ieeetransactionsonreliability,vol.64,no.4,december2015 … tr 2015, no_4_weiwen... · 2015. 12....

IEEE TRANSACTIONS ON RELIABILITY, VOL. 64, NO. 4, DECEMBER 2015 1367

Leveraging Degradation Testing and ConditionMonitoring for Field Reliability AnalysisWith Time-Varying Operating Missions

Weiwen Peng, Yan-Feng Li, Yuan-Jian Yang, Jinhua Mi, and Hong-Zhong Huang, Member, IEEE

Abstract—Traditionally, degradation testing and conditionmon-itoring are used separately to investigate field reliability. Barriersare naturally formed between these two types of methods due tocondition-discrepancies between lab testing and field monitoring,as well as time-varyingmissions among product population groups.In this paper, a joint framework for field reliability analysis is pre-sented by integrating degradation testing data as well as missionoperating information with condition monitoring observations. Acoherent modeling strategy is introduced for the information in-tegration by gradually adopting random effects, dynamic covari-ates, and marker processes into a baseline stochastic degradationmodel. In detail, random effects are introduced to cope with theinherent unit-to-unit variation. Dynamic covariates are adopted todeal with the external condition heterogeneity. Marker processesare used to account for the time-varying missions. To facilitate in-formation integration and reliability analysis, the Bayesianmethodis used to implement parameter estimation and degradation anal-ysis. The reliability assessment of products' populations, degrada-tion prediction, and residual life prediction of individual productsare investigated. Finally, an illustrative example for field degra-dation analysis of oil debris in a lubrication system of a machinetool's spindle system is presented. The effectiveness of informationintegration and the capability of degradation inference are demon-strated through this example.

Index Terms—Field reliability, degradation model, random ef-fect, dynamic covariate, Bayesian method.

ACRONYMS AND ABBREVIATIONS

RUL Remaining useful lifeALT Accelerated life testADT Accelerated degradation testPDF Probability density functionCDF Cumulative distribution functionMCMC Markov chain Monte Carlo

Manuscript received April 01, 2014; revised October 27, 2014; accepted June08, 2015. Date of publication June 19, 2015; date of current version November25, 2015. This workwas supported in part by theNational Natural Science Foun-dation of China under the contract number 11272082 and in part the NationalScience and Technology Major Project of China under the contract number2013ZX04013-011. Associate Editor: R. Yeh.The authors are with the Institute of Reliability Engineering, Univer-

sity of Electronic Science and Technology of China, Chengdu, Sichuan,611731, China (e-mail: [email protected]; [email protected];[email protected]; [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TR.2015.2443858

NOTATION

Degradation process of a productDegradation observation at timeDegradation incrementShape function of a gamma processRate parameter of a gamma processPDF of a gamma distributionGamma functionFailure threshold of a degradation processFailure time of a productPDF of a degradation observationCDF of a failure timeReliability functionProbabilityVector of accelerated variablesVector of usage environment variablesVariable of mission typeMission durationMission intensityVector of parameters for a degradation testing model

Model parameters for the shape function of aGamma processVector of parameters for an ADT model

Model parameters associated with acceleratedvariablesVector of parameters for a field degradation model

Model parameters associated with mission operatingvariablesModel parameters associated with usageenvironment variablesPrior distributionLikelihood functionPosterior distributionDegradation testing dataADT dataField observation data

0018-9529 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

1368 IEEE TRANSACTIONS ON RELIABILITY, VOL. 64, NO. 4, DECEMBER 2015

I. INTRODUCTION

F IELD reliability has long become a critical issue forboth manufacturers and customers of modern products.

