an integrated procedure for bayesian reliability inference...

Research ArticleAn Integrated Procedure for Bayesian Reliability InferenceUsing MCMC

Jing Lin12

1 Division of Operation and Maintenance Engineering Lulea University of Technology 97187 Lulea Sweden2 Lulea Railway Research Centre (JVTC) 97187 Lulea Sweden

Correspondence should be addressed to Jing Lin janetlinltuse

Received 5 August 2013 Revised 27 November 2013 Accepted 28 November 2013 Published 14 January 2014

Academic Editor Luigi Portinale

Copyright copy 2014 Jing Lin This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The recent proliferation of Markov chain Monte Carlo (MCMC) approaches has led to the use of the Bayesian inference in a widevariety of fields To facilitateMCMC applications this paper proposes an integrated procedure for Bayesian inference usingMCMCmethods from a reliability perspectiveThe goal is to build a framework for related academic research and engineering applicationsto implementmodern computational-basedBayesian approaches especially for reliability inferencesTheprocedure developed hereis a continuous improvement process with four stages (Plan Do Study and Action) and 11 steps including (1) data preparation (2)prior inspection and integration (3) prior selection (4)model selection (5) posterior sampling (6)MCMCconvergence diagnostic(7) Monte Carlo error diagnostic (8) model improvement (9) model comparison (10) inference making (11) data updating andinference improvement The paper illustrates the proposed procedure using a case study

1 Introduction

The recent proliferation of Markov Chain Monte Carlo(MCMC) approaches has led to the use of the Bayesianinference in a wide variety of fields including behaviouralscience finance human health process control ecologicalrisk assessment and risk assessment of engineered systems[1] Discussions of MCMC-related methodologies and theirapplications in Bayesian Statistics now appear throughoutthe literature [2 3] For the most part studies in reliabilityanalysis focus on the following topics and their cross-applications (1) hierarchical reliability models [4ndash7] (2)complex system reliability analysis [8ndash10] (3) faulty treeanalysis [11 12] (4) accelerated failure models [13ndash17] (5)reliability growth models [18 19] (6) masked system relia-bility [20] (7) software reliability engineering [21 22] (8)reliability benchmark problems [23 24] However most ofthe literature emphasizes themodelrsquos development no studiesoffer a full framework to accommodate academic researchand engineering applications seeking to implement moderncomputational-based Bayesian approaches especially in thearea of reliability

To fill the gap and to facilitateMCMC applications from areliability perspective this paper proposes an integrated pro-cedure for the Bayesian inferenceThe remainder of the paperis organized as follows Section 2 outlines the integratedprocedure this comprises a continuous improvement processincluding four stages and 11 sequential steps Sections 3 to 8discuss the procedure focusing on (1) prior elicitation (2)model construction (3) posterior sampling (4) MCMC con-vergence diagnostic (5) Monte Carlo error diagnostic (6)model comparison Section 9 gives examples and discusseshow to use the procedure Finally Section 10 offers conclu-sions

2 Description of Procedure

The proposed procedure uses the Bayesian reliability infere-nce to determine system (or unit) reliability and failure dis-tribution to support the optimisation of maintenance strate-gies and so forth

The general procedure begins with the collection ofreliability data (see Figure 1) These are the observed val-ues of a physical process such as various ldquolifetime datardquo

Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2014 Article ID 264920 16 pageshttpdxdoiorg1011552014264920

2 Journal of Quality and Reliability Engineering

(6) MCMCconvergencediagnostic

(7) Monte Carloerror diagnosticYes Accept

No

No

Posterior results

(2) Priorsrsquo inspectionand integration

(3) Prior selection

(4) Model selection

(5) Posterior sampling (8) Model improvement

(9) Model comparison

Bayesian modelaverage

MCMC modelaverage

Reliability data andso forth

Prior knowledge

Historyinformation

Currentdata set

(11) Data updating andinference improvement

Do

Plan

(10) Inference making

Action

(1) Data preparation

Study

Figure 1 An integrated procedure for Bayesian reliability inference via MCMC

The data may be subject to uncertainties such as imprecisemeasurement censoring truncated information and inter-pretation errors Reliability data are found in the ldquocurrent datasetrdquo they contain original data and include the evaluationmanipulation andor organisation of data samples At ahigher level in the collection of data a wide variety of ldquohis-torical informationrdquo can be obtained including the results ofinspecting and integrating this ldquoinformationrdquo thereby addingto ldquoprior knowledgerdquo The final level is reliability inferencewhich is the process of making a conclusion based on ldquopost-erior resultsrdquo

Using the definitions shown in Figure 1 we propose anintegrated procedure which constructs a full framework forthe standardized process of Bayesian reliability inference Asshown in Figure 1 the procedure is composed of a continuousimprovement process including four stages (Plan Do Studyand Action) which will be discussed later in this section and11 sequential steps (1) data preparation (2) prior inspectionand integration (3) prior selection (4) model selection (5)posterior sampling (6) MCMC convergence diagnostic (7)Monte Carlo error diagnostic (8) model improvement (9)

model comparison (10) inference making (11) data updatingand inference improvement

Step 1 (data preparation) The original data sets for ldquohistoryinformationrdquo and ldquocurrent datardquo related to reliability studiesneed to be acquired evaluated andmerged In this way ldquohis-tory informationrdquo can be transferred to ldquoprior knowledgerdquoand ldquocurrent datardquo can become ldquoreliability datardquo in later steps

Step 2 (prior inspection and integration) During this stepldquoprior knowledgerdquo receives a second andmore extensive treat-ment including a reliability consistency check a credencetest and a multisource integration This step improves priorreliability data

Step 3 (prior selection) This step uses the results achievedin Step 2 to determine the modelrsquos form and parameters forinstance selecting informative or noninformative priors orunknown parameters and their distributed forms

Step 4 (model selection) This step determines a reliabilitymodel (parametric or nonparametric) selecting from 119899

Journal of Quality and Reliability Engineering 3

candidates for the studied systemunits It considers bothldquoreliability datardquo and the inspection integration and selectionof priors to implement the 119894th (119894 = 1 119894 + 1 119899)

Step 5 (posterior sampling) In this step we determine asamplingmethod (eg Gibbs samplingMetropolis-Hastingssampling etc) to implement MCMC simulation for themodelrsquos posterior calculations

Step 6 (MCMC convergence diagnostic) In this step wecheck whether the Markov chains have reached convergenceIf they have we move on to the next step if they have notwe return to Step 5 and redetermine the iteration times ofposterior sampling or rechoose the sampling methods if theresults still cannot be satisfied we return to Steps 3 and 4 andredetermine the prior selection and model selection

Step 7 (Monte Carlo error diagnostic) We need to decideif the Monte Carlo error is small enough to be accepted inthis step As discussed in Step 6 if it is accepted we go onto the next step if it is not we return to Step 5 and redecidethe iteration times of the posterior sampling or rechoose thesampling methods if the results still cannot be accepted wego back to Steps 3 and 4 and recalculate the prior selectionand model selection

Step 8 (model improvement) Here we choose the 119894 + 1thcandidate model and restart from Step 4

Step 9 (model comparison) After implementing 119899 candidatemodels we need to (1) compare the posterior results todetermine the most suitable model or (2) adopt the averageposterior estimations (using the Bayesian model average orthe MCMCmodel average) as the final results

Step 10 (inference making) After achieving the posteriorresults in Step 9we can performBayesian reliability inferenceto determine system (or unit) reliability find the failure dis-tribution optimise maintenance strategies and so forth

Step 11 (data updating and inference improvement) Alongwith the passage of time new ldquocurrent datardquo can be obtainedrelegating ldquopreviousrdquo inference results to ldquohistorical datardquo Byupdating ldquoreliability datardquo and ldquoprior knowledgerdquo and res-tarting at Step 1 we can improve the reliability inference

In summary by using this step-by-step method we cancreate a continuous improvement process for the Bayesianreliability inference

Note that Steps 1 2 and 3 are assigned to the ldquoPlanrdquostage when data for MCMC implementation are prepared Inaddition a part of Steps 1 2 and 3 refers to the elicitation ofprior knowledge Steps 4 and 5 are both assigned to the ldquoDordquostage where the MCMC sampling is carried out Steps 6 to 9are treated as the ldquoStudyrdquo stage in these steps the samplingresults are checked and compared in addition knowledge isaccumulated and improved upon by implementing variouscandidate reliability models The ldquoActionrdquo stage consists ofSteps 10 and 11 at this point a continuously improvedloop can be obtained In other words by implementing the

step-by-step procedure we can accumulate and graduallyupdate prior knowledge Equally posterior results will beimproved upon and become increasingly robust therebyimproving the accuracy of the inference results

Also note that this paper will focus on six steps and theirrelationship to MCMC inference implementation (1) priorelicitation (2) model construction (3) posterior sampling(4) MCMC convergence diagnostic (5) Monte Carlo errordiagnostic (6) model comparison

3 Elicitation of Prior KnowledgeIn the proposed procedure the elicitation of prior knowledgeplays a crucial role It is related to Steps 1 2 and 3 and is partof the Plan Stage as shown in Figure 1

In practice prior information is derived from a varietyof data sources and is also considered ldquohistorical datardquo (orldquoexperience datardquo) Those data taking various forms requirevarious processing methods Although in the first step ldquohis-torical informationrdquo can be transferred to ldquoprior knowledgerdquothis is not enough Credible prior information and properforms of these data are necessary to compute the modelrsquosposterior probabilities especially in the case of a smallsample set Meanwhile either noncredible or improper priordata may cause instability in the estimation of the modelrsquosprobabilities or lead to convergence problems in MCMCimplementationThis section will discuss some relevant priorelicitation problems in Bayesian reliability inference namelyincluding acquiring priors performing a consistency checkand credence test fusing multisource priors and selectingwhich priors to use in MCMC implementation

31 Acquisition of Prior Knowledge In Bayesian reliabilityanalysis prior knowledge comes from a wide range of histor-ical information As shown in Figure 2 data sources include(1) engineering design data (2) component test data (3)system test data (4) operational data from similar systems (5)field data in various environments (6) computer simulations(7) related standards and operation manuals (8) experiencedata from similar systems (9) expert judgment and personalexperience Of these the first seven yield objective prior dataand the last two provide subjective prior data

Prior data also take a variety of forms including reliabilitydata the distribution of reliability parameters momentsconfidence intervals quantiles and upper and lower limits

32 Inspection of Priors In Bayesian analysis different priorinformation will lead to similar results when the data sampleis sufficiently large While the selection of priors and theirform has little influence on posterior inferences in practiceparticularly with a small data sample we know that someprior information is associated with the current observedreliability data However we are not sure whether the priordistributions are the same as the posterior distributions Inother words we cannot confirm that all posterior distribu-tions converge and are consistent (a consistency check issue)Even if they pass a consistency check we can only say thatthey are consistent under a certain specified confidence inter-val Therefore an important prerequisite for applying any


