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 fully Bayesian approach for combining multi-level information in multi-state fault tree quantification

This paper presents a fully Bayesian approach that simultaneously combines non-overlapping (in time) basic event and higher-level event failure data in fault tree quantification with multi-state events. Such higher-level data often correspond to train, subsystem or system failure events. The fully Bayesian approach also automatically propagates the highest-level data to lower levels in the fault tree. A simple example illustrates our approach.     

A Bayesian approach to modeling and predicting pitting flaws in steam generator tubes

Steam generators in nuclear power plants have experienced varying degrees of under-deposit pitting corrosion. A probabilistic model to accurately predict pitting damage is necessary for effective life-cycle management of steam generators. This paper presents an advanced probabilistic model of pitting corrosion characterizing the inherent randomness of the pitting process and measurement uncertainties of the in-service inspection (ISI) data obtained from eddy current (EC) inspections. A Markov chain Monte Carlo simulation-based Bayesian method, enhanced by a data augmentation technique, is developed for estimating the model parameters. The proposed model is able to predict the actual pit number, the actual pit depth as well as the maximum pit depth, which is the main interest of the pitting corrosion model. The study also reveals the significance of inspection uncertainties in the modeling of pitting flaws using the ISI data: Without considering the probability-of-detection issues and measurement errors, the leakage risk resulted from the pitting corrosion would be under-estimated, despite the fact that the actual pit depth would usually be over-estimated.     

A fast Bayesian approach using adaptive densifying approximation technique accelerated MCMC

A novel algorithm is presented in this study to improve the efficiency and accuracy of Bayesian approach for fast sampling of posterior distributions of the unknown structure parameters. This algorithm can save a computational cost by resolving the efficiency problem in Bayesian identifications. In this algorithm, an approximation model based on the radial basis function is first used to replace the actual joint posterior distribution of the unknown parameters. An adaptive densifying technique is then introduced to increase the accuracy of the approximation model by reconstructing them with densified samples. Finally, the marginal posterior distributions for each parameter with fine accuracy can be efficiently achieved using the Markov Chain Monte Carlo method based on the present densified approximation model. Two numerical examples are investigated to demonstrate that the present algorithm can achieve significant computational gains without sacrificing the accuracy.     

A fully Bayesian approach for combining multilevel failure information in fault tree quantification and optimal follow-on resource allocation

This paper presents a fully Bayesian approach that simultaneously combines non-overlapping (in time) basic event and higher-level event failure data in fault tree quantification. Such higher-level data often correspond to train, subsystem or system failure events. The fully Bayesian approach also automatically propagates the highest-level data to lower levels in the fault tree. A simple example illustrates our approach. The optimal allocation of resources for collecting additional data from a choice of different level events is also presented. The optimization is achieved using a genetic algorithm.     

Bayesian time–domain approach for modal updating using ambient data

The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian time–domain approach for modal updating is presented which is based on an approximation of a conditional probability expansion of the response. It allows one to obtain not only the optimal values of the updated modal parameters but also their associated uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed in a subsequent step with the updating of a theoretical finite-element model based on modal estimates. The proposed approach requires only one set of response data. It is found that the updated PDF can be well approximated by a Gaussian distribution centered at the optimal parameters at which the updated PDF is maximized. Examples using simulated data are presented to illustrate the proposed method.     

A fully Bayesian multivariate approach to before-after safety evaluation

This paper presents a fully Bayesian multivariate approach to before-after safety evaluation. Although empirical Bayes (EB) methods have been widely accepted as statistically defensible safety evaluation tools in observational before-after studies for more than a decade, EB has some limitations such that it requires a development and calibration of reliable safety performance functions (SPFs) and the uncertainty in the EB safety effectiveness estimates may be underestimated when a fairly large reference group is not available. This is because uncertainty (standard errors) of the estimated regression coefficients and dispersion parameter in SPFs is not reflected in the final safety effectiveness estimate of EB.Fully Bayesian (FB) methodologies in safety evaluation are emerging as the state-of-the-art methods that have a potential to overcome the limitations of EB in that uncertainty in regression parameters in the FB approach is propagated throughout the model and carries through to the final safety effectiveness estimate. Nonetheless, there have not yet been many applications of fully Bayesian methods in before-after studies. Part of reasons is the lack of documentation for a step-by-step FB implementation procedure for practitioners as well as an increased complexity in computation. As opposed to the EB methods of which steps are well-documented in the literature for practitioners, the steps for implementing before-after FB evaluations have not yet been clearly established, especially in more general settings such as a before-after study with a comparison group/comparison groups. The objectives of this paper are two-fold: (1) to develop a fully Bayesian multivariate approach jointly modeling crash counts of different types or severity levels for a before-after evaluation with a comparison group/comparison groups and (2) to establish a step-by-step procedure for implementing the FB methods for a before-after evaluation with a comparison group/comparison groups.The fully Bayesian multivariate approach introduced in this paper has additional advantages over the corresponding univariate approaches (whether classical or Bayesian) in that the multivariate approach can recover the underlying correlation structure of the multivariate crash counts and can also lead to a more precise safety effectiveness estimate by taking into account correlations among different crash severities or types for estimation of the expected number of crashes. The new method is illustrated with the multivariate crash count data obtained from expressways in Korea for 13 years to assess the safety effectiveness of decreasing the posted speed limit.     