For manufacturers, field reliability analysis is often used toestimate field returns within a given warranty period [1]. Aprecise estimation of field reliability is essential for the op-timization of monetary reserves for warranty claims. In thelong run, manufacturers will benefit from the field reliabilityanalysis by eliminating potential weaknesses, and by deliveringmore competitive products. On the other hand, for customers,field reliability analysis is generally adopted to estimate theremaining useful life (RUL) of products [2]. A real-time pre-diction of RUL is one of the key factors for the optimization ofsystem operation, and the planning of preventive maintenance.Under the support of real-time prediction of RUL, customerswill obtain an excellent availability of products with cost-ef-fective capital goods. Commonly, reliability tests are used bymanufacturers to extrapolate products' reliabilities under fieldconditions (i.e. typical working conditions in the service stageof a product), while condition monitoring is used by customersto estimate products' RULs under unit-specific conditions. Agap is naturally formed between these two types of methods,as well as the corresponding reliability data when they are usedseparately.Unfortunately, neither a precise estimation nor a real-time

prediction can be achieved for the field reliability of complexproducts nowadays, if the degradation testing and conditionmonitoring are used separately. Take a machine tool as anexample. Due to continually increasing competition in the ma-chine tool industry, manufacturers of machine tools are undergreat pressure to deliver products with high reliability andavailability under limited time. Accelerated life tests (ALT) andaccelerated degradation tests (ADT) are used by manufacturersto investigate the field reliability of machine tools. However,due to differences of usage environments and operating mis-sions between lab tests and field operations, large discrepanciesare often found between the results of lab test inference andfield failure analysis. This difference is large because thereliability of machine tools is mainly influenced by two groupsof factors: the usage environment related factors such as tem-perature, moisture, and vibration; and the operating missionrelated factors such as work-piece material, cutting speed, anddepth of cut. These factors are often dynamic or time-varyingunder field conditions. The misspecification or simplificationof these factors generally leads to a biased inference of fieldreliability. On the other hand, for users of machine tools, thereduction of the total cost of ownership, and the improvementof availability, have become critical issues to make machinetools more profitable and productive. Condition monitoring andRUL prediction are adopted to investigate the field reliabilityof machine tools. Because machine tools are complex systemswith multiple interactive components, indicators of systemreliability can sometimes hardly be monitored. The indicatorsof system reliability for the machine tools generally includemanufacturing precision, position precision, and oil debris.The observations of these indicators are often sparse becausecontinuous monitoring of these indicators is either technicallyimpractical or economically unaffordable. Most of these indi-cators are observed and measured discontinuously when themachine tools are idle, generating fragmented observations of

these indicators. In this situation, results of field reliability es-timation using these sparse measurements under time-varyingconditions are generally imprecise for further utilization.Lab test methods for field reliability analysis are impacted

by the uncertainty introduced by time-varying field conditions,while condition monitoring methods are challenged by the largevariance resulting from fragmented field observations. It wouldbe useful to combine these methods to investigate their corre-lations, and to mitigate their limitations. Ye et al. [3] raised thequestion about how heterogeneities in operating environmentsaffect the predictions of field failures and the planning of labtests. They introduced a model that linked the lab failure timedistribution and field failure time distribution. Improved fieldreliability estimation and optimized ALT planning have beendemonstrated through real life examples. Meeker et al. [4] pre-sented a method for field reliability prediction based on the ALTresults and field usage information. However, both methods arelimited to field reliability analysis with lifetime data. Consid-ering field reliability assessment based on degradation analysis,Liao and Elsayed [5] developed a method to relate ADT exper-iments to field applications by incorporating stochastic stressesinto ADT models. However, only lab test data and field stressinformation can be integrated in their method. Recently, Liaoand Tian [6] introduced a framework to integrate ADT modelsand Bayesian updating techniques for the RUL prediction ofindividual units under time-varying operating conditions. AnADT model was used by them to investigate degradation anal-ysis with time-varying operating conditions. However, the ef-fects of unit-specific usage environments and time-varying mis-sions cannot be differentiated under their model. Unfortunately,in practical engineering, the field reliability of a product can beaffected by many factors. Such as the example of machine toolsdescribed above, different factors contribute differently to fieldreliability. As a result, the inherent unit-to-unit variation, the ex-ternal unit-specific usage environments, and time-varying mis-sions should be treated separately in field degradation modeling.The models summarized above are not adequate to handle thischallenging problem. It then motivates the research presented inthis paper.This paper presents an integrated framework for field reli-