Obj

ectiv

e

Engineering design

Component test

Systems test

Operational data from similar systems

Field data under various environments

Computer simulations

Related standards and operation manual

Experience data from similar systems

Expert judgment and personal experiencefrom studied system

Information

Subj

ectiv

e

Knowledge

Parametersrsquo distributions

Parametersrsquo moments

Parametersrsquo confidence intervals

Parametersrsquo quantiles

Parametersrsquo upper limits

Parametersrsquo lower limits

Transfer

Reliability data

Figure 2 Data source transfer from historical to prior knowledge

prior information is to confirm its credibility by performinga consistency check and credence test

As noted by Li [25] the consistency check of prior andposterior distributions was first studied from a statisticalviewpoint by Walker [26] Under specified conditions pos-terior distributions are not only consistent with those of thepriors but they have an asymptotic normality which couldsimplify their calculation Li [25] also notes that studies onthe consistency check of priors have focused on checking themoments and confidence intervals of reliability parametersas well as historical data A number of checking method-ologies have been developed including robustness analysissignificance test Smirnov test rank-sum test and mood testMore studies have been reviewed by Ghosal [27] and Choiet al [28]

The credibility of prior information can be viewed as theprobability that it and the collected data come from the samepopulation Li [25] lists the following methods to perform acredence test frequency method bootstrap method rank-sum text and so forth However in the case of a smallsample or simulated data the above methods are not suitablebecause even if data pass the credence test selecting differentpriors will lead to different results We therefore suggesta comprehensive use of the above methods to ensure thesuperiority of Bayesian inference

33 Fusion of Prior Information Due to the complexity of thesystem not to mention the diversification of test equipmentand methodologies prior information can come from manysources As all priors can pass a consistency check an inte-grated fusion estimation based on the smoothness of cred-ibility is sometimes necessary In such situations a commonchoice is parallel fusion estimation or serial fusion estimationachieved by determining the reliability parametersrsquo weightedcredibility [25] However as the credibility computationcan be difficult other methods to fuse the priors may becalled for In the area of reliability related research studiesinclude the following Savchuk andMartz [29] develop Bayes

estimators for the true binomial survival probability whenthere are multiple sources of prior information Ren et al[30] adopt Kullback information as the distance measurebetween different prior information and fusion distributionsminimizing the sum to get the combined prior distributionlooking at aerospace propulsion as a case study Liu et al [31]discuss a similar fusion problem in a complex system andsuggest [32] a fusion approach based on expert experiencewith the analytic hierarchy process (AHP) Fang [33] pro-poses using multisource information fusion techniques withFuzzy-Bayesian for reliability assessment Zhou et al [34]propose a Bayes fusion approach for assessment of spaceflightproducts integrating degradation data and field lifetime datawith Fisher information and the Weiner process In generalthe most important thing for multisource integration is todetermine the weights of the different priors

34 Selection of Priors Based onMCMC In Bayesian reliabil-ity inference two kinds of priors are very useful the conjugateprior and the noninformative prior To apply MCMC meth-ods however the ldquolog-concave priorrdquo is recommended

The conjugate prior family is very popular because it isconvenient for mathematical calculation The concept alongwith the term ldquoconjugate priorrdquo was introduced by Howardand Robert [35] in their work on Bayesian decision theory Ifthe posterior distributions are in the same family as the priordistributions the prior and posterior distributions are calledconjugate distributions and the prior is called a conjugateprior For instance the Gaussian family is a conjugate of itself(or a self-conjugate) with respect to a Gaussian likelihoodfunction if the likelihood function is Gaussian choosing aGaussian prior distribution over the mean distribution willensure that the posterior distribution is also Gaussian Thismeans that the Gaussian distribution is a conjugate priorfor the likelihood function which is also Gaussian Otherexamples include the following the conjugate distributionof a normal distribution is a normal or inverse-normaldistribution the Poisson and the exponential distributionrsquos


conjugate both have aGammadistribution while theGammadistribution is a self-conjugate the binomial and the negativebinomial distributionrsquos conjugate both have a Beta distribu-tion the polynomial distributionrsquos conjugate is a Dirichletdistribution

Noninformative prior refers to a prior for which we onlyknow certain parametersrsquo value ranges or their importancefor example there may be a uniform distribution A non-informative prior can also be called a vague prior flat priordiffuse prior ignorance prior and so forth There are manydifferent ways to determine the distribution of a nonin-formative prior including Bayes hypothesis Jeffreyrsquos rulereference prior inverse reference prior probability-matchingprior maximum entropy prior relative likelihood approachcumulative distribution function Monte Carlo methodbootstrap method random weighting simulation methodHaar invariant measurement Laplace prior Lindley rulegeneralized maximum entropy principle and the use ofmarginal distributions From another perspective the typesof prior distribution also include informative prior hierarchi-cal prior Power prior and nonparameter prior processes

At this point there are no set rules for selecting priordistributions Regardless of the manner used to determinea priorrsquos distribution the selected prior should be bothreasonable and convenient for calculation Of the above theconjugate prior is a common choice To facilitate the calcu-lation of MCMC especially for adaptive rejection samplingand Gibbs sampling a popular choice is log-concave priordistribution Log-concave prior distribution refers to a priordistribution in which the natural logarithm is concave thatis the second derivative is nonpositive Common logarithmicconcavity prior distributions include the normal distribu-tion family logistic distribution Studentrsquos-119905 distribution theexponential distribution family the uniform distribution ona finite interval greater than the gamma distribution witha shape parameter greater than 1 and Beta distributionwith a value interval (0 1) As logarithmic concavity priordistributions are very flexible this paper recommends theiruse in reliability studies

4 Model Construction

To apply MCMC methods we divide the reliability modelsinto four categories parametric semiparametric frailty andother nontraditional reliability models

Parametric modelling offers straightforward modellingand analysis techniques Common choices include Bayesianexponential model Bayesian Weibull model Bayesianextreme value model Bayesian log-normal model andGammamodel Lin et al [36] present a reliability study usingthe Bayesian parametric framework to explore the impact ofa railway train wheelrsquos installed position on its service lifetimeand to predict its reliability characteristics They apply aMCMC computational scheme to obtain the parametersrsquoposterior distributions Besides the hierarchical reliabilitymodels mentioned above other parametric models includeBayesian accelerated failure models (AFM) Bayesianreliability growth models and Bayesian faulty tree analysis(FTA)

Semiparametric reliability models have become quitecommon and are well accepted in practice since they offer amore general modelling strategy with fewer assumptions Inthesemodels the failure rate is described in a semiparametricform or the priors are developed by a stochastic processWithrespect to the semiparametric failure rate Lin and Asplund[37] adopt the piecewise constant hazardmodel to analyze thedistribution of the locomotive wheelsrsquo lifetime The appliedhazard function is sometimes called a piecewise exponentialmodel it is convenient because it can accommodate variousshapes of the baseline hazard over a number of intervals In astudy of the processes of priors Ibrahim et al [38] examinethe gamma process beta process correlated prior processesand the Dirichlet process separately using an MCMC com-putational scheme By introducing the gamma process of thepriorrsquos increment Lin et al [39] propose its reliability whenapplied to the Gibbs sampling scheme

In reliability inference most studies are implementedunder the assumption that individual lifetimes are indepen-dent and identically distributed (iid) However Cox pro-portional hazard (CPH) models can sometimes not be usedbecause of the dependence of data within a group Take trainwheels as an example because they have the same operatingconditions the wheels mounted on a particular locomo-tive may be dependent In a different context some datamay come from multiple records of wheels installed in thesame position but on other locomotives Modelling depen-dence has received considerable attention especially in caseswhere the datasets may come from related subjects of thesame group [40 41] A key development in modelling suchdata is to build frailtymodels in which the data are condition-ally independent When frailties are considered the depen-dence within subgroups can be considered an unknownand unobservable risk factor (or explanatory variable) ofthe hazard function In a recent reliability study Lin andAsplund [37] consider a gamma shared frailty to explore theunobserved covariatesrsquo influence on the wheels on the samelocomotive

Some nontraditional Bayesian reliability models are bothinteresting and helpful For instance Lin et al [10] point outthat in traditional methods of reliability analysis a complexsystem is often considered to be composed of several subsys-tems in series The failure of any subsystem is usually consid-ered to lead to the failure of the entire system However somesubsystemsrsquo lifetimes are long enough or they never fail dur-ing the life cycle of the entire system In addition such subsys-temsrsquo lifetimes will not be influenced equally under differentcircumstances For example the lifetimes of some screws willfar exceed the lifetime of the compressor in which they areplaced However the failure of the compressorrsquos gears maydirectly lead to its failure In practice such interferences willaffect the modelrsquos accuracy but are seldom considered intraditional analysis To address these shortcomings Lin etal [10] present a new approach to reliability analysis forcomplex systems in which a certain fraction of subsystemsis defined as a ldquocure fractionrdquo based on the consideration thatsuch subsystemsrsquo lifetimes are long enough or they never failduring the life cycle of the entire system this is called the curerate model


5 Posterior Sampling

To implement MCMC calculations Markov chains requirea stationary distribution There are many ways to constructthese chains During the last decade the following MonteCarlo (MC) based sampling methods for evaluating high-dimensional posterior integrals have been developed MCimportance sampling Metropolis-Hastings sampling Gibbssampling and other hybrid algorithms In this section weintroduce two common samplings (1) Metropolis-Hastingssampling the best known MCMC sampling algorithm and(2) Gibbs sampling the most popular MCMC samplingalgorithm in the Bayesian computation literature which isactually a special case of Metropolis-Hastings sampling

51 Metropolis-Hastings Sampling Metropolis-Hastings sa-mpling is a well-known MCMC sampling algorithm whichcomes from importance sampling It was first developed byMetropolis et al [42] and later generalized by Hastings [43]Tierney [44] gives a comprehensive theoretical exposition ofthe algorithm Chib and Greenberg [45] provide an excellenttutorial on it

Suppose that we need to create a sample using theprobability density function119901(120579) Let119870 be a regular constantthis is a complicated calculation (eg a regular factor inBayesian analysis) and is normally an unknown parameterThen let 119901(120579) = ℎ(120579)119870 Metropolis sampling from 119901(120579) canbe described as follows

Step 1 Choose an arbitrary starting point 1205790 and set ℎ(120579

0) gt

0

Step 2 Generate a proposal distribution 119902(1205791 1205792) which

represents the probability for 1205792to be the next transfer

value as the current value is 1205791 The distribution 119902(120579

1 1205792) is

named as the candidate generating distribution This can-didate generating distribution is symmetric which meansthat 119902(120579

1 1205792) = 119902(120579

2 1205791) Now based on the current 120579

generate a candidate point 120579lowast from 119902(1205791 1205792)

Step 3 For the specified candidate point 120579lowast calculate thedensity ratio 120572 with 120579lowastand the current value 120579

119905minus1as follows

120572 (120579119905minus1 120579lowast) =

119901 (120579lowast)

119901 (120579119905minus1)

=

ℎ (120579lowast)

ℎ (120579119905minus1)

(1)

The ratio 120572 refers to the probability to accept 120579lowast where theconstant119870 can be neglected

Step 4 If 120579lowast increases the probability density so that 120572 gt 1then accept 120579lowast and let 120579

119905= 120579lowast Otherwise if 120572 lt 1 then let

120579119905= 120579119905minus1

and go to Step 2

The acceptance probability 120572 can be written as

120572 (120579119905minus1 120579lowast) = min(1

ℎ (120579lowast)

