Bayesian inference with overlapping data: Reliability estimation of multi-state on-demand continuous life metric systems with uncertain evidence
Introduction
Many system reliability analysis methodologies focus on system failure logic (such as that represented in reliability block diagrams or fault-trees) to express system failure probability in terms of subordinate component failure probabilities. Such methodologies are explained in virtually all texts on systems reliability: for example Hoyland and Rausand [1], Hiromitsu and Henley [2] and the Nuclear Regulatory Committee (NRC) Probability Risk Assessment Guide [3]. These techniques promote system reliability as a function of component reliabilities, which in turn direct focus to reliability testing and data collection at component level. Component data are then used to develop component level reliability estimates, with which system level reliability values are calculated. This approach automatically precludes useful system and sub-system data, which is referred to here as higher level data as it appears ‘higher’ in many visualization methodologies such as fault trees.
Generally, systems can be of two types: ‘demand-based’ or ‘continuously operating’ (noting that mixed types are also possible). Demand-based or on-demand systems are subjected to discrete demands or trials and respond by operating (or existing) within certain discrete states. The simplest of on-demand systems are ‘binary-state’ systems, where components are either in the ‘functional’ or ‘failed’ states. ‘Multi-state’ systems involve components that can be classified by order of severity in various degraded states ranging from ‘functional’ to ‘failed.’ Methods for reliability analysis of such systems can be found in the literature (see for example [4] and [5]). Systems based on continuous life metrics are those whose failure probability is an explicit function of an independent life variable such as time or distance.
Jackson and Mosleh [6], [7], [8], [9] developed Bayesian methodologies for incorporating higher level data in on-demand systems and continuous life metric systems. A system often involves multiple sensors (sensors are broadly defined as monitoring points through data gathering devices; human, machine, or otherwise). This means that reliability data sets are often overlapping in nature. Sets of overlapping data meet the following criteria:
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Simultaneity – the data sets are drawn from observations or demands that occur at the same time; and
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Correspondence – the data sets result from the same system or process.
Initially, only approximate Bayesian methodologies were developed for Bayesian analysis of higher-level data [10], [11], [12], [13]. These methods have been generalized further, particularly for continuous multi-state systems in [14], [15] that place bounds on parametermoments, noting that the underlyhing mathematics remains particularly onerous. Multiple methodologies that can incorporate higher-level non-overlapping data have since been developed and are discussed in detail below [16], [17], [18]. Jackson and Mosleh discuss the error that is introduced when incorrectly analyzing overlapping data by constraining it to be considered as non-overlapping [6], [7], [8], [9]. Graves et al. [19] proposed a method that incorporates overlapping data for multi-state on-demand systems. The methodology considers each demand in isolation (i.e. sensor states for each demand must be known), but cannot incorporate data that summarizes multiple demands on the system. The methodology proposed by Jackson and Mosleh [9] is completely generalized and able to consider multiple demands for multi-state systems.
This paper develops fully Bayesian methodologies for incorporating uncertain overlapping higher level data using techniques discussed above. In the case of on-demand systems, uncertain data manifests itself in terms of the uncertainty in number of observed failures from demands. For continuously operating systems, it is manifested in terms of the uncertainty in the time at which failure is detected. The latter scenario for continuously operating systems, a likelihood function that is not only computationally simpler than that proposed by Jackson and Mosleh in [7], but correctly replicates reality as all detection times have uncertainty expressed as the accuracy of the timing devices. This allows utility in applications where system reliability is periodically checked over specific time intervals (such as the case of maintenance schedules) or at specific points of operations (such as the case of sequential systems checks of space-craft after completion of various mission stages).