ability analysis with a Bayesian fusion strategy and a modelcoupling technique. Under this framework, the strengths of labtest methods and condition monitoringmethods are synthesized.Degradation testing data and in situ mission operating infor-mation are integrated with condition monitoring observationsto facilitate the reliability assessment of products' populationand real-time RUL prediction for individual products. A jointmodel for multi-source information fusion is constructed bycoupling the random effects, time-varying covariates, andmarker processes into a degradation model. The effects ofunit-to-unit variation, heterogeneous usage conditions, andtime-varying missions are modeled coherently. The problemsof fragmented degradation observations and simplified in-fluence factors are handled properly. The remainder of thispaper is organized as follows. Section II presents a literaturereview, and some discussion of field reliability analysis. Themodels for degradation modeling and information integrationare presented step-by-step in Section III. Parameter estimation,population reliability estimation, and individual RUL predic-tion are described in Section IV. In Section V, an illustrativeexample is presented to demonstrate the proposed method.

PENG et al.: LEVERAGING DEGRADATION TESTING AND CONDITION MONITORING 1369

Finally, Section VI concludes this paper, and highlights severalpoints for future research.

II. RELATED WORKS

Degradation testing and condition monitoring are two typesof methods for reliability analysis. In the past decades, variousmethods have been introduced to carry out field reliability esti-mation or RUL prediction or both. There are two classical linesof research that carry out the investigation of field reliability es-timation and RUL prediction.One research line follows the gradual development and im-

provement of degradation models. Various degradation modelsfor different data types have been introduced to handle differentengineering applications. These models include degradationpath models [7], [8], Markovian models [9], [10], and stochasticprocess based models [11]–[22]. For the stochastic processbased models, the models based onWiener processes [11]–[15],gamma processes [16]–[18], and inverse Gaussian processes[19]–[22] have been studied to facilitate degradation basedreliability analysis. Among the various degradation models,the paper by Lu and Meeker [7] has been recognized as oneof the pioneering research works. They introduced a generaldegradation path model, which was composed by an actualdegradation path function, and a random error term. They alsoderived the time-to-failure distribution based on this model,and introduced the methods for parameter estimation andreliability assessment. Many papers were published followingtheir work. Comprehensive reviews about these degradationmodels for reliability assessment and RUL prediction wereseparately presented by Ye and Xie [23], and Si et al. [2]. Indetail, the Wiener process and its extensions have been widelystudied for degradation modeling. Ye et al. [14] proposed adegradation model using a random effects Wiener process withmeasurement errors. The heterogeneity of degradation rateamong product population, together with the imperfectness ofdegradation measurements, were investigated in their model.In addition, Si et al. [13] proposed a nonlinear drift diffusionprocess for degradation modeling and RUL prediction basedon a Wiener process. A time-space transformation was adoptedby them to derive the analytical approximation of the failuretime distribution. Wang et al. [15] proposed a generalizedWiener process model for degradation modeling. Variousexisting Wiener process degradation models were treated astheir model's limiting cases. For the development of gammaprocess based degradation models, Lawless and Crowder [17]incorporated random effects and covariates into the gammaprocess for degradation modeling with unit-specific variabilityand explanatory variables. A closed form of the time to failuredistribution was derived by them as well. Wang et al. [18]introduced a change-point gamma and Wiener process fora degradation process with change points or multi-phases.Real-time reliability assessment using this change-point degra-dation model was also studied. In general, stochastic processdegradation models are often used due to the flexibility ofthese models for the incorporation of nonlinear degradation,unit-specific variability, and time-varying covariates. Amongthem, the Wiener process degradation models were the mostused, because of their tractability and simplicity. These modelswere suitable for non-monotonic degradations. For monotonicdegradations, the gamma process degradation models were

often used. However, the investigation of gamma processdegradation models and their implementations for degradationanalysis considering the inherent unit-to-unit variation, theexternal unit-specific usage environments, and time-varyingmission operating heterogeneity have not been well studied yet.The other line of research focuses on the integration of

