Bayesian inference with overlapping data: Reliability estimation of multi-state on-demand continuous life metric systems with uncertain evidence

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Highlights

  • Application of Bayesian techniques for the analysis of overlapping multi-state data for both discrete and continuous systems.

  • Development of methodology for incorporation of uncertain data.

  • Modification of likelihood function for time based continuous systems resulting in substantial mathematical simplification.

  • Discussion of uncertain data as it applies to extant forms of analysis and observation error.

Abstract

A Bayesian system reliability estimation methodology for multiple overlapping uncertain data sets within complex multi-state on-demand and continuous life metric systems is presented in this paper. Data sets are overlapping if they are drawn from the same process at the same time, with reliability data from sensors attached to a system at different functional and physical levels being a prime example. Treating overlapping data as non-overlapping loses or incorrectly infers information on system reliability. Methodologies for system reliability analysis of certain overlapping data sets have previously been proposed. These methodologies, and the approach presented in this paper, are able to incorporate overlapping uncertain evidence from systems with a detailed understanding of the system logic represented using fault-trees, reliability block diagrams or equivalent representations. The method presented here builds on approaches that have already been developed by the authors that allow incorporation of exact or certain data sets.

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:

  • 1.

    Simultaneity – the data sets are drawn from observations or demands that occur at the same time; and

  • 2.

    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

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