Bayesian Approach to Assessing Uncertainty and Calculating a Reference Value in Key Comparison Experiments

International experiments called Key Comparisons pose an interesting statistical problem, the estimation of a quantity called a Reference Value. There are many possible forms that this estimator can take. Recently, this topic has received much international attention. In this paper, it is argued that a fully Bayesian approach to this problem is compatible with the current practice of metrology, and can easily be used to create statistical models which satisfy the varied properties and assumptions of these experiments.     

Bayesian inference in probabilistic risk assessment—The current state of the art

Markov chain Monte Carlo (MCMC) approaches to sampling directly from the joint posterior distribution of aleatory model parameters have led to tremendous advances in Bayesian inference capability in a wide variety of fields, including probabilistic risk analysis. The advent of freely available software coupled with inexpensive computing power has catalyzed this advance. This paper examines where the risk assessment community is with respect to implementing modern computational-based Bayesian approaches to inference. Through a series of examples in different topical areas, it introduces salient concepts and illustrates the practical application of Bayesian inference via MCMC sampling to a variety of important problems.     

Investigation of time and weather effects on crash types using full Bayesian multivariate Poisson lognormal models

Previous research shows that various weather elements have significant effects on crash occurrence and risk; however, little is known about how these elements affect different crash types. Consequently, this study investigates the impact of weather elements and sudden extreme snow or rain weather changes on crash type. Multivariate models were used for seven crash types using five years of daily weather and crash data collected for the entire City of Edmonton. In addition, the yearly trend and random variation of parameters across the years were analyzed by using four different modeling formulations. The proposed models were estimated in a full Bayesian context via Markov Chain Monte Carlo simulation. The multivariate Poisson lognormal model with yearly varying coefficients provided the best fit for the data according to Deviance Information Criteria. Overall, results showed that temperature and snowfall were statistically significant with intuitive signs (crashes decrease with increasing temperature; crashes increase as snowfall intensity increases) for all crash types, while rainfall was mostly insignificant. Previous snow showed mixed results, being statistically significant and positively related to certain crash types, while negatively related or insignificant in other cases. Maximum wind gust speed was found mostly insignificant with a few exceptions that were positively related to crash type. Major snow or rain events following a dry weather condition were highly significant and positively related to three crash types: Follow-Too-Close, Stop-Sign-Violation, and Ran-Off-Road crashes. The day-of-the-week dummy variables were statistically significant, indicating a possible weekly variation in exposure. Transportation authorities might use the above results to improve road safety by providing drivers with information regarding the risk of certain crash types for a particular weather condition.     

遗传智能采样技术的贝叶斯理论识别滑动轴承-转子系统不平衡量

轴承转子系统不平衡量识别过程中,在输出响应和模型中存在的不确定性参数一般采用概率法描述,通过贝叶斯理论获得不平衡量的联合后验概率密度分布时涉及大量采样.针对采样效率,提出了基于遗传智能采样技术改进贝叶斯理论.首先,以代价函数作为指示因子通过信赖域模型管理方法不断更新先验空间使其覆盖高密度后验空间,然后通过智能布点技术和样本遗传策略以有限的样本点集中呈现在联合后验概率密度分布的高密度区域,提高信赖域上关键区域的精度,从而加快收敛速度,减小耗时的正问题调用次数.最后将其应用于识别具有不平衡量先验信息和带有随机噪声的测试响应的滑动轴承-转子系统的不平衡量,获得不平衡量的均值、置信区间.案例显示能准确快速地抽样,提高了贝叶斯识别的计算效率.     