ℎ (120579119905minus1)

) (2)

Following the above steps generate a Markov chain withthe sampling points 120579

0 1205791 120579

119896 Thetransfer probability

from 120579119905to 120579119905+1

is related to 120579119905but not related to 120579

0 1205791 120579

119905minus1

After experiencing a sufficiently long burn-in period theMarkov chain reaches a steady state and the sampling points120579119905+1 120579

119905+119899from 119901(120579) are obtained

Metropolis-Hastings samplingwas promoted byHastings[43] The candidate generating distribution can adopt anyform and does not need to be symmetric In Metropolis-Hastings sampling the acceptance probability 120572 can bewritten as

120572 (120579119905minus1 120579lowast) = min(1

ℎ (120579lowast) 119902 (120579lowast 120579119905minus1)

ℎ (120579119905minus1) 119902 (120579119905minus1 120579lowast)

) (3)

In the above equation as 119902(120579lowast 120579119905minus1) = 119902(120579

119905minus1 120579lowast) and

Metropolis-Hastings sampling becomes Metropolis sam-plingTheMarkov transfer probability function can thereforebe

119901 (120579 120579lowast) =

119902 (120579119905minus1 120579lowast) ℎ (120579

lowast) gt ℎ (120579

119905minus1)

119902 (120579119905minus1 120579lowast)

ℎ (120579lowast)

ℎ (120579119905minus1)

ℎ (120579lowast) lt ℎ (120579

119905minus1)

(4)

From another perspective when using Metropolis-Hastingssampling say we need to generate the candidate point 120579lowast Inthis case generate an arbitrary 120583 from a uniform distribution119880(0 1) Set 120579

119905= 120579lowast if120583 le 120572 (120579

119905minus1 120579lowast) and 120579

119905= 120579119905minus1

otherwise

52 Gibbs Sampling Metropolis-Hastings sampling is conve-nient for lower-dimensional numerical computation How-ever if 120579 has a higher dimension it is not easy to choosean appropriate candidate generating distribution By usingGibbs sampling we only need to know the full condi-tional distribution Therefore it is more advantageous inhigh-dimensional numerical computation Gibbs sampling isessentially a special case of Metropolis-Hastings samplingas the acceptance probability equals one It is currently themost popular MCMC sampling algorithm in the Bayesianreliability inference literature Gibbs sampling is based on theideas of Grenander [46] but the formal term comes fromS Geman and D Geman [47] to analyze lattice in imageprocessing A landmarkwork forGibbs sampling in problemsof Bayesian inference is Gelfand and Smith [48] Gibbssampling is also called heat bath algorithm in statisticalphysics A similar idea data augmentation is introduced byTanner and Wong [49]

Gibbs sampling belongs to the Markov update mecha-nism and adopts the ideology of ldquodivide and conquerrdquo Itsupposes that all other parameters are fixed and knowninferring a set of parameters Let 120579

119894be a random parameter

or several parameters in the same group For the 119895th groupthe conditional distribution is 119891(120579

119895) To carry out Gibbs

sampling the basic scheme is as follows

Step 1 Choose an arbitrary starting point 120579(0) = (120579(0)1

120579(0)

119896)


Step 2 Generate 120579(1)1

from the conditional distribution119891(1205791|

120579(0)

2 120579

(0)

119896) and generate 120579(1)

2from the conditional distribu-

tion 119891(1205792| 120579(1)

1 120579(0)

3 120579

(0)

119896)

Step 3 Generate 120579(1)119895

from 119891(120579119895| 120579(1)

1 120579

(1)

119895minus1 120579(0)

119895+1 120579

(0)

119896)

Step 4 Generate 120579(1)119896

from 119891(120579119896| 120579(1)

1 120579(1)

2 120579

(1)

119896minus1)

As shown above one-step transitions from 120579(0) to 120579(1) =(120579(1)

1 120579

(1)

119896) where 120579(1)can be viewed as a one-time accom-

plishment of the Markov chain

Step 5 Go back to Step 2

After 119905 iterations 120579(119905) = (120579(119905)1 120579

(119905)

119896) can be obtained

and each component 120579(1) 120579(2) 120579(3) will be achieved TheMarkov transition probability function is

119901 (120579 120579lowast)

= 119891 (1205791| 1205792 120579

119896) 119891 (1205792| 120579lowast

1 1205793 120579

119896) sdot sdot sdot

119891 (120579119896| 120579lowast

1 120579

lowast

119896minus1)

(5)

Starting from different 120579(0) as 119905 rarr infin the marginal dis-tribution of 120579(119905) can be viewed as a stationary distributionbased on the theory of the ergodic average In this case theMarkov chain is seen as converging and the sampling pointsare seen as observations of the sample

For both methods it is not necessary to choose thecandidate generating distribution but it is necessary to dosampling from the conditional distribution There are manyother ways to do sampling from the conditional distri-bution including sample-importance resampling rejectionsampling and adaptive rejection sampling (ARS)

6 Markov Chain Monte CarloConvergence Diagnostic

Because of the Markov chainrsquos ergodic property all statisticsinferences are implemented under the assumption that theMarkov chain convergesTherefore theMarkovChainMonteCarlo convergence diagnostic is very importantWhen apply-ing MCMC for reliability inference if the iteration times aretoo small the Markov chain will not ldquoforgetrdquo the influence ofthe initial values if the iteration times are simply increased toa large number there will be insufficient scientific evidenceto support the results causing a waste of resources

Markov chain Monte Carlo convergence diagnostic isa hot topic for Bayesian researchers Efforts to determineMCMC convergence have been concentrated in two areasThe first is theoretical and the second is practical For theformer the Markov transition kernel of the chain is analyzedin an attempt to predetermine a number of iterations thatwill insure convergence in a total variation distance within aspecified tolerance of the true stationary distribution Relatedstudies include Polson [50] Roberts and Rosenthal [51] andMengersen and Tweedie [52]While approaches like these are

promising they typically involve sophisticated mathematicsas well as laborious calculations that must be repeated forevery model under consideration As for practical studiesmost research is focused on developing diagnostic toolsincluding Gelman and Rubin [53] Raftery and Lewis [54]Garren and Smith [55] and Roberts and Rosenthal [56]When these tools are used sometimes the bounds obtainedare quite loose even so they are considered both reasonableand feasible

At this point more than 15 MCMC convergence diag-nostic tools have been developed In addition to a basictool which provides intuitive judgments by tracing a timeseries plot of the Markov chain other examples include toolsdeveloped byGelman andRubin [53] Raftery and Lewis [54]Garren and Smith [55] Brooks and Gelman [57] Geweke[58] Johnson [59]Mykland et al [60] Ritter andTanner [61]Roberts [62] Yu [63] Yu andMykland [64] Zellner andMin[65] Heidelberger and Welch [66] and Liu et al [67]

We can divide diagnostic tools into categories dependingon the following (1) if the target distribution needs to bemonitored (2) if the target distribution needs to calculate asingle Markov chain or multiple chains (3) if the diagnosticresults depend on the output of the Monte Carlo algorithmnot on other information from the target distribution

It is not wise to rely on one convergence diagnostictechnique and researchers suggest amore comprehensive useof different diagnostic techniques Some suggestions include(1) simultaneously diagnosing the convergence of a set ofparallel Markov chains (2) monitoring the parametersrsquo auto-correlations and cross-correlations (3) changing the parame-ters of themodel or the samplingmethods (4) using differentdiagnostic methods and choosing the largest preiterationtimes as the formal iteration times (5) combining the resultsobtained from the diagnostic indicatorsrsquo graphs

Six tools have been achieved by computer programsThe most widely used are the convergence diagnostic toolsproposed by Gelman and Rubin [53] and Raftery and Lewis[54] the latter is an extension of the former Both are basedon the theory of analysis of variance (ANOVA) both usemultiple Markov chains and both use the output to performthe diagnostic

61 Gelman-Rubin Diagnostic In traditional literature oniterative simulations many researchers suggest that to guar-antee the Markov chainrsquos diagnostic ability multiple parallelchains should be used simultaneously to do the iterativesimulation for one target distribution In this case afterrunning the simulation for a certain period it is necessaryto determine whether the chains have ldquoforgottenrdquo the initialvalue and if they converge Gelman and Rubin [53] point outthat lack of convergence can sometimes be determined frommultiple independent sequences but cannot be diagnosedusing simulation output from any single sequence They alsofind that more information can be obtained during one singleMarkov chainrsquos replication process than in multiple chainsrsquoiterations Moreover if the initial values are more dispersedthe status of nonconvergence is more easily foundThereforethey propose a method using multiple replications of thechain to decide whether it becomes stationary in the second


half of each sample path The idea behind this is an implicitassumption that convergence will be achieved within the firsthalf of each sample path the validity of this assumption istested by the Gelman-Rubin diagnostic or the variance ratiomethod

Based on normal theory approximations to exactBayesian posterior inference Gelman-Rubin diagnostic inv-olves two steps In Step 1 for a target scalar summary 120593 selectan overdispersed starting point from its target distribution120587(120593) Then generate 119898 Markov chains for 120587(120593) whereeach chain has 2119899 times iterations Delete the first 119899 timesiterations and use the second 119899 times iterations for analysisIn Step 2 apply the between-sequence variance 119861119899 andthe within-sequence variance 119882 to compare the correctedscale reduction factor (CSRF) CSRF is calculated by 119898Markov chains and the formed 119898119899 values in the sequenceswhich stem from 120587(120593) By comparing the CSRF value theconvergence diagnostic can be implemented In addition

119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593)

2

120593119895=

1

119899

119899

sum

119905=1

120593119895119905 120593

=

1

119898

119898

sum

119895=1

120593119895

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895)

2

(6)

where120593119895119905denotes the 119905th of the 119899 iterations of120593 in chain 119895 and

119895 = 1 119898 119905 = 1 119899 Let 120593 be a random variable of 120587(120593)with mean 120583 and variance 1205902 under the target distributionSuppose that 120583 has some unbiased estimator To explainthe variability between 120583 and Gelman and Rubin constructStudentrsquos 119905-distribution with a mean 120583 and variance asfollows

120583 = 120593 =

119899 minus 1

119899

119882 + (1 +

1

119898

)

119861

119899

(7)

where is a weighted average of 119882 and 119861 The aboveestimation will be unbiased if the starting points of thesequences are drawn from the target distribution Howeverit will be overestimated if the starting distribution is overdis-persed Therefore is also called a ldquoconservativerdquo estimateMeanwhile because the iteration number 119899 is limited thewithin-sequence variance119882 can be too low leading to falselydiagnosing convergence As 119899 rarr infin both and119882 shouldbe close to1205902 In other words the scale of current distributionof 120593 should be decreasing as 119899 is increasing

Denoteradic119904= radic120590

2 as the scale reduction factor (SRF)

By applying119882 radic119901= radic119882 becomes the potential scale

reduction factor (PSRF) By applying a correct factor df(dfminus2) for PSRF a correct scale reduction factor (CSRF) can beobtained as follows

radic119888= radic(

119882

)

dfdf minus 2

= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

dfdf minus 2

(8)

where df represents the degree of freedom in Studentrsquos 119905-distribution Following Fisher [68] df asymp 2Var() Thediagnostic based on CSRF can be implemented as followsif 119888gt 1 it indicates that the iteration number 119899 is too

small When 119899 is increasing will become smaller and 119882will become larger If