multi-source information for field reliability analysis. Padgettand Tomlinson [12] presented a method for lifetime inferenceby integrating degradation measures and failure times obtainedthrough accelerated tests. Meeker et al. [4] introduced a use-ratemodel for field reliability analysis by integrating ALT dataand field usage information. Hong and Meeker [24] furtherpresented a method for field failure prediction by developinga cumulative exposure model to integrate dynamic usage in-formation and failure time data collected in the field. All thesemethods were based on lifetime data analysis. Consideringthe field reliability analysis through degradation analysis,Gebraeel et al. [25] proposed a method for RUL prediction byincorporating information about the reliability characteristicsof a product's population and real-time sensor information ofspecific products interested. Gebraeel and Pan [26] furtherextended the method to incorporate the real-time status ofenvironmental conditions. Recently, Chen and Tsui [27] ex-tended this information integration method to a more specificdegradation situation. Within their work, degradation processeswith change points, unit-specific variance, and correlateddegradation prediction were modeled and investigated underthe Bayesian information integration framework. Meanwhile,Liao and Tian [6] proposed a Bayesian framework for theRUL prediction of individual products by further extending theinformation integration method as mentioned above. In general,the information integration method for field reliability analysisis heading to a more specific and practical situation. Varioustypes of information have been incorporated to facilitate fieldreliability estimation or RUL prediction or both. The inherentunit-to-unit variation and the time-varying usage environmentsare investigated. However, the methods summarized abovewere all based on the Wiener process degradation model dueto its simplicity and tractability. Few works focused on othertypes of degradation models; a rare exception was the workpresented by Wang et al. [28]. They proposed a Bayesian eval-uation method by integrating ADT test data and field failuredata, where a calibrating factor was incorporated to link themodel for ADT and failure observations. Both Wiener processand gamma process degradation models were studied in theirwork. However, only lab test information and field observationwere incorporated in their work. Information of field conditionscan hardly be incorporated in their model, which makes theapplication of this method limited, especially for the situationwhere field reliability is affected greatly by field conditionsand operating missions such as the situation of machine toolsintroduced above. Accordingly, the integration method for fieldreliability analysis by incorporating degradation tests, conditionmonitoring, and mission operating information deserve furtherstudy.Based on the literature review presented above, the contri-

butions of the proposed method lie in the following aspects.From the line of research about degradation modeling for fieldreliability analysis, other than introducing a specific modelsolely for degradation analysis with time varying operatingmissions, a group of degradation models raising from baseline


degradation, to lab degradation, and finally into field degrada-tion, are constructed. A link between degradation test modelsand condition monitoring models is presented. It is imple-mented by gradually incorporating random effects, dynamiccovariates, and marker processes into a baseline degradationmodel. The unit-to-unit variation and the effects of usageconditions and time-varying missions are modeled coherently.Moreover, a gamma process degradation model is investigated,and mathematically tractable conditional distributions for bothRUL prediction and degradation inference are obtained. Fromthe line of information integration for field reliability analysis,a coherent method for integrating degradation test, conditionmonitoring, and mission operating information, is presented.Together with the proposed model, the integration methodis extended to the gamma process degradation model, whichis suitable for monotonic degradation modeling. Moreover,information about usage environments and operation missionsis treated separately in the proposed model. In detail, theinformation about usage environments is integrated throughdynamic covariates, and the information about operation mis-sions is incorporated through marker processes. It can facilitatethe identification of the major impact factor to the degradationprocess. Especially for complex engineering systems, such asthe machine tools introduced above, different factors affect thedegradation process differently. It can also deliver a deeperunderstanding of the connection between lab tests and fieldobservations, which in turn improve field reliability assess-ment, and RUL prediction for manufacturers, and customersrespectively.

III. MODELS

To relate lab test data to condition monitoring data, the effectsof unit-to-unit variability, usage conditions, and time-varyingmissions should be modeled coherently. In this section, themodel proposed in this paper is constructed by graduallyintegrating random effects, dynamic covariates, and markerprocesses into a baseline degradation model.

A. Baseline Degradation Model

In this paper, we investigate the degradation process withcontinuous state and continuous time. For a product withdegradation process , given the degradationthreshold , the failure time of this product is defined as

. We consider the degradation pro-cesses with -independent non-negative increments observedfor many engineering systems today [23]. For illustration of theproposed method, the gamma process is used as the baselinedegradation process model. A basic gamma process model isthen defined for the degradation process with .It has the following properties.a) The degradation increments

are -independent.

b) The degradation increment follows a gamma dis-tribution , where is a mono-tone increasing function with .

