Bayesian networks in reliability

Over the last decade, Bayesian networks (BNs) have become a popular tool for modelling many kinds of statistical problems. We have also seen a growing interest for using BNs in the reliability analysis community. In this paper we will discuss the properties of the modelling framework that make BNs particularly well suited for reliability applications, and point to ongoing research that is relevant for practitioners in reliability.     

A framework to integrate software behavior into dynamic probabilistic risk assessment

Software plays an increasingly important role in modern safety-critical systems. Although, research has been done to integrate software into the classical probabilistic risk assessment (PRA) framework, current PRA practice overwhelmingly neglects the contribution of software to system risk. Dynamic probabilistic risk assessment (DPRA) is considered to be the next generation of PRA techniques. DPRA is a set of methods and techniques in which simulation models that represent the behavior of the elements of a system are exercised in order to identify risks and vulnerabilities of the system. The fact remains, however, that modeling software for use in the DPRA framework is also quite complex and very little has been done to address the question directly and comprehensively. This paper develops a methodology to integrate software contributions in the DPRA environment. The framework includes a software representation, and an approach to incorporate the software representation into the DPRA environment SimPRA. The software representation is based on multi-level objects and the paper also proposes a framework to simulate the multi-level objects in the simulation-based DPRA environment. This is a new methodology to address the state explosion problem in the DPRA environment. This study is the first systematic effort to integrate software risk contributions into DPRA environments.     

The mathematical formulation for the event sequence diagram framework

The Event Sequence Diagram (ESD) framework allows modeling of dynamic situations of interest to PRA analysts. A qualitative presentation of the framework was given in an earlier article. The mathematical formulation for the components of the ESD framework is described in this article. The formulation was derived from the basic probabilistic dynamics equations. For tackling certain dynamic non-Markovian situations, the probabilistic dynamics framework was extended. The mathematical treatment of dependencies among fault trees in a multi layered ESD framework is also presented.     

Bayesian decision analysis and reliability certification

Reliability certification is set as a problem of Bayesian Decision Analysis. Uncertainties about the system reliability are quantified by assuming the parameters of the models describing the stochastic behavior of components as random variables. A utility function quantifies the relative value of each possible level of system reliability after having been accepted or the opportunity loss of the same level if the system has been rejected. A decision about accepting or rejecting the system can be made either on the basis of the existing prior assessment of uncertainties or after obtaining further information through testing of the components or the system at a cost. The concepts of value of perfect information, expected value of sample information and the expected net gain of sampling are specialized to the reliability of a multicomponent system to determine the optimum component testing scheme prior to deciding on the system's certification. A component importance ranking is proposed on the basis of the expected value of perfect information about the reliability of each component. The proposed approach is demonstrated on a single component system failing according to a Poisson random process and with natural conjugate prior probability density functions (pdf) for the failure rate and for a multicomponent system under general assumptions.     

A generic method for estimating system reliability using Bayesian networks

This study presents a holistic method for constructing a Bayesian network (BN) model for estimating system reliability. BN is a probabilistic approach that is used to model and predict the behavior of a system based on observed stochastic events. The BN model is a directed acyclic graph (DAG) where the nodes represent system components and arcs represent relationships among them. Although recent studies on using BN for estimating system reliability have been proposed, they are based on the assumption that a pre-built BN has been designed to represent the system. In these studies, the task of building the BN is typically left to a group of specialists who are BN and domain experts. The BN experts should learn about the domain before building the BN, which is generally very time consuming and may lead to incorrect deductions. As there are no existing studies to eliminate the need for a human expert in the process of system reliability estimation, this paper introduces a method that uses historical data about the system to be modeled as a BN and provides efficient techniques for automated construction of the BN model, and hence estimation of the system reliability. In this respect K2, a data mining algorithm, is used for finding associations between system components, and thus building the BN model. This algorithm uses a heuristic to provide efficient and accurate results while searching for associations. Moreover, no human intervention is necessary during the process of BN construction and reliability estimation. The paper provides a step-by-step illustration of the method and evaluation of the approach with literature case examples.     

Bayesian networks inference algorithm to implement Dempster Shafer theory in reliability analysis

This paper deals with the use of Bayesian networks to compute system reliability. The reliability analysis problem is described and the usual methods for quantitative reliability analysis are presented within a case study. Some drawbacks that justify the use of Bayesian networks are identified. The basic concepts of the Bayesian networks application to reliability analysis are introduced and a model to compute the reliability for the case study is presented. Dempster Shafer theory to treat epistemic uncertainty in reliability analysis is then discussed and its basic concepts that can be applied thanks to the Bayesian network inference algorithm are introduced. Finally, it is shown, with a numerical example, how Bayesian networks’ inference algorithms compute complex system reliability and what the Dempster Shafer theory can provide to reliability analysis.     

A model for Bayesian software reliability analysis

Two problems which are of great interest in relation to software reliability are the prediction of future times to failure and the calculation of the optimal release time. An important assumption in software reliability analysis is that the reliability grows whenever bugs are found and removed. In this paper we present a model for software reliability analysis using the Bayesian statistical approach in order to incorporate in the analysis prior assumptions such as the (decreasing) ordering in the assumed constant failure rates of prescribed intervals. We use as prior model the product of gamma functions for each pair of subsequent interval constant failure rates, considering as the location parameter of the first interval the failure rate of the following interval. In this way we include the failure rate ordering information. Using this approach sequentially, we predict the time to failure for the next failure using the previous information obtained. Using also the relevant predictive distributions obtained, we calculate the optimal release time for two different requirements of interest: (a) the probability of an in‐service failure in a prescribed time t; (b) the cost associated with a single or more failures in a prescribed time t. Finally a numerical example is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

贝叶斯网络推理在信息系统安全风险评估中的应用

提出了一种基于贝叶斯网络推理的安全风险评估方法。从实际出发建立信息系统的贝叶斯网络模型，根据专家给出的先验信息，结合获得的证据信息，运用Pearl方法完成对模型的评估，给出在特定条件下模型的计算——线性推理算法。最后，以实例分析信息系统安全评估的实现过程，结果表明，该方法可行、有效。     

Risk-based reconfiguration of safety monitoring system using dynamic Bayesian network

To prevent an abnormal event from leading to an accident, the role of its safety monitoring system is very important. The safety monitoring system detects symptoms of an abnormal event to mitigate its effect at its early stage. As the operation time passes by, the sensor reliability decreases, which implies that the decision criteria of the safety monitoring system should be modified depending on the sensor reliability as well as the system reliability. This paper presents a framework for the decision criteria (or diagnosis logic) of the safety monitoring system. The logic can be dynamically modified based on sensor output data monitored at regular intervals to minimize the expected loss caused by two types of safety monitoring system failure events: failed-dangerous (FD) and failed-safe (FS). The former corresponds to no response under an abnormal system condition, while the latter implies a spurious activation under a normal system condition. Dynamic Bayesian network theory can be applied to modeling the entire system behavior composed of the system and its safety monitoring system. Using the estimated state probabilities, the optimal decision criterion is given to obtain the optimal diagnosis logic. An illustrative example of a three-sensor system shows the merits and characteristics of the proposed method, where the reasonable interpretation of sensor data can be obtained.     

Application of Bayesian network to the probabilistic risk assessment of nuclear waste disposal

The scenario in a risk analysis can be defined as the propagating feature of specific initiating event which can go to a wide range of undesirable consequences. If we take various scenarios into consideration, the risk analysis becomes more complex than do without them. A lot of risk analyses have been performed to actually estimate a risk profile under both uncertain future states of hazard sources and undesirable scenarios. Unfortunately, in case of considering specific systems such as a radioactive waste disposal facility, since the behaviour of future scenarios is hardly predicted without special reasoning process, we cannot estimate their risk only with a traditional risk analysis methodology. Moreover, we believe that the sources of uncertainty at future states can be reduced pertinently by setting up dependency relationships interrelating geological, hydrological, and ecological aspects of the site with all the scenarios. It is then required current methodology of uncertainty analysis of the waste disposal facility be revisited under this belief.In order to consider the effects predicting from an evolution of environmental conditions of waste disposal facilities, this paper proposes a quantitative assessment framework integrating the inference process of Bayesian network to the traditional probabilistic risk analysis. We developed and verified an approximate probabilistic inference program for the specific Bayesian network using a bounded-variance likelihood weighting algorithm. Ultimately, specific models, including a model for uncertainty propagation of relevant parameters were developed with a comparison of variable-specific effects due to the occurrence of diverse altered evolution scenarios (AESs). After providing supporting information to get a variety of quantitative expectations about the dependency relationship between domain variables and AESs, we could connect the results of probabilistic inference from the Bayesian network with the consequence evaluation model addressed. We got a number of practical results to improve current knowledge base for the prioritization of future risk-dominant variables in an actual site.     

Complex system reliability modelling with Dynamic Object Oriented Bayesian Networks (DOOBN)

Nowadays, the complex manufacturing processes have to be dynamically modelled and controlled to optimise the diagnosis and the maintenance policies. This article presents a methodology that will help developing Dynamic Object Oriented Bayesian Networks (DOOBNs) to formalise such complex dynamic models. The goal is to have a general reliability evaluation of a manufacturing process, from its implementation to its operating phase. The added value of this formalisation methodology consists in using the a priori knowledge of both the system's functioning and malfunctioning. Networks are built on principles of adaptability and integrate uncertainties on the relationships between causes and effects. Thus, the purpose is to evaluate, in terms of reliability, the impact of several decisions on the maintenance of the system. This methodology has been tested, in an industrial context, to model the reliability of a water (immersion) heater system.     

Dynamic modeling of the tradeoff between productivity and safety in critical engineering systems

Short-term tradeoffs between productivity and safety often exist in the operation of critical facilities such as nuclear power plants, offshore oil platforms, or simply individual cars. For example, interruption of operations for maintenance on demand can decrease short-term productivity but may be needed to ensure safety. Operations are interrupted for several reasons: scheduled maintenance, maintenance on demand, response to warnings, subsystem failure, or a catastrophic accident. The choice of operational procedures (e.g. timing and extent of scheduled maintenance) generally affects the probabilities of both production interruptions and catastrophic failures. In this paper, we present and illustrate a dynamic probabilistic model designed to describe the long-term evolution of such a system through the different phases of operation, shutdown, and possibly accident. The model's parameters represent explicitly the effects of different components' performance on the system's safety and reliability through an engineering probabilistic risk assessment (PRA). In addition to PRA, a Markov model is used to track the evolution of the system and its components through different performance phases. The model parameters are then linked to different operations strategies, to allow computation of the effects of each management strategy on the system's long-term productivity and safety. Decision analysis is then used to support the management of the short-term trade-offs between productivity and safety in order to maximize long-term performance. The value function is that of plant managers, within the constraints set by local utility commissions and national (e.g. energy) agencies. This model is illustrated by the case of outages (planned and unplanned) in nuclear power plants to show how it can be used to guide policy decisions regarding outage frequency and plant lifetime, and more specifically, the choice of a reactor tripping policy as a function of the state of the emergency core cooling subsystem.     

Safety analysis in process facilities: Comparison of fault tree and Bayesian network approaches

Safety analysis in gas process facilities is necessary to prevent unwanted events that may cause catastrophic accidents. Accident scenario analysis with probability updating is the key to dynamic safety analysis. Although conventional failure assessment techniques such as fault tree (FT) have been used effectively for this purpose, they suffer severe limitations of static structure and uncertainty handling, which are of great significance in process safety analysis. Bayesian network (BN) is an alternative technique with ample potential for application in safety analysis. BNs have a strong similarity to FTs in many respects; however, the distinct advantages making them more suitable than FTs are their ability in explicitly representing the dependencies of events, updating probabilities, and coping with uncertainties. The objective of this paper is to demonstrate the application of BNs in safety analysis of process systems. The first part of the paper shows those modeling aspects that are common between FT and BN, giving preference to BN due to its ability to update probabilities. The second part is devoted to various modeling features of BN, helping to incorporate multi-state variables, dependent failures, functional uncertainty, and expert opinion which are frequently encountered in safety analysis, but cannot be considered by FT. The paper concludes that BN is a superior technique in safety analysis because of its flexible structure, allowing it to fit a wide variety of accident scenarios.     

Reliability graph with general gates: an intuitive and practical method for system reliability analysis

In this paper, we propose an intuitive and practical method for system reliability analysis. Among the existing methods for system reliability analysis, reliability graph is particularly attractive due to its intuitiveness, even though it is not widely used for system reliability analysis. We provide an explanation for why it is not widely used, and propose a new method, named reliability graph with general gates, which is an extension of the conventional reliability graph. An evaluation method utilizing existing commercial or free software tools are also provided. We conclude that the proposed method is intuitive, easy-to-use, and practical while as powerful as fault tree analysis, which is currently the most widely used method for system reliability analysis.     

Crash risk analysis for Shanghai urban expressways: A Bayesian semi-parametric modeling approach

Urban expressway systems have been developed rapidly in recent years in China; it has become one key part of the city roadway networks as carrying large traffic volume and providing high traveling speed. Along with the increase of traffic volume, traffic safety has become a major issue for Chinese urban expressways due to the frequent crash occurrence and the non-recurrent congestions caused by them. For the purpose of unveiling crash occurrence mechanisms and further developing Active Traffic Management (ATM) control strategies to improve traffic safety, this study developed disaggregate crash risk analysis models with loop detector traffic data and historical crash data. Bayesian random effects logistic regression models were utilized as it can account for the unobserved heterogeneity among crashes. However, previous crash risk analysis studies formulated random effects distributions in a parametric approach, which assigned them to follow normal distributions. Due to the limited information known about random effects distributions, subjective parametric setting may be incorrect. In order to construct more flexible and robust random effects to capture the unobserved heterogeneity, Bayesian semi-parametric inference technique was introduced to crash risk analysis in this study. Models with both inference techniques were developed for total crashes; semi-parametric models were proved to provide substantial better model goodness-of-fit, while the two models shared consistent coefficient estimations. Later on, Bayesian semi-parametric random effects logistic regression models were developed for weekday peak hour crashes, weekday non-peak hour crashes, and weekend non-peak hour crashes to investigate different crash occurrence scenarios. Significant factors that affect crash risk have been revealed and crash mechanisms have been concluded.     

Building prior distributions to support Bayesian reliability growth modelling using expert judgement

A sound methodology for the elicitation of subjective expert judgement is a pre-requisite for specifying prior distributions for the parameters of reliability growth models. In this paper, we describe an elicitation process that is developed to ensure valid data are collected by suggesting how possible bias might be identified and managed. As well as discussing the theory underpinning the elicitation process, the paper gives practical guidance concerning its implementation during reliability growth testing. The collection of subjective data using the proposed elicitation process is embedded within a Bayesian reliability growth modelling framework and reflections upon its practical use are described.     

Modelling the reliability of search and rescue operations with Bayesian Belief Networks

This paper uses a Bayesian Belief Networks (BBN) methodology to model the reliability of Search And Rescue (SAR) operations within UK Coastguard (Maritime Rescue) coordination centres. This is an extension of earlier work, which investigated the rationale of the government's decision to close a number of coordination centres. The previous study made use of secondary data sources and employed a binary logistic regression methodology to support the analysis. This study focused on the collection of primary data through a structured elicitation process, which resulted in the construction of a BBN. The main findings of the study are that statistical analysis of secondary data can be used to complement BBNs. The former provided a more objective assessment of associations between variables, but was restricted in the level of detail that could be explicitly expressed within the model due to a lack of available data. The latter method provided a much more detailed model, but the validity of the numeric assessments was more questionable. Each method can be used to inform and defend the development of the other. The paper describes in detail the elicitation process employed to construct the BBN and reflects on the potential for bias.     

Prediction of road accidents: A Bayesian hierarchical approach

In this paper a novel methodology for the prediction of the occurrence of road accidents is presented. The methodology utilizes a combination of three statistical methods: (1) gamma-updating of the occurrence rates of injury accidents and injured road users, (2) hierarchical multivariate Poisson-lognormal regression analysis taking into account correlations amongst multiple dependent model response variables and effects of discrete accident count data e.g. over-dispersion, and (3) Bayesian inference algorithms, which are applied by means of data mining techniques supported by Bayesian Probabilistic Networks in order to represent non-linearity between risk indicating and model response variables, as well as different types of uncertainties which might be present in the development of the specific models.     

Bayesian updating of reliability of civil infrastructure facilities based on condition-state data and fault-tree model

This paper considers a difficult but practical circumstance of civil infrastructure management—deterioration/failure data of the infrastructure system are absent while only condition-state data of its components are available. The goal is to develop a framework for estimating time-varying reliabilities of civil infrastructure facilities under such a circumstance. A novel method of analyzing time-varying condition-state data that only reports operational/non-operational status of the components is proposed to update the reliabilities of civil infrastructure facilities. The proposed method assumes that the degradation arrivals can be modeled as a Poisson process with unknown time-varying arrival rate and damage impact and that the target system can be represented as a fault-tree model. To accommodate large uncertainties, a Bayesian algorithm is proposed, and the reliability of the infrastructure system can be quickly updated based on the condition-state data. Use of the new method is demonstrated with a real-world example of hydraulic spillway gate system.