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 joint-probability approach to crash prediction models

Many road safety researchers have used crash prediction models, such as Poisson and negative binomial regression models, to investigate the associations between crash occurrence and explanatory factors. Typically, they have attempted to separately model the crash frequencies of different severity levels. However, this method may suffer from serious correlations between the model estimates among different levels of crash severity. Despite efforts to improve the statistical fit of crash prediction models by modifying the data structure and model estimation method, little work has addressed the appropriate interpretation of the effects of explanatory factors on crash occurrence among different levels of crash severity. In this paper, a joint probability model is developed to integrate the predictions of both crash occurrence and crash severity into a single framework. For instance, the Markov chain Monte Carlo (MCMC) approach full Bayesian method is applied to estimate the effects of explanatory factors. As an illustration of the appropriateness of the proposed joint probability model, a case study is conducted on crash risk at signalized intersections in Hong Kong. The results of the case study indicate that the proposed model demonstrates a good statistical fit and provides an appropriate analysis of the influences of explanatory factors.     

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.     

基于多分辨率分析的结构物理参数识别贝叶斯估计方法:方法推导与验证

在观测噪声和模型误差等不确定性因素的影响下,结构物理参数识别问题是一个不确定性问题.针对此问题,该文从结构运动微分方程出发,利用小波多分辨率分析原理,建立结构多尺度动力方程,由该方程以结构激励和响应信息在多尺度上的细节信号和最大尺度上的概貌信号为观测量推得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...     

Bayesian tolerance intervals for the balanced two-factor nested random effects model

Statistical intervals, properly calculated from sample data, are likely to be substantially more informative to decision makers than obtaining a point estimate alone and are often of paramount interest to practitioners and thus management (and are usually a great deal more meaningful than statistical significance or hypothesis tests). Wolfinger (1998, J Qual Technol 36:162–170) presented a simulation-based approach for determining Bayesian tolerance intervals in a balanced one-way random effects model. In this note the theory and results of Wolfinger are extended to the balanced two-factor nested random effects model. The example illustrates the flexibility and unique features of the Bayesian simulation method for the construction of tolerance intervals.      

A Bayesian model for measurements on a set of instruments

A Bayesian model is proposed based on randomizing the systematic errors of the instruments. Conditions are identified under which the randomization reduces the expected bias in estimating a measured quantity. __________ Translated from Izmeritel’naya Tekhnika, No. 3, pp. 22–25, March, 2007.     

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.     

An evaluation of graduated driver licensing programs in North America using a meta-analytic approach

Most jurisdictions in North America have some version of graduated driver licensing (GDL). A sound body of evidence documenting the effectiveness of GDL programs in reducing collisions, fatalities and injuries among novice drivers is available. However, information about the relative importance of individual components of GDL is lacking. The objectives of this study are to calculate a summary statistic of GDL effectiveness and to identify the most effective components of GDL programs using a meta-analytic approach. Data from 46 American States, the District of Columbia and 11 Canadian jurisdictions are used and were obtained from the Fatality Analysis Reporting System (FARS) for the U.S. and from Transport Canada's Traffic Accident Information Database (TRAID) for Canada. The timeframe of this evaluation is 1992 through 2006, inclusive. Relative fatality risks and their 95% confidence intervals were calculated using fatality counts and population data for target and comparison groups, both in a pre-implementation and post-implementation period in each jurisdiction. The target groups were 16-, 17-, 18- and 19-year-old drivers. The comparison group was 25–54-year-old drivers. The relative fatality risks of all jurisdictions were summarized using the random effects DerSimonian and Laird model. Meta-regression using Restricted Maximum Likelihood (REML) and Markov Chain Monte Carlo (MCMC) Gibbs sampling was also conducted. Strong evidence in support of GDL was found. GDL had a positive and significant impact on the relative fatality risk of 16-year-old drivers (reduction of 19.1%). Significant effects were found for meta-regression models with 16-, 18- and 19-year-old drivers. These effects include length of night restriction in the learner stage, country, driver education in the learner stage and in the intermediate stage, whether night restrictions are lifted in the intermediate stage for work purposes, passenger restriction in the intermediate stage, whether passenger restrictions in the intermediate stage are lifted if passengers are family members, and whether there is an exit test in the intermediate stage. In conclusion, several GDL program components have an important effect on the relative fatality risk of novice drivers. These results help understand how such effects are achieved.