The probability density function (PDF) of a gamma distribu-tion for a random variable with mean

and variance is

(1)

where is the shape parameter, and is the inverse scaleparameter.The degradation process is described as

. Its degradation increment is given aswith . The PDF of the

degradation observation is obtained as

(2)

Because the failure time of this degradation process is de-fined as , the cumulative distribu-tion function (CDF) of the failure time is obtained as

(3)

where , which is a lowerincomplete gamma function.

B. Degradation Model for Testing DataDegradation tests or ADT are generally used to obtain lab test

data, and to assess system reliability by mimicking normal fieldconditions and nominal operating missions. Commonly, theseconditions and missions are well-controlled. Due to the uncer-tainty introduced in the design process or manufacturing processor both, there is unit-to-unit variation among products, whichleads to the dispersion of degradation curves. It is common tointroduce a unit-specific random effect into the baseline degra-dation model to account for the unit-to-unit variation. Two sce-narios are considered in this paper: the normal degradation tests,and the accelerated degradation tests.For the scenario of normal degradation tests, a random effect

is incorporated in the basic degradation model by assuming thatthe inverse scale parameter follows a gamma distribution as inLawless and Crowder [17], and Tsai et al. [29]. In this way, boththe variance among the product's population and the variancewithin a product's degradation curve can be well modelled. Thatis, with . Then the PDF ofthe degradation observation is given as (4) at the bottom of thepage.It can be derived that follows an -distribu-

tion with parameters , and [17], i.e.,. Accordingly, the reliability function is ob-

tained as shown in (5) at the bottom of the next page, where

(4)


is the CDF of a -distribution withparameters and

is an incomplete beta func-tion, and is a betafunction.As for the scenario of ADT, the baseline degradation

model is extended to an ADT modelas , where is a vector of accel-erated variables. The accelerated variables are the stressfactors used in the ADT, such as temperature-cycle, use-rate,voltage-stress, and so on [30]. The function is obtainedby incorporating the effects of accelerated variables into theshape function . The effects of accelerated variables aregenerally formulated by a parametric regression function, aCox-proportional hazards model, or some other accelerationfunctions such as a power-law model, an Arrhenius reactionrate model, or an inverse-log model, to name a few. Therandom effect is incorporated in this ADT degradation modelby assuming that the inverse scale parameter follows a gammadistribution. Similarly, the conditional PDF of the degradationobservation and the conditional reliability function are obtainedas shown in (6) and (7) at the bottom of the page.

C. Degradation Model for Field Observations

A degradation model for field observations is a model to de-scribe the monitored degradation observations by consideringthe effects of usage conditions and time-varying missions. It isaimed to integrate condition monitoring data and mission op-erating information. Based on the engineering imperative pre-sented by the case of machine tools introduced above, the in-formation of field observations is categorized into three parts:

the monitored degradation observations, the usage environmentinformation, and the operating mission information. This cate-gorization is aimed to model the effects of usage environmentsand operating mission separately on the observed degradation.Because the usage environments are generally static or well-or-dered situations, they are treated as non-stochastic covariatesin this paper. Because the operating missions are generally un-predictable, and affected by many factors, they are modeled bymarker processes in this paper.The variables of usage environments are generally mea-

surements of conditions that affect the products' reliability,such as moisture, vibration, pressure, and so on [31]. Theeffects of these variables are introduced into the degrada-tion model through the modification of the rate parameter

[28]. The degradation process model is then given as, where is formulated

following the idea of ADT models, and is the accel-erated variable in the field conditions if it is observed. As aresult, the conditional PDF of the degradation observation andthe conditional reliability function are obtained as shown in (8)and (9) at the bottom of the page.For the incorporation of mission operating information,

marker processes are used in this paper. By marker processes,we mean that the missions are characterized by various indexes,and these indexes are used as markers to differentiate differentmissions. In detail, the missions' type, duration, and intensityare taken into consideration. The information about these in-dexes is recorded and incorporated into the degradation model.The specification of mission operating information is originatedfrom the fact that modern products and engineering systemscan implement various missions. Each mission is characterizedby a specific duration of time, and a particular stress intensity.

(5)

(6)

(7)

(8)

(9)


Different degradation levels or rates may be observed in dif-ferent missions. Suppose there are types of missions thatcan be carried out by a product. The marker process of missiontype is a serial of numbers within the integers . It actsas a marker to differentiate different mission types. For a spe-cific mission with , the marker processes ofmission duration and mission intensity are serialsof real numbers in their respective intervals. The processes

and are used to describe the detailed information ofmission operations experienced by the product within a seriesof missions. The degradation characteristics of a product arethen differentiated mission by mission. By taking account ofthe marker processes, the degradation model of the product isgiven as a model of degradation increment within a mission asshown in (10) at the bottom of the page, where

is the specific time that the mission starts, and isthe degradation observation at time point .The key point for degradation modeling with mission oper-

ating information is to modify the shape function by incorpo-rating the effect of mission intensity into the shape functionas or if the accelerated variablesare also presented in the field observations. Meanwhile, becausethe shape function is a function of both the operation time andthe mission type , the characteristic of -independent degra-dation increments is only valid within the time interval of a spe-cific mission . The degradation observations are accumulatedby the degradation increments generated in all completed mis-sions. In other word, it is impossible to derive a conditionalPDF for the degradation observation at any time point as (4),(6), and (8). The only way to describe the degradation observa-tion is through the degradation increments conditionally on theavailable mission operating information of a specificmission .As a result, the conditional PDF of the degradation observation,and the conditional reliability of the product are obtained for amission as shown in (11) and (12) at the bottom of the page,where .

IV. PARAMETER ESTIMATION, AND RELIABILITY INFERENCE

The models for degradation tests, condition monitoring, andmission operating information are presented above. A coherentframework is then constructed to integrate the information forfield reliability inference in this section. The Bayesian methodis adopted to implement parameter estimation and reliability in-ference. Four aspects are highlighted in this section: 1) deriva-tion of the prior distribution, 2) formulization of the likelihoodfunction, 3) construction of the joint posterior distribution, and4) calculating or updating the field reliability inference.

A. Prior Distribution

The prior distribution is a representation of quantified priorinformation. Based on the degradation model presented in (4),(6), and (10), all model parameters are summarized as shown in(13) at the bottom of the page.Due to the models for the degradation test, condition moni-

toring, and mission operating information being derived hierar-chically, their model parameters are coupled together as shownin (13), where is included in , and is furtherincluded in . This approach can facilitate the combina-tion of multiple-source information, and the updating of priordistributions. Two types of prior distributions are commonlyadopted. The first type is the prior distributions derived fromsubjective prior information. The second type is the prior distri-butions transformed from analysis results of a previous data set.Generally, two kinds of advantages are achieved for field re-

liability analysis by using these two types of prior distributions.For the prior distributions derived from subjective information,the incorporation of prior information can improve the preci-sion of field reliability analysis, and further add value to thecorresponding decision making. This kind of prior informationis generally in the form of expert judgment which contains theengineer's experience and expertise. Wang and Zhang [32] havedemonstrated the advantage of incorporating subjective expert

(10)

(11)

(12)

(13)


judgment for residual life prediction. For specific techniquesabout the elicitation of experts' probability, and the derivationof the prior distribution, please refer to the work by O'Haganet al. [33].For the prior distributions transformed from the analysis re-

sults of the previous data set, a continual updating strategy canbe formed for field reliability by making the posterior distribu-tions of model parameters for the previous observed data as theprior distributions of model parameters for the present observeddata. This kind of prior information is commonly in the form ofa probability density function which contains all the informationaggregated to the present time. Various works for field reliabilityanalysis have utilized this advantage of prior distributions, in-cluding [5], [26], [28], and [34]. In this paper, the continual up-dating of model parameters is employed to gradually integratedegradation test data, ADT data, and condition monitoring datawith mission operating information. A further demonstration ofthe prior distribