A Bayesian model averaging method for software reliability modeling and assessment

The software reliability modeling is of great significance in improving software quality and managing the software development process. However, the existing methods are not able to accurately model software reliability improvement behavior because existing single model methods rely on restrictive assumptions and combination models cannot well deal with model uncertainties. In this article, we propose a Bayesian model averaging (BMA) method to model software reliability. First, the existing reliability modeling methods are selected as the candidate models, and the Bayesian theory is used to obtain the posterior probabilities of each reliability model. Then, the posterior probabilities are used as weights to average the candidate models. Both Markov Chain Monte Carlo (MCMC) algorithm and the Expectation–Maximization (EM) algorithm are used to evaluate a candidate model's posterior probability and for comparison purpose. The results show that the BMA method has superior performance in software reliability modeling, and the MCMC algorithm performs better than EM algorithm when they are used to estimate the parameters of BMA method.     

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.     

一种基于改进MCMC算法的模型修正方法

标准马尔可夫链蒙特卡罗(MCMC)算法不易收敛、拒绝率高,使其应用受到限制。在贝叶斯方法中引入最大熵值法来估计参数的后验概率密度函数最大值,进而将布谷鸟算法中新鸟巢更新的思想融入Metropolis-Hasting(MH)抽样算法得到改进的MH抽样算法,同时使用支持向量机(SVM)建立待修正参数与有限元模型输出之间的代理模型,以提高模型修正的计算效率。分别使用三自由度线性系统和平面桁架模型来验证本文方法的有效性,结果表明:修正后样本的马尔可夫链混合性能好,停滞概率低,修正后参数相对误差均小于2%。     

A stochastic simulation method for uncertainty quantification in the linearized inverse conductivity problem

This paper considers the inverse problem in electrical impedance tomography with non‐informative prior information on the required conductivity function. The problem is approached with a Newton‐type iterative algorithm where the solution of the linearized approximation is estimated using Bayesian inference. The novelty of this work focuses on maximum a posteriori estimation assuming a model that incorporates the linearization error as a random variable. From an analytical expression of this term, we employ Monte Carlo simulation in order to characterize its probability distribution function. This simulation entails sampling an improper prior distribution for which we propose a stable scheme on the basis of QR decomposition. The simulation statistics show that the error on the linearized model is not Gaussian, however, to maintain computational tractability, we derive the posterior probability density function of the solution by imposing a Gaussian kernel approximation to the error density. Numerical results obtained through this approach indicate the superiority of the new model and its respective maximum a posteriori estimator against the conventional one that neglects the impact of the linearization error. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Reliability Analysis for Degradation of Locomotive Wheels using Parametric Bayesian Approach

This paper undertakes a reliability study using a Bayesian survival analysis framework to explore the impact of a locomotive wheel's installed position on its service lifetime and to predict its reliability characteristics. The Bayesian Exponential Regression Model, Bayesian Weibull Regression Model and Bayesian Log‐normal Regression Model are used to analyze the lifetime of locomotive wheels using degradation data and taking into account the position of the wheel. This position is described by three different discrete covariates: the bogie, the axle and the side of the locomotive where the wheel is mounted. The goal is to determine reliability, failure distribution and optimal maintenance strategies for the wheel. The results show that: (i) under specified assumptions and a given topography, the position of the locomotive wheel could influence its reliability and lifetime; (ii) the Bayesian Log‐normal Regression Model is a useful tool. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A polynomial chaos efficient global optimization approach for Bayesian optimal experimental design

This paper proposes a global optimization framework to address the high computational cost and non convexity of Optimal Experimental Design (OED) problems. To reduce the computational burden and the presence of noise in the evaluation of the Shannon expected information gain (SEIG), this framework proposes the coupling of Laplace approximation and polynomial chaos expansions (PCE). The advantage of this procedure is that PCE allows large samples to be employed for the SEIG estimation, practically vanishing the noisy introduced by the sampling procedure. Consequently, the resulting optimization problem may be treated as deterministic. Then, an optimization approach based on Kriging surrogates is employed as the optimization engine to search for the global solution with limited computational budget. Four numerical examples are investigated and their results are compared to state-of-the-art stochastic gradient descent algorithms. The proposed approach obtained better results than the stochastic gradient algorithms in all situations, indicating its efficiency and robustness in the solution of OED problems.     

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 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.     

A Bayesian analysis of reliability improvement experiments

The number of effects can be studied in a log‐location‐scale regression model used in analyzing a reliability improvement experiment is restricted to the number of runs, which is usually small. In many real examples, only the main effects and a few 2‐factor interactions are considered. In this work, we propose using a Bayesian approach to analyze reliability improvement experiments. By specifying a prior on the effects, the number of effects can be studied is no longer restricted to the number of runs, and aliased effects can all be identified and estimated simultaneously. We analyze 2 real data sets to demonstrate the proposed approach. The results show that when complex interactions are present, the proposed approach can provide a more reliable result.     

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.     

Bayesian analysis for estimating statistical parameter distributions of elasto-viscoplastic material models

High temperature design methods rely on constitutive models for inelastic deformation and failure typically calibrated against the mean of experimental data without considering the associated scatter. Variability may arise from the experimental data acquisition process, from heat-to-heat material property variations, or both and need to be accurately captured to predict parameter bounds leading to efficient component design. Applying the Bayesian Markov Chain Monte Carlo (MCMC) method to produce statistical models capturing the underlying uncertainty in the experimental data is an area of ongoing research interest. This work varies aspects of the Bayesian MCMC method and explores their effect on the posterior parameter distributions for a uniaxial elasto-viscoplastic damage model using synthetically generated reference data. From our analysis with the uniaxial inelastic model we determine that an informed prior distribution including different types of test conditions results in more accurate posterior parameter distributions. The parameter posterior distributions, however, do not improve when increasing the number of similar experimental data. Additionally, changing the amount of scatter in the data affects the quality of the posterior distributions, especially for the less sensitive model parameters. Moreover, we perform a sensitivity study of the model parameters against the likelihood function prior to the Bayesian analysis. The results of the sensitivity analysis help to determine the reliability of the posterior distributions and reduce the dimensionality of the problem by fixing the insensitive parameters. The comprehensive study described in this work demonstrates how to efficiently apply the Bayesian MCMC methodology to capture parameter uncertainties in high temperature inelastic material models. Quantifying these uncertainties in inelastic models will improve high temperature engineering design practices and lead to safer, more effective component designs.     

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.