119888is close to 1 (eg

119888lt 12) we can

conclude that each of the 119898 sets of 119899 simulated observationsis close to the target distribution and the Markov chain canbe viewed as converging

62 Brooks-GelmanDiagnostic Although theGelman-Rubindiagnostic is very popular its theory has several defectsTherefore Brooks and Gelman [57] extend the method in thefollowing way

First in the above equation df(df minus 2) represents theratio of the variance between the Student 119905-distribution andthe normal distribution Brooks and Gelman [57] point outsome obvious defects in the equation For instance if theconvergence speed is slow or df lt 2 CSRF could be infiniteand may even be negative Therefore they set up a new andcorrect factor for PSRF the new CSRF becomes

radiclowast

119888= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

df + 3df + 1

(9)

Second Brooks and Gelman [57] propose a new and moreeasily implemented way to calculate PSRF The first stepis similar to Gelman-Rubinrsquos diagnostic Using 119898 chainsrsquosecond 119899 iterations obtain an empirical interval 100(1 minus120572) after each chainrsquos second 119899 iteration Then 119898 empiricalintervals can be achieved within a sequence denoted by 119897In the second step determine the total empirical intervalsfor sequences from 119898119899 estimates of 119898 chains denoted by 119871Finally calculate the PSRF as follows

radiclowast

119901= radic119897

119871

(10)

The basic theory behind the Gelman-Rubin and Brooks-Gelman diagnostics is the same The difference is that wecompare the variance in the former and the interval lengthin the latter

Third Brooks and Gelman [57] point out that the valueof CSRF being close to one is a necessary but not sufficientcondition for MCMC convergence Additional condition isthat both 119882 and should stabilize as a function of 119899 Thatis to say if 119882 and have not reached the steady stateCSRF could still be one In other words before convergence119882 lt and both should be close to one Therefore asan alternative Brooks and Gelman [57] propose a graphicalapproach tomonitoring convergenceDivide the119898 sequencesinto batches of length 119887Then calculate (119896) 119882(119896) and

119888(119896)

based upon the latter half of the observations of a sequence oflength 2119896119887 for 119896 = 1 119899119887 Plotradic(119896) radic119882(119896) and

119888(119896)

as a function of 119896 on the same plot Approximate convergenceis attained if

119888(119896) is close to one and at the same time both

(119896) and119882(119896) stabilize at one


Fourth and finally Brooks and Gelman [57] discuss themultivariate situation Let 120593 denote the parameter vector andcalculate the following

119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)

Let 1205821be maximum characteristic root of119882minus1119861119899 then the

PSRF can be expressed as

radic119877119901= radic119899 minus 1

119899

+

119898 + 1

119898

1205821 (12)

7 Monte Carlo Error Diagnostic

When implementing the MCMCmethod besides determin-ing the Markov chainsrsquo convergence diagnostic we mustcheck two uncertainties related to theMonte Carlo point esti-mation statistical uncertainty and Monte Carlo uncertainty

Statistical uncertainty is determined by the sample dataand the adoptedmodel Once the data are given and themod-els are selected the statistical uncertainty is fixed For max-imum likelihood estimation (MLE) statistical uncertaintycan be calculated by the inverse square root of the Fisherinformation For Bayesian inference statistical uncertainty ismeasured by the parametersrsquo posterior standard deviation

Monte Carlo uncertainty stems from the approximationof the modelrsquos characteristics which can be measured by asuitable standard error (SE) Monte Carlo standard error ofthemean also calledMonte Carlo error (MC error) is a well-known diagnostic tool In this case define MC error as theratio of the samplersquos standard deviation and the square rootof the sample volume which can be written as

SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)

Obviously as 119899 becomes larger MC error will be smallerMeanwhile the average of the sample data will be closer tothe average of the population

As in the MCMC algorithm we cannot guarantee thatall the sampled points are independent and identically dis-tributed (iid) we must correct the sequencersquos correlationTo this end we introduce the autocorrelation function andsample size inflation factor (SSIF)

Following the samplingmethods introduced in Section 5define a sampling sequence 120579

1 120579

119899with length 119899 Suppose

there are autocorrelations which exist among the adjacentsampling points this means that 120588(120579

119894 120579119894+1) = 0 Furthermore

120588(120579119894 120579119894+119896) = 0 Then the autocorrelation coefficient 120588

119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=

sum119899minus119896

119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)

where 120579 = 1119899sum119899119905=1120579119905 Following the above discussion the

MC error with consideration of autocorrelations in MCMCimplementation can be written as

SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)

In the above equation SDradic119899 represents the MC errorshown in SE[119868(119910)] Meanwhile radic(1 + 120588)(1 minus 120588) representsthe SSIF SDradic119899 is helpful to determine whether the samplevolume 119899 is sufficient and SSIF reveals the influence ofthe autocorrelations on the sample datarsquos standard deviationTherefore by following each parameterrsquos MC error we canevaluate the accuracy of its posterior

The core idea of the Monte Carlo method is to viewthe integration of some function 119891(119909) as an expectationof the random variable therefore the sampling methodsimplemented on the random variable are very important Ifthe sampling distribution is closer to 119891(119909) the MC error willbe smaller In other words by increasing the sample volumeor improving the adopted models the statistical uncertaintycould be reduced the improved samplingmethods could alsoreduce the Monte Carlo uncertainty

8 Model Comparison

In Step 8 we might have several candidate models whichcould pass the MCMC convergence diagnostic and the MCerror diagnostic Thus model comparison is a crucial partof reliability inference Broadly speaking discussions of thecomparison of Bayesianmodels focus onBayes factorsmodeldiagnostic statistics measure of fit and so forth In a morenarrow sense the concept of model comparison refers toselecting a better model after comparing several candidatemodels The purpose of doing model comparison is notto determine the modelrsquos correctness Rather it is to findout why some models are better than others (eg whichparametric model or non-parametric model which priordistribution which covariates which family of parameters forapplication etc) or to obtain an average estimation based ontheweighted estimate of themodel parameters and stemmingfrom the posterior probability of model comparison (egmodel average)

In Bayesian reliability inference the model comparisonmethods can be divided into three categories

(1) separate estimation [69ndash82] including posterior pre-dictive distribution Bayes factors (BF) and its app-roximate estimation-Bayesian information criterion(BIC) deviance information criterion (DIC) pseudo-Bayes factor (PBF) and conditional predictive ordi-nate (CPO) It also includes some estimations based


on the likelihood theory using the same as BF butoffering more flexibility for instance harmonic meanestimator importance sampling reciprocal impor-tance estimator bridge sampling candidate estima-tor Laplace-Metropolis estimator data augmentationestimator and so forth

(2) comparative estimation including different distancemeasures [83ndash87] entropy distance Kullback-Leiblerdivergence (KLD) 119871-measure and weighted 119871-mea-sure

(3) simultaneous estimation [49 88ndash92] including rev-ersible Jump MCMC (RJMCMC) birth-and-deathMCMC (BDMCMC) and fusionMCMC (FMCMC)

Related reference reviews are given by Kass and Raftery [70]Tatiana et al [91] and Chen and Huang [93]

This section introduces three popular comparison meth-ods used in Bayesian reliability studies BF BIC and DICBF is the most traditional method BIC is BFrsquos approximateestimation and DIC improves BIC by dealing with theproblem of the parametersrsquo degree of freedom

81 Bayes Factors (BF) Suppose that119872 represents 119896modelswhich need to be compared The data set 119863 stems from119872119894(119894 = 1 119896) and 119872

1 119872

119896are called competing

models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the

distribution of119863 with consideration of the 119894th model and itsunknown parameter vector 120579 of dimension 119901

119894 also called the

likelihood function of119863 with a specified model Under priordistribution 120587(120579

119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal

distributions of119863 are found by integrating out the parametersas follows

119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)

where Θ119894represents the parameter data set for the 119894th mode

As in the data likelihood function the quantity 119901(119863 |

119872119894) = 119871(119863 | 119872

119894) is called model likelihood Suppose we

have some preliminary knowledge about model probabilities120587(119872119894) after considering the given observed data set 119863 the

posterior probability of 119894th model being the best model isdetermined as

119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)

The integration part of the above equation is also called theprior predictive density or marginal likelihood where 119901(119863)is a nonconditional marginal likelihood of119863

Suppose there are two models 1198721and 119872

2 Let BF

12

denote the Bayes factors equal to the ratio of the posteriorodds of the models to the prior odds

BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)

The above equation shows that BF12

equals the ratio of themodel likelihoods for the two models Thus it can be writtenas

119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)

We can also say that BF12

shows the ratio of posterior oddsof the model119872

1and the prior odds of119872

1 In this way the

collection of model likelihoods 119901(119863 | 119872119894) is equivalent to the

model probabilities themselves (since the prior probabilities120587(119872119894) are known in advance) and hence could be considered

as the key quantities needed for Bayesian model choiceJeffreys [94] recommends a scale of evidence for inter-

preting Bayes factors Kass and Raftery [70] provide a similarscale alongwith a complete review of Bayes factors includingtheir interpretation computation or approximation androbustness to the model-specific prior distributions andapplications in a variety of scientific problems

82 Bayesian Information Criterion (BIC) Under some situa-tions it is difficult to calculate BF especially for those modelswhich consider different random effects or adopt diffusionpriors or a large number of unknown and informative priorsTherefore we need to calculate BFrsquos approximate estimationThe Bayesian information criterion (BIC) is also calledthe Schwarz information criterion (SIC) and is the mostimportant method to get BFrsquos approximate estimation Thekey point of BIC is to obtain the approximate estimation of119901(119863 | 119872

119894) It is proved by Volinsky and Raftery [72] that

ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)

Then we get SIC as follows

ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)

As discussed above considering two models 1198721and 119872

2

BIC12represents the likelihood ratio test statistic with model

sample size 119899 and the modelrsquos complexity as penalty It can bewritten as

BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive

Axels 1 2 3 Bogies I II

(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)

Figure 3 Locomotive wheelsrsquo installation positions

where 119901119894is proportional to the modelrsquos sample size and

complexityKass and Raftery [70] discuss BICrsquos calculation program

Kass and Wasserman [73] show how to decide 119899 Volinskyand Raftery [72] discuss the way to choose 119899 if the data arecensored If 119899 is large enough BFrsquos approximate estimationcan be written as

BF12asymp exp (minus05BIC

12) (23)

Obviously if the BIC is smaller we should consider model1198721 otherwise119872

2should be considered

83 Deviance InformationCriterion (DIC) Traditionalmeth-ods for model comparison consider two main aspects themodelrsquos measure of fit and the modelrsquos complexity Normallythe modelrsquos measure of fit can be increased by increasing themodelrsquos complexity For this reason most model comparisonmethods are committed balancing both two points To utilizeBIC the number 119901 of free parameters of the model mustbe calculated However for complex Bayesian hierarchicalmodels it is very difficult to get 119901rsquos exact number ThereforeSpiegelhalter et al [74] propose the deviance informationcriterion (DIC) to compare Bayesian models Celeux et al[95] discuss DIC issues for a censored data set this paper andother researchersrsquo discussion of it are representative literaturein the DIC field in recent years

DIC utilizes deviance to evaluate the modelrsquos measureof fit and it utilizes the number of parameters to evaluateits complexity Note that it is consistent with the Akaikeinformation criterion (AIC) which is used to compareclassical models [96]

Let119863(120579) denote the Bayesian deviance and

119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)

Let119901119889denote themodelrsquos effective number of parameters and

119901119889= 119863 (120579) minus 119863 (120579)

= minusint 2 ln (119901 (119863 | 120579)) 119889120579 minus (minus2 ln (119901 (119863 | 120579))) (25)

Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)

Select the model with a lower DIC value As DIC lt 5 thedifference between models can be ignored

9 Discussions with a Case Study

In this section we discuss a case study for a locomotivewheelrsquos degradation data to illustrate the proposed procedureThe case was first discussed by Lin et al [36]

91 Background The service life of a railway wheel can besignificantly reduced due to failure or damage leading toexcessive cost and accelerated deterioration This case studyfocuses on the wheels of the locomotive of a cargo trainWhile two types of locomotives with the same type of wheelsare used in cargo trains we consider only one

There are two bogies for each locomotive and three axelsfor each bogie (Figure 3)The installed position of the wheelson a particular locomotive is specified by a bogie number (IIImdashnumber of bogies on the locomotive) an axel number (12 3mdashnumber of axels for each bogie) and the side of thewheelon the alxe (right or left) where each wheel is mounted

The diameter of a new locomotive wheel in the studiedrailway company is 1250mm In the companyrsquos currentmaintenance strategy a wheelrsquos diameter is measured afterrunning a certain distance If it is reduced to 1150mm thewheel is replaced by a new one Otherwise it is reprofiled orother maintenance strategies are implemented A thresholdlevel for failure is defined as 100mm (= 1250mm ndash 1150mm)The wheelrsquos failure condition is assumed to be reached if thediameter reaches 100mm

The companyrsquos complete database also includes the diam-eters of all locomotive wheels at a given observation timethe total running distances corresponding to their ldquotime tobe maintainedrdquo and the wheelsrsquo bill of material (BOM) datafrom which we can determine their positions

Two assumptions are made (1) for each censored datumit is supposed that the wheel is replaced (2) degradation islinear Only one locomotive is considered in this exampleto ensure that (1) all wheelrsquos maintenance strategies are thesame (2) the alxe load and running speed are obviouslyconstant and (3) the operational environments includingroutes climates and exposure are common for all wheels

The data set contains 46 datum points (119899 = 46) of a singlelocomotive throughout the periodNovember 2010 to January2012We take the following steps to obtain locomotivewheelsrsquolifetime data (see Figure 4)

(i) Establish a threshold level 119871119904 where 119871

119904= 100mm

(1250mm ndash 1150mm)

12 Journal of Quality and Reliability EngineeringD

egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)

Figure 4 Plot of the wheel degradation data one example

(ii) Transfer observed 90 records of wheel diametersat reported time 119905 degradation data this equals to1250mm minus the corresponding observed diame-ter

(iii) Assume a liner degradation path and construct adegradation line119871

119894(eg119871

1 1198712) using the origin point

and the degradation data (eg 1198611 1198612)

(iv) Set 119871119904= 119871119894 get the point of intersection and the

corresponding lifetimes data (eg1198631 1198632)

To explore the impact of a locomotive wheelrsquos installedposition on its service lifetime and to predict its reliabilitycharacteristics the Bayesian exponential regression modelBayesianWeibull regressionmodel and Bayesian log-normalregression model are used to establish the wheelrsquos lifetimeusing degradation data and taking into account the positionof the wheel For each reported datum a wheelrsquos installationposition is documented and asmentioned above positioningis used in this study as a covariateThewheelrsquos position (bogieaxel and side) or covariateX is denoted by 119909

1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)

Correspondingly the covariatesrsquo coefficients are representedby 1205731 1205732 and 120573

3 In addition 120573

0is defined as random effect

The goal is to determine reliability failure distribution andoptimal maintenance strategies for the wheel

92 Plan During the Plan Stage we first collect the ldquocurrentdatardquo as mentioned in Section 91 including the diametermeasurements of the locomotive wheel total distances cor-responding to the ldquotime to maintenancerdquo and the wheelrsquosbill of material (BOM) data Then see Figure 4 we note theinstalled position and transfer the diameter into degradationdata which becomes ldquoreliability datardquo during the ldquodata prepa-rationrdquo process

We consider the noninformative prior for the constructedmodels and select the vague prior with log-concave formwhich has been determined to be a suitable choice as anoninformative prior selection For exponential regression120573 sim 119873(0 00001) for Weibull regression 120572 sim 119866(02 02)

120573 sim 119873(0 00001) for lognormal regression 120591 sim 119866(1001) 120573 sim 119873(0 00001)

93 Do In the Do Stage we set up the three models notedabove for the degradation analysis Bayesian exponentialregression model Bayesian Weibull regression model andBayesian log-normal regression model For our calculationswe implement Gibbs sampling

The joint likelihood function for the exponential regres-sion model Weibull regression model and log-normalregression model is given as follows (Lin et al [36])

(i) exponential regression model

119871 (120573 | 119863) =119899

prod

119894=1

[exp (x1015840i120573) exp (minus exp (x1015840

i120573) 119905119894)]120592119894

times [exp (minus exp (x1015840i120573) 119905119894)]1minus120592119894

= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)

(ii) Weibull regression model

119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)

(iii) log-normal regression model

119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)

94 Study After checking the MCMC convergence diagnos-tic and accepting the Monte Carlo error diagnostic for allthree models in the Study Stage we compare the model withthe three DIC values After comparing the DIC values weselect the Bayesian log-normal regression model as the mostsuitable one (see Table 1)

Accordingly the locomotive wheelsrsquo reliability functionsare achieved


119877 (119905119894| X) = exp [minus exp (minus5862 minus 0072119909

1

minus00321199092minus 0012119909

3) times 119905119894]

(30)


Table 1 DIC summaries

Model 119863(120579) 119863(120579) 119901119889

DICExponential 64898 64503 395 65293Weibull 47222 46739 483 47705Log-normal 44203 43687 516 44719

Table 2 MTTF statistics based on Bayesian log-normal regressionmodel

Bogie Axel Side 120583119894

MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203


119877 (119905119894| X) = exp lfloor minus exp (minus6047 minus 0078119909

1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)

95 Action With the chosen modelrsquos posterior results in theAction Stage wemake ourmaintenance predictions (Table 2)and apply them to the proposed maintenance inspectionlevel This in turn allows us to evaluate and optimise thewheelrsquos replacement and maintenance strategies (includingthe reprofiling interval inspection interval and lubricationinterval)

As more data are collected in the future the old ldquocurrentdata setrdquo will be replaced by new ldquocurrent datardquo meanwhilethe results achieved in this casewill become ldquohistory informa-tionrdquo which will be transferred to be ldquoprior knowledgerdquo anda new cycle will start With this step-by-step method we cancreate a continuous improvement process for the locomotivewheelrsquos reliability inference

10 Conclusions

This paper has proposed an integrated procedure for Bayesianreliability inference using Markov chain Monte Carlo meth-odsThe goal is to build a full framework for related academicresearch and engineering applications to implement moderncomputational-basedBayesian approaches especially for reli-ability inference The suggested procedure is a continuousimprovement process with four stages (Plan Do Study andAction) and 11 sequential steps including (1) data preparation(2) priorsrsquo inspection and integration (3) prior selection(4) model selection (5) posterior sampling (6) MCMCconvergence diagnostic (7)Monte Carlo error diagnostic (8)model improvement (9) model comparison (10) inferencemaking (11) data updating and inference improvement

The paper illustrates the proposed procedure using a casestudy It concludes that the procedure can be used to performBayesian reliability inference to determine system (or unit)reliability failure distribution and to support maintenancestrategies optimization and so forth

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Lulea Railway ResearchCentre (Jarnvagstekniskt Centrum Sweden) for initiatingthe research study and Swedish Transport Administration(Trafikverket) for providing financial support

References

[1] D L Kelly and C L Smith ldquoBayesian inference in probabilisticrisk assessmentmdashthe current state of the artrdquo Reliability Engi-neering and System Safety vol 94 no 2 pp 628ndash643 2009

[2] P Congdon Bayesian Statistical Modeling John Wiley amp SonsChichester UK 2001

[3] P Congdon Applied Bayesian Modeling John Wiley amp SonsChichester UK 2003

[4] D G Robinson ldquoA hierarchical bayes approach to system reli-ability analysisrdquo Tech Rep Sandia 2001-3513 Sandia NationalLaboratories 2001

[5] V E Johnson ldquoA hierarchical model for estimating the reliabil-ity of complex systemsrdquo in Bayesian Statistics 7 J M BernardoEd Oxford University Press Oxford UK 2003

[6] A G Wilson ldquoHierarchical Markov Chain Monte Carlo(MCMC) for bayesian system reliabilityrdquo Encyclopedia of Statis-tics in Quality and Reliability 2008

[7] J LinM Asplund andA Parida ldquoBayesian parametric analysisfor reliability study of locomotive wheelsrdquo in Proceedings of the59th Annual Reliability and Maintainability Symposium (RAMSrsquo13) Orlando Fla USA 2013

[8] G Tont L Vladareanu M S Munteanu and D G TontldquoHierarchical Bayesian reliability analysis of complex dynamicalsystemsrdquo in Proceedings of the 9thWSEAS International Confer-ence on Applications of Electrical Engineering (AEE rsquo10) pp 181ndash186 March 2010


[9] J Lin ldquoA two-stage failure model for Bayesian change pointanalysisrdquo IEEETransactions on Reliability vol 57 no 2 pp 388ndash393 2008

[10] J Lin M L Nordenvaad and H Zhu ldquoBayesian survivalanalysis in reliability for complex system with a cure fractionrdquoInternational Journal of Performability Engineering vol 7 no 2pp 109ndash120 2011

[11] M Hamada H F Martz C S Reese T Graves V Johnsonand A G Wilson ldquoA fully Bayesian approach for combiningmultilevel failure information in fault tree quantification andoptimal follow-on resource allocationrdquo Reliability Engineeringand System Safety vol 86 no 3 pp 297ndash305 2004

[12] T L Graves M S Hamada R Klamann A Koehler and HF Martz ldquoA fully Bayesian approach for combining multi-levelinformation in multi-state fault tree quantificationrdquo ReliabilityEngineering and System Safety vol 92 no 10 pp 1476ndash14832007

[13] L Kuo and B Mallick ldquoBayesian semiparametric inference forthe accelerated failure-time modelrdquo The Canadian Journal ofStatistics vol 25 no 4 pp 457ndash472 1997

[14] S Walker and B K Mallick ldquoA Bayesian semiparametricaccelerated failure time modelrdquo Biometrics vol 55 no 2 pp477ndash483 1999

[15] T Hanson andWO Johnson ldquoA Bayesian semiparametric AFTmodel for interval-censored datardquo Journal of Computationaland Graphical Statistics vol 13 no 2 pp 341ndash361 2004

[16] S K Ghosh and S Ghosal ldquoSemiparametric accelerated failuretime models for censored datardquo in Bayesian Statistics and ItsApplications S K Upadhyay Ed Anamaya Publishers NewDelhi India 2006

[17] A Komarek and E Lesaffre ldquoThe regression analysis of cor-related interval-censored data illustration using acceleratedfailure time models with flexible distributional assumptionsrdquoStatistical Modelling vol 9 no 4 pp 299ndash319 2009

[18] G Y Li Q G Wu and Y H Zhao ldquoOn bayesian analysis ofbinomial reliability growthrdquoThe Japan Statistical Society vol 32no 1 pp 1ndash14 2002

[19] J Y Tao Z M Ming and X Chen ldquoThe bayesian reliabilitygrowth models based on a new dirichlet prior distributionrdquo inReliability Risk and Safety C G Soares Ed Chapter 237 CRCPress 2009

[20] L Kuo andT Y Yang ldquoBayesian reliabilitymodeling formaskedsystem lifetime datardquo Statistics and Probability Letters vol 47no 3 pp 229ndash241 2000

[21] K Ilkka ldquoMethods and problems of software reliability esti-mationrdquo Tech Rep VTT Working Papers 1459-7683 VTTTechnical Research Centre of Finland 2006

[22] Y Tamura H Takehara and S Yamada ldquoComponent-orientedreliability analysis based on hierarchical bayesian model for anopen source softwarerdquoAmerican Journal of Operations Researchvol 1 pp 25ndash32 2011

[23] G I Schueller and H J Pradlwarter ldquoBenchmark study on reli-ability estimation in higher dimensions of structural systemsmdashan overviewrdquo Structural Safety vol 29 no 3 pp 167ndash182 2007

[24] S K Au J Ching and J L Beck ldquoApplication of subset sim-ulation methods to reliability benchmark problemsrdquo StructuralSafety vol 29 no 3 pp 183ndash193 2007

[25] R Li Bayes reliability studies for complex system [Doctor thesis]National University of Defense Technology 1999 Chinese

[26] A M Walker ldquoOn the asymptotic behavior of posterior distri-butionsrdquo Journal of Royal Statistics Society B vol 31 no 1 pp80ndash88 1969

[27] S Ghosal ldquoA review of consistency and convergence of poste-rior distribution Technical reportrdquo in Proceedings of NationalConference in Bayesian Analysis Benaras Hindu UniversityVaranashi India 2000

[28] T Choi R V B C Ramamoorthi and S Ghosal EdsPushing the Limits of Contemporary Statistics Contributions inHonor of Jayanta K Ghosh Institute of Mathematical StatisticsBeachwood Ohio USA 2008

[29] V P Savchuk andH FMartz ldquoBayes reliability estimation usingmultiple sources of prior information binomial samplingrdquoIEEE Transactions on Reliability vol 43 no 1 pp 138ndash144 1994

[30] K J Ren M D Wu and Q Liu ldquoThe combination of priordistributions based on kullback informationrdquo Journal of theAcademy of Equipment of Command amp Technology vol 13 no4 pp 90ndash92 2002

[31] Q Liu J Feng and J-L Zhou ldquoApplication of similar systemreliability information to complex system Bayesian reliabilityevaluationrdquo Journal of Aerospace Power vol 19 no 1 pp 7ndash102004

[32] Q Liu J Feng and J Zhou ldquoThe fusion method for priordistribution based on expertrsquos informationrdquo Chinese SpaceScience and Technology vol 24 no 3 pp 68ndash71 2004

[33] G H Fang Research on the multi-source information fusiontechniques in the process of reliability assessment [Doctor thesis]Hefei University of Technology China 2006

[34] Z B Zhou H T Li X M Liu G Jin and J L Zhou ldquoAbayes information fusion approach for reliability modeling andassessment of spaceflight long life productrdquo Journal of SystemsEngineering vol 32 no 11 pp 2517ndash2522 2012

[35] R Howard and S Robert Applied Statistical Decision TheoryDivision of Research Graduate School of Business Administra-tion Harvard University 1961

[36] J Lin M Asplund and A Parida ldquoReliability analysis fordegradation of locomotive wheels using parametric bayesianapproachrdquo Quality and Reliability Engineering International2013

[37] J Lin and M Asplund ldquoBayesian semi-parametric analysis forlocomotive wheel degradation using gamma frailtiesrdquo Journalof Rail and Rapid Transit 2013

[38] J G Ibrahim M H Chen and D Sinha Bayesian SurvivalAnalysis 2001

[39] J Lin Y-Q Han and H-M Zhu ldquoA semi-parametric propor-tional hazards regressionmodel based on gamma process priorsand its application in reliabilityrdquo Chinese Journal of SystemSimulation vol 22 pp 5099ndash5102 2007

[40] S K Sahu D K Dey H Aslanidou and D Sinha ldquoA weibullregression model with gamma frailties for multivariate survivaldatardquo Lifetime Data Analysis vol 3 no 2 pp 123ndash137 1997

[41] H Aslanidou D K Dey and D Sinha ldquoBayesian analysisof multivariate survival data using Monte Carlo methodsrdquoCanadian Journal of Statistics vol 26 no 1 pp 33ndash48 1998

[42] N Metropolis A W Rosenbluth M N Rosenbluth A HTeller and E Teller ldquoEquation of state calculations by fastcomputing machinesrdquo The Journal of Chemical Physics vol 21no 6 pp 1087ndash1092 1953

[43] W K Hastings ldquoMonte carlo sampling methods using Markovchains and their applicationsrdquo Biometrika vol 57 no 1 pp 97ndash109 1970

[44] L Tierney ldquoMarkov chains for exploring posterior distribu-tionsrdquoTheAnnals of Statistics vol 22 no 4 pp 1701ndash1762 1994


[45] S Chib and E Greenberg ldquoUnderstanding the metropolis-hastings algorithmrdquo Journal of American Statistician vol 49 pp327ndash335 1995

[46] U Grenander Tutorial in Pattern Theory Division of AppliedMathematics Brow University Providence RI USA 1983

[47] S Geman andDGeman ldquoStochastic relaxation Gibbs distribu-tions and the Bayesian restoration of imagesrdquo IEEETransactionson Pattern Analysis and Machine Intelligence vol 6 no 6 pp721ndash741 1984

[48] A E Gelfand and A F M Smith ldquoSampling-based approachesto calculation marginal densitiesrdquo Journal of American Asscioa-tion vol 85 pp 398ndash409 1990

[49] M A Tanner and W H Wong ldquoThe calculation of posteriordistributions by data augmentation (with discussion)rdquo Journalof American Statistical Association vol 82 pp 805ndash811 1987

[50] N G Polson ldquoConvergence of markov chain conte carloalgorithmsrdquo in Bayesian Statistics 5 Oxford University PressOxford UK 1996

[51] G O Roberts and J S Rosenthal ldquoMarkov-chain Monte Carlosome practical implications of theoretical resultsrdquo CanadianJournal of Statistics vol 26 no 1 pp 4ndash31 1997

[52] K LMengersen and R L Tweedie ldquoRates of convergence of theHastings and Metropolis algorithmsrdquo The Annals of Statisticsvol 24 no 1 pp 101ndash121 1996

[53] AGelman andD B Rubin ldquoInference from iterative simulationusing multiple sequences (with discussion)rdquo Statistical Sciencevol 7 pp 457ndash511 1992

[54] A E Raftery and S Lewis ldquoHow many iterations in the gibbssamplerrdquo inBayesian Statistics 4 pp 763ndash773OxfordUniversityPress Oxford UK 1992

[55] S T Garren and R L Smith Convergence Diagnostics forMarkov Chain Samplers Department of Statistics University ofNorth Carolina 1993


[57] S P Brooks and A Gelman ldquoGeneral methods for monitoringconvergence of iterative simulationsrdquo Journal of Computationaland Graphical Statistics vol 7 no 4 pp 434ndash455 1998

[58] J Geweke ldquoEvaluating the accuracy of sampling-based app-roaches to the calculation of posterior momentsrdquo in BayesianStatistics 4 J M Bernardo J Berger A P Dawid and A F MSmith Eds pp 169ndash193 Oxford University Press Oxford UK1992

[59] V E Johnson ldquoStudying convergence of markov chain montecarlo algorithms using coupled sampling pathsrdquo Tech RepInstitute for Statistics and Decision Sciences Duke University1994

[60] P Mykland L Tierney and B Yu ldquoRegeneration in markovchain samplersrdquo Journal of the American Statistical Associationno 90 pp 233ndash241 1995

[61] C Ritter and M A Tanner ldquoFacilitating the gibbs samplerthe gibbs stopper and the griddy gibbs samplerrdquo Journal of theAmerican Statistical Association vol 87 pp 861ndash868 1992

[62] G O Roberts ldquoConvergence diagnostics of the gibbs samplerrdquoin Bayesian Statistics 4 pp 775ndash782 Oxford University PressOxford UK 1992

[63] B Yu Monitoring the Convergence of Markov Samplers Basedon Estimated L1 Error Department of Statistics University ofCalifornia Berkeley Calif USA 1994

[64] B Yu and P Mykland Looking at Markov Samplers throughCusum Path Plots A Simple Diagnostic Idea Department ofStatistics University of California Berkeley Calif USA 1994

[65] A Zellner and C K Min ldquoGibbs sampler convergence criteriardquoJournal of the American Statistical Association vol 90 pp 921ndash927 1995

[66] P Heidelberger and P DWelch ldquoSimulation run length controlin the presence of an initial transientrdquoOperations Research vol31 no 6 pp 1109ndash1144 1983

[67] C Liu J Liu andD B Rubin ldquoA variational control variable forassessing the convergence of the gibbs samplerrdquo in Proceedingsof the American Statistical Association Statistical ComputingSection pp 74ndash78 1992

[68] J F Lawless Statistical Models and Method for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[69] A Gelman J B Carlin H S Stern and D B Rubin BayesianData Analysis Chapman and HallCRC Press New York NYUSA 2004

[70] R A Kass and A E Raftery ldquoBayes factorrdquo Journal of AmericanStattistical Association vol 90 pp 773ndash795 1995

[71] A Gelfand andD Dey ldquoBayesianmodel choice asymptotic andexact calculationsrdquo Journal of Royal Statistical Society B vol 56pp 501ndash514 1994

[72] C TVolinsky andA E Raftery ldquoBayesian information criterionfor censored survivalmodelsrdquoBiometrics vol 56 no 1 pp 256ndash262 2000

[73] R E Kass and L Wasserman ldquoA reference bayesian test fornested hypotheses and its relationship to the schwarz criterionrdquoJournal of American Statistical Association vol 90 pp 928ndash9341995

[74] D J Spiegelhalter N G Best B P Carlin and A van der LindeldquoBayesian measures of model complexity and fitrdquo Journal of theRoyal Statistical Society B vol 64 no 4 pp 583ndash639 2002

[75] S Geisser and W Eddy ldquoA predictive approach to modelselectionrdquo Journal of American Statistical Association vol 74pp 153ndash160 1979

[76] A E Gelfand D K Dey and H Chang ldquoModel deter-minating using predictive distributions with implementationvia sampling-based methods (with discussion)rdquo in BayesianStatistics 4 Oxford University Press Oxford UK 1992

[77] M Newton and A Raftery ldquoApproximate bayesian inference bythe weighted likelihood bootstraprdquo Journal of Royal StatisticalSociety B vol 56 pp 1ndash48 1994

[78] P J Lenk and W S Desarbo ldquoBayesian inference for finitemixtures of generalized linear models with random effectsrdquoPsychometrika vol 65 no 1 pp 93ndash119 2000

[79] X-L Meng andW HWong ldquoSimulating ratios of normalizingconstants via a simple identity a theoretical explorationrdquoStatistica Sinica vol 6 no 4 pp 831ndash860 1996

[80] S Chib ldquoMarginal likelihood from the Gibbs outputrdquo Journal ofthe American Statistical Association vol 90 pp 1313ndash1321 1995

[81] S Chib and I Jeliazkov ldquoMarginal likelihood from themetropolis-hastings outputrdquo Journal of the American StatisticalAssociation vol 96 no 453 pp 270ndash281 2001

[82] S M Lewis and A E Raftery ldquoEstimating Bayes factors viaposterior simulation with the Laplace-Metropolis estimatorrdquoJournal of the American Statistical Association vol 92 no 438pp 648ndash655 1997

[83] S K Sahu and R C H Cheng ldquoA fast distance-based approachfor determining the number of components in mixturesrdquoCanadian Journal of Statistics vol 31 no 1 pp 3ndash22 2003


[84] KMengersen andC P Robert ldquoTesting formixtures a bayesianentropic approachrdquo in Bayesian Statistics 5 Oxford UniversityPress Oxford UK 1996

[85] J G Ibrahim and P W Laud ldquoA predictive approach to theanalysis of designed experimentsrdquo Journal of the AmericanStatistical Association vol 89 pp 309ndash319 1994

[86] A E Gelfand and S K Ghosh ldquoModel choice a minimumposterior predictive loss approachrdquo Biometrika vol 85 no 1pp 1ndash11 1998

[87] M-H Chen D K Dey and J G Ibrahim ldquoBayesian criterionbased model assessment for categorical datardquo Biometrika vol91 no 1 pp 45ndash63 2004

[88] P J Green ldquoReversible jumpMarkov chainmonte carlo compu-tation and Bayesian model determinationrdquo Biometrika vol 82no 4 pp 711ndash732 1995

[89] C Robert and G Casella Monte Carlo Statistical MethodsSpringer New York NY USA 2004

[90] M Stephens ldquoBayesian analysis of mixture models with anunknown number of componentsmdashan alternative to reversiblejump methodsrdquo Annals of Statistics vol 28 no 1 pp 40ndash742000

[91] M Tatiana F S Sylvia and D A Georg Comparison ofBayesian Model Selection Based on MCMC with an Applicationto GARCH-Type Models Vienna University of Economics andBusiness Administration 2003

[92] P C Gregory ldquoExtra-solar planets via Bayesian fusionMCMCrdquoinAstrostatistical Challenges for the NewAstronomy J M HilbeEd Springer Series in Astrostatistics Chapter 7 Springer NewYork NY USA 2012

[93] H Chen and S Huang ldquoA comparative study on modelselection and multiple model fusionrdquo in Proceedings of the 7thInternational Conference on Information Fusion (FUSION rsquo05)pp 820ndash826 July 2005

[94] H Jeffreys Theory of Probability Oxford Univiersity PressOxford UK 1961

[95] G Celeux F Forbes C P Robert and D M Tittering-ton ldquoDeviance information criteria for missing data modelsrdquoBayesian Analysis vol 1 no 4 pp 651ndash674 2006

[96] H Akaike ldquoInformation theory and an extension of the maxi-mum likelihood principlerdquo in Proceedings of the 2nd Interna-tional Symposium on Information Theory Tsahkadsor 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



(6) MCMCconvergencediagnostic

(7) Monte Carloerror diagnosticYes Accept

No

No

Posterior results

(2) Priorsrsquo inspectionand integration

(3) Prior selection

(4) Model selection

(5) Posterior sampling (8) Model improvement

(9) Model comparison

Bayesian modelaverage

MCMC modelaverage

Reliability data andso forth

Prior knowledge

Historyinformation

Currentdata set

(11) Data updating andinference improvement

Do

Plan

(10) Inference making

Action

(1) Data preparation

Study

Figure 1 An integrated procedure for Bayesian reliability inference via MCMC

The data may be subject to uncertainties such as imprecisemeasurement censoring truncated information and inter-pretation errors Reliability data are found in the ldquocurrent datasetrdquo they contain original data and include the evaluationmanipulation andor organisation of data samples At ahigher level in the collection of data a wide variety of ldquohis-torical informationrdquo can be obtained including the results ofinspecting and integrating this ldquoinformationrdquo thereby addingto ldquoprior knowledgerdquo The final level is reliability inferencewhich is the process of making a conclusion based on ldquopost-erior resultsrdquo

Using the definitions shown in Figure 1 we propose anintegrated procedure which constructs a full framework forthe standardized process of Bayesian reliability inference Asshown in Figure 1 the procedure is composed of a continuousimprovement process including four stages (Plan Do Studyand Action) which will be discussed later in this section and11 sequential steps (1) data preparation (2) prior inspectionand integration (3) prior selection (4) model selection (5)posterior sampling (6) MCMC convergence diagnostic (7)Monte Carlo error diagnostic (8) model improvement (9)

model comparison (10) inference making (11) data updatingand inference improvement

Step 1 (data preparation) The original data sets for ldquohistoryinformationrdquo and ldquocurrent datardquo related to reliability studiesneed to be acquired evaluated andmerged In this way ldquohis-tory informationrdquo can be transferred to ldquoprior knowledgerdquoand ldquocurrent datardquo can become ldquoreliability datardquo in later steps

Step 2 (prior inspection and integration) During this stepldquoprior knowledgerdquo receives a second andmore extensive treat-ment including a reliability consistency check a credencetest and a multisource integration This step improves priorreliability data

Step 3 (prior selection) This step uses the results achievedin Step 2 to determine the modelrsquos form and parameters forinstance selecting informative or noninformative priors orunknown parameters and their distributed forms

Step 4 (model selection) This step determines a reliabilitymodel (parametric or nonparametric) selecting from 119899




















Obj

ectiv

e

Engineering design

Component test

Systems test







Information

Subj

ectiv

e

Knowledge







Transfer

Reliability data

























0) gt

0




1 1205792) is


1 1205792) = 119902(120579





120572 (120579119905minus1 120579lowast) =

119901 (120579lowast)

119901 (120579119905minus1)

=

ℎ (120579lowast)

ℎ (120579119905minus1)

(1)




120579119905= 120579119905minus1

and go to Step 2


120572 (120579119905minus1 120579lowast) = min(1

ℎ (120579lowast)

ℎ (120579119905minus1)

) (2)


0 1205791 120579


from 120579119905to 120579119905+1


0 1205791 120579

119905minus1




120572 (120579119905minus1 120579lowast) = min(1



) (3)




119901 (120579 120579lowast) =

119902 (120579119905minus1 120579lowast) ℎ (120579


119905minus1)

119902 (120579119905minus1 120579lowast)

ℎ (120579lowast)

ℎ (120579119905minus1)

ℎ (120579lowast) lt ℎ (120579

119905minus1)

(4)


119905= 120579lowast if120583 le 120572 (120579

119905minus1 120579lowast) and 120579

119905= 120579119905minus1

otherwise








120579(0)

119896)




120579(0)

2 120579

(0)

119896) and generate 120579(1)


tion 119891(1205792| 120579(1)

1 120579(0)

3 120579

(0)

119896)

Step 3 Generate 120579(1)119895

from 119891(120579119895| 120579(1)

1 120579

(1)

119895minus1 120579(0)

119895+1 120579

(0)

119896)

Step 4 Generate 120579(1)119896

from 119891(120579119896| 120579(1)

1 120579(1)

2 120579

(1)

119896minus1)


1 120579

(1)





(119905)



119901 (120579 120579lowast)

= 119891 (1205791| 1205792 120579

119896) 119891 (1205792| 120579lowast

1 1205793 120579


119891 (120579119896| 120579lowast

1 120579

lowast

119896minus1)

(5)















119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593)

2

120593119895=

1

119899

119899

sum

119905=1

120593119895119905 120593

=

1

119898

119898

sum

119895=1

120593119895

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895)

2

(6)



120583 = 120593 =

119899 minus 1

119899

119882 + (1 +

1

119898

)

119861

119899

(7)






radic119888= radic(

119882

)

dfdf minus 2

= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

dfdf minus 2

(8)




119888lt 12) we can




radiclowast

119888= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

df + 3df + 1

(9)


radiclowast

119901= radic119897

119871

(10)



119888(119896)


119888(119896)






119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)




119899

+

119898 + 1

119898

1205821 (12)





SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)




1 120579



119894 120579119894+1) = 0 Furthermore


119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=


119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)



SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)



8 Model Comparison











1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























Obj

ectiv

e

Engineering design

Component test

Systems test







Information

Subj

ectiv

e

Knowledge







Transfer

Reliability data

























0) gt

0




1 1205792) is


1 1205792) = 119902(120579





120572 (120579119905minus1 120579lowast) =

119901 (120579lowast)

119901 (120579119905minus1)

=

ℎ (120579lowast)

ℎ (120579119905minus1)

(1)




120579119905= 120579119905minus1

and go to Step 2


120572 (120579119905minus1 120579lowast) = min(1

ℎ (120579lowast)

ℎ (120579119905minus1)

) (2)


0 1205791 120579


from 120579119905to 120579119905+1


0 1205791 120579

119905minus1




120572 (120579119905minus1 120579lowast) = min(1



) (3)




119901 (120579 120579lowast) =

119902 (120579119905minus1 120579lowast) ℎ (120579


119905minus1)

119902 (120579119905minus1 120579lowast)

ℎ (120579lowast)

ℎ (120579119905minus1)

ℎ (120579lowast) lt ℎ (120579

119905minus1)

(4)


119905= 120579lowast if120583 le 120572 (120579

119905minus1 120579lowast) and 120579

119905= 120579119905minus1

otherwise








120579(0)

119896)




120579(0)

2 120579

(0)

119896) and generate 120579(1)


tion 119891(1205792| 120579(1)

1 120579(0)

3 120579

(0)

119896)

Step 3 Generate 120579(1)119895

from 119891(120579119895| 120579(1)

1 120579

(1)

119895minus1 120579(0)

119895+1 120579

(0)

119896)

Step 4 Generate 120579(1)119896

from 119891(120579119896| 120579(1)

1 120579(1)

2 120579

(1)

119896minus1)


1 120579

(1)





(119905)



119901 (120579 120579lowast)

= 119891 (1205791| 1205792 120579

119896) 119891 (1205792| 120579lowast

1 1205793 120579


119891 (120579119896| 120579lowast

1 120579

lowast

119896minus1)

(5)















119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593)

2

120593119895=

1

119899

119899

sum

119905=1

120593119895119905 120593

=

1

119898

119898

sum

119895=1

120593119895

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895)

2

(6)



120583 = 120593 =

119899 minus 1

119899

119882 + (1 +

1

119898

)

119861

119899

(7)






radic119888= radic(

119882

)

dfdf minus 2

= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

dfdf minus 2

(8)




119888lt 12) we can




radiclowast

119888= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

df + 3df + 1

(9)


radiclowast

119901= radic119897

119871

(10)



119888(119896)


119888(119896)






119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)




119899

+

119898 + 1

119898

1205821 (12)





SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)




1 120579



119894 120579119894+1) = 0 Furthermore


119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=


119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)



SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)



8 Model Comparison











1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























0) gt

0




1 1205792) is


1 1205792) = 119902(120579





120572 (120579119905minus1 120579lowast) =

119901 (120579lowast)

119901 (120579119905minus1)

=

ℎ (120579lowast)

ℎ (120579119905minus1)

(1)




120579119905= 120579119905minus1

and go to Step 2


120572 (120579119905minus1 120579lowast) = min(1

ℎ (120579lowast)

ℎ (120579119905minus1)

) (2)


0 1205791 120579


from 120579119905to 120579119905+1


0 1205791 120579

119905minus1




120572 (120579119905minus1 120579lowast) = min(1



) (3)




119901 (120579 120579lowast) =

119902 (120579119905minus1 120579lowast) ℎ (120579


119905minus1)

119902 (120579119905minus1 120579lowast)

ℎ (120579lowast)

ℎ (120579119905minus1)

ℎ (120579lowast) lt ℎ (120579

119905minus1)

(4)


119905= 120579lowast if120583 le 120572 (120579

119905minus1 120579lowast) and 120579

119905= 120579119905minus1

otherwise








120579(0)

119896)




120579(0)

2 120579

(0)

119896) and generate 120579(1)


tion 119891(1205792| 120579(1)

1 120579(0)

3 120579

(0)

119896)

Step 3 Generate 120579(1)119895

from 119891(120579119895| 120579(1)

1 120579

(1)

119895minus1 120579(0)

119895+1 120579

(0)

119896)

Step 4 Generate 120579(1)119896

from 119891(120579119896| 120579(1)

1 120579(1)

2 120579

(1)

119896minus1)


1 120579

(1)





(119905)



119901 (120579 120579lowast)

= 119891 (1205791| 1205792 120579

119896) 119891 (1205792| 120579lowast

1 1205793 120579


119891 (120579119896| 120579lowast

1 120579

lowast

119896minus1)

(5)















119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593)

2

120593119895=

1

119899

119899

sum

119905=1

120593119895119905 120593

=

1

119898

119898

sum

119895=1

120593119895

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895)

2

(6)



120583 = 120593 =

119899 minus 1

119899

119882 + (1 +

1

119898

)

119861

119899

(7)






radic119888= radic(

119882

)

dfdf minus 2

= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

dfdf minus 2

(8)




119888lt 12) we can




radiclowast

119888= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

df + 3df + 1

(9)


radiclowast

119901= radic119897

119871

(10)



119888(119896)


119888(119896)






119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)




119899

+

119898 + 1

119898

1205821 (12)





SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)




1 120579



119894 120579119894+1) = 0 Furthermore


119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=


119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)



SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)



8 Model Comparison











1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120579(0)

2 120579

(0)

119896) and generate 120579(1)


tion 119891(1205792| 120579(1)

1 120579(0)

3 120579

(0)

119896)

Step 3 Generate 120579(1)119895

from 119891(120579119895| 120579(1)

1 120579

(1)

119895minus1 120579(0)

119895+1 120579

(0)

119896)

Step 4 Generate 120579(1)119896

from 119891(120579119896| 120579(1)

1 120579(1)

2 120579

(1)

119896minus1)


1 120579

(1)





(119905)



119901 (120579 120579lowast)

= 119891 (1205791| 1205792 120579

119896) 119891 (1205792| 120579lowast

1 1205793 120579


119891 (120579119896| 120579lowast

1 120579

lowast

119896minus1)

(5)















119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593)

2

120593119895=

1

119899

119899

sum

119905=1

120593119895119905 120593

=

1

119898

119898

sum

119895=1

120593119895

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895)

2

(6)



120583 = 120593 =

119899 minus 1

119899

119882 + (1 +

1

119898

)

119861

119899

(7)






radic119888= radic(

119882

)

dfdf minus 2

= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

dfdf minus 2

(8)




119888lt 12) we can




radiclowast

119888= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

df + 3df + 1

(9)


radiclowast

119901= radic119897

119871

(10)



119888(119896)


119888(119896)






119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)




119899

+

119898 + 1

119898

1205821 (12)





SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)




1 120579



119894 120579119894+1) = 0 Furthermore


119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=


119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)



SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)



8 Model Comparison











1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593)

2

120593119895=

1

119899

119899

sum

119905=1

120593119895119905 120593

=

1

119898

119898

sum

119895=1

120593119895

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895)

2

(6)



120583 = 120593 =

119899 minus 1

119899

119882 + (1 +

1

119898

)

119861

119899

(7)






radic119888= radic(

119882

)

dfdf minus 2

= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

dfdf minus 2

(8)




119888lt 12) we can




radiclowast

119888= radic(

119899 minus 1

119899

+

119898 + 1

119898119899

119861

119882

)

df + 3df + 1

(9)


radiclowast

119901= radic119897

119871

(10)



119888(119896)


119888(119896)






119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)




119899

+

119898 + 1

119898

1205821 (12)





SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)




1 120579



119894 120579119894+1) = 0 Furthermore


119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=


119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)



SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)



8 Model Comparison











1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119861

119899

=

1

119898 minus 1

119898

sum

119895=1

(120593119895minus 120593) (120593119895minus 120593)

1015840

119882 =

1

119898 (119899 minus 1)

119898

sum

119895=1

119899

sum

119905=1

(120593119895119905minus 120593119895) (120593119895119905minus 120593119895)

1015840

(11)




119899

+

119898 + 1

119898

1205821 (12)





SE [119868 (119910)] =SD [119868 (119910)]radic119899

=[

[

1

119899

(

1

119899 minus 1

119899

sum

119894=1

(ℎ (119910 | 119909119894)) minus 119868 (119910))

2

]

]

12

(13)




1 120579



119894 120579119894+1) = 0 Furthermore


119896can be

calculated by

120588119896=

Cov (120579119905 120579119905+119896)

Var (120579119905)

=


119905=1(120579119905minus 120579) (120579

119905minus119896minus 120579)


119905=1(120579119905minus 120579)

2 (14)



SE (120579) = SDradic119899

radic

1 + 120588

1 minus 120588

(15)



8 Model Comparison











1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















1 119872


models Let 119891(119863 | 120579119894119872119894) = 119871(120579

119894| 119863119872

119894) denote the




119894| 119872119894) and sum119896

119894=1120587(120579119894| 119872119894) = 1 the marginal


119901 (119863 | 119872119894) = int

Θ119894

119891 (119863 | 120579119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894 (16)



119872119894) = 119871(119863 | 119872




119901 (119872119894| 119863)

=

119901 (119863 | 119872119894) 120587 (119872

119894)

119901 (119863)

=

120587 (119872119894) int119891 (119863 | 120579

119894119872119894) 120587 (120579119894| 119872119894) 119889120579119894

sum119896

119895=1[120587 (119872

119895) int119891 (119863 | 120579

119895119872119895) 120587 (120579

119895| 119872119895) 119889120579119895]

(17)



2 Let BF

12


BF12=

119901 (1198721| 119863)

119901 (1198722| 119863)

times

120587 (1198722)

120587 (1198721)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

(18)



119901 (1198721| 119863)

119901 (1198722| 119863)

=

119901 (119863 | 1198721)

119901 (119863 | 1198722)

times

120587 (1198721)

120587 (1198722)

(19)




1 In this way the







ln119901 (119863 | 119872119894) asymp ln119891 (119863 | 120579

119894119872119894) minus

119901119894

2

ln (119899) (20)


ln BF12

= ln119901 (119863 | 1198721) minus ln119901 (119863 | 119872

2)

= ln119891 (119863 | 12057911198721) minus ln119891 (119863 | 120579

21198722) minus

1199011minus 1199012

2

ln (119899) (21)


2



BIC12= minus2 ln BF

12

= minus2 ln(119891(119863 |

12057911198721)

119891 (119863 |12057921198722)

) + (1199011minus 1199012) ln (119899)

(22)


Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Locomotive


(a)

1 2 3 1 2 3

Right Right

Left Left

II I IIII II

(b)






12) (23)






119863 (120579) = minus2 log (119901 (119863 | 120579)) (24)


119901119889= 119863 (120579) minus 119863 (120579)


Then

DIC = 119863(120579) + 2119901119889= 119863 (120579) + 119901119889

(26)











119904= 100mm



egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










egra

datio

n (m

m)

100

0

B1

L1

L2

B2

Ls

D1 D2

Distance (km)




119894(eg119871






1(bogie I119909

1= 1

bogie II 1199091= 2) 119909

2(axel 1 119909

2= 1 axel 2 119909

2= 2

and axel 3 1199092= 3) and 119909

3(right 119909

3= 1 left 119909

3= 2)











119871 (120573 | 119863) =119899

prod

119894=1


i120573) 119905119894)]120592119894


= exp[119899

sum

119894=1

120592119894x1015840i120573] exp[minus

119899

sum

119894=1

exp (x1015840i120573) 119905119894]

(27)


119871 (120572120573 | 119863)

= 120572sum119899

119894=1120592119894 exp

119899

sum

119894=1

120592119894[x1015840i120573 + 120592119894 (120572 minus 1) ln (119905119894)

minus exp (x1015840i120573) 119905120572

119894]

(28)


119871 (120573 120591 | 119863)

= (2120587120591minus1)

minus(12)sum119899

119894=1120592119894 expminus120591

2

119899

sum

119894=1

120592119894[(ln (119905

119894) minus x1015840i120573)]

2

times

119899

prod

119894=1

119905minus120592119894

1198941 minus Φ[

ln (119905119894) minus x1015840i120573120591minus12

]

1minus120592119894

(29)





1

minus00321199092minus 0012119909

3) times 119905119894]

(30)



Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Model 119863(120579) 119863(120579) 119901119889




MTTF (times103 km)

I (1199091= 1)

1 (1199092= 1) Right (119909

3= 1) 59532 38703

Left (1199093= 2) 59543 38746

2 (1199092= 2) Right (119909

3= 1) 59740 39516

Left (1199093= 2) 59751 39560

3 (1199092= 3) Right (119909

3= 1) 59947 40343

Left (1199093= 2) 59958 40387

II (1199091= 2)

1 (1199092= 1) Right (119909

3= 1) 60205 41397

Left (1199093= 2) 60216 41443

2 (1199092= 2) Right (119909

3= 1) 60413 42267

Left (1199093= 2) 60424 42314

3 (1199092= 3) Right (119909

3= 1) 60621 43156

Left (1199093= 2) 60632 43203



1

minus01461199092minus 0050119909

3) times 1199051008

119894rfloor

(31)


119877 (119905119894| X)

= 1 minus Φ[

ln (119905119894) minus (5864 + 0067119909

1+ 002119909

2+ 0001119909

3)

(1875)minus12

]

(32)



10 Conclusions





Acknowledgments


References






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









an integrated procedure for bayesian reliability inference...

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