Section snippets
System reliability analysis and sensors
System reliability analysis generally revolves around understanding constituent component reliability characteristics and relating these characteristics to system level performance via some form of system logic. In any test or analysis activity, data is gathered through sensors which will be placed at various ‘places’ within this system logical framework. Even the simplest, single component reliability test involves a sensor – which could be the direct observation of a reliability or test
Bayesian analysis of higher level data for on-demand systems
Fully Bayesian techniques have been developed by Johnson et al. [16], Hamada et al. [17] and Graves et al. [18] for non-overlapping higher level data in on demand systems. The Graves et al. method incorporates multi-state systems and uses Dirichlet prior distributions at component and system levels to incorporate test data of the form of observed component states out of a given number of trials to generate posterior component distributions. It generalizes the other two methods that are strictly
Bayesian analysis of higher level data: continuous time-based systems
The generalization of continuous, time-based Bayesian analysis methodologies is considerably more involved than that for on-demand systems. Where the likelihood function of overlapping data for on-demand system primarily encompasses determining a larger set of plausible state vector combinations to incorporate uncertain evidence, continuous time-based systems require the likelihood function itself to be modified to incorporate sensor time to failure detection inaccuracies. However, this
Continuous time-based systems and measurement inaccuracies
Inaccuracies associated with time based measurement of reliability data are often associated with observation intervals. There are two main scenarios where these inaccuracies are prevalent: routine checks by operators and measurement by digital time-pieces.
In the first scenario where systems are routinely physically checked by operators, there are time intervals where the system remains unobserved. Typically, operators will check several sensors, components or sub-systems in such a relatively
Treatment of uncertain data
The analysis of uncertain data, especially in a Bayesian context, has been the subject of significant literature (for example see [33]). This paper does not seek to replicate or extend previous work. However, the methodology outlined above inherently involves uncertain data analysis, and therefore warrants an examination with respect to its relation to existing frameworks.
Uncertain data in a Bayesian context conventionally refers to error associated with data collection. In any case, a system
Discussion
This paper sought to develop and examine the likelihood functions developed by Jackson and Mosleh [6], [7], [8], [9] (for the purpose of analysis of sets of overlapping systemic data) in the context of uncertain evidence.
For continuous (generally time-based) systems, a particular case of uncertain, ‘interval-based’ evidences was examined. This case is the most prevalent form of time-based evidence in the context of reliability engineering as it applies to both routine human inspections of
References (41)
- et al.
Bayesian inference with overlapping data for systems with continuous life metrics
Reliab Eng Syst Saf
(2012) - et al.
A fully Bayesian approach for combining multilevel failure information in fault tree quantification and optimal follow-on resource allocation
Reliab Eng Syst Saf
(2004) - et al.
A fully Bayesian approach for combining multi-level information in multi-state fault tree quantification
Reliab Eng Syst Saf
(2007) - et al.
Using simultaneous higher-level and partial lower-level data in reliability assessments
Reliab Eng Syst Saf
(2008) - et al.
Pitfalls in risk calculations
Reliab Eng
(1981) - et al.
Foundations of probabilistic inference with uncertain evidence
Int J Approx Reason
(2005) - et al.
Bayesian analysis with consideration of data uncertainty in a specific scenario
Reliab Eng Syst Saf
(2003) - et al.
System reliability theory: models, statistical methods and applications
(2008) - et al.
Probablistic risk assessment and management for engineers and scientists
(2000) - Hickman, JW. PRA procedures guide: a guide to the performance of probabilistic risk assessments for nuclear power...
Multi state system reliability: assessment, optimization, and applications
Interval-valued reliability analysis of multi-state systems
IEEE Trans Reliab,
Downwards inference: Bayesian analysis of overlapping higher-level data sets of complex binary-state on-demand systems
Proc Inst Mech Eng Part O: J Risk Reliab
Bayesian inference with overlapping data: methodology for reliability estimation of multi-state on-demand systems
Proc Inst Mech Eng Part O: J Risk Reliab
Incorporating component and system test data into the same assessment: a Bayesian approach
Oper Res
Bayesian reliability analysis of series systems of binomial subsystems and components
Technometrics
Bayesian reliability analysis of complex series/parallel systems of binomial subsystems and components
Technometrics
Bayesian estimation of system reliability
Scand J Stat
Multistate systems reliability theory with applications
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Director, Acuitas Reliability Pty Ltd.