth data.     

An Efficient Sampling Technique for Bayesian Inference With Computationally Demanding Models

In the environmental sciences, a large knowledge base is typically available on an investigated system or at least on similar systems. This makes the application of Bayesian inference techniques in environmental modeling very promising. However, environmental systems are often described by complex, computationally demanding simulation models. This strongly limits the application of Bayesian inference techniques, because numerical implementation of these techniques requires a very large number of simulation runs. The development of efficient sampling techniques that attempt to approximate the posterior distribution with a relatively small parameter sample can extend the range of applicability of Bayesian inference techniques to such models. In this article a sampling technique is presented that tries to achieve this goal. The proposed technique combines numerical techniques typically applied in Bayesian inference, including posterior maximization, local normal approximation, and importance sampling, with copula techniques for the construction of a multivariate distribution with given marginals and correlation structure and with low-discrepancy sampling. This combination improves the approximation of the posterior distribution by the sampling distribution and improves the accuracy of results for small sample sizes. The usefulness of the proposed technique is demonstrated for a simple model that contains the major elements of models used in the environmental sciences. The results indicate that the proposed technique outperforms conventional techniques (random sampling from simpler distributions or Markov chain Monte Carlo techniques) in cases in which the analysis can be limited to a relatively small number of parameters.     

A spectral representation based model for Monte Carlo simulation

A new model is proposed for generating samples of real-valued stationary Gaussian processes. The model is based on the spectral representation theorem stating that a weakly stationary process can be viewed as a superposition of harmonics with random properties. The classical use of this theorem for Monte Carlo simulation is based on models consisting of a superposition of harmonics with fixed frequencies but random amplitude and phase. The resulting samples have the same period depending on the discretization of the frequency band. In contrast, the proposed model consists of a superposition of harmonics with random amplitude, phase, and frequency so that different samples have different periods depending on the particular sample values of the harmonic frequencies.

A band limited Gaussian white noise process is used to illustrate the proposed Monte Carlo simulation algorithm and demonstrate that the estimates of the covariance function based on the samples of the proposed model are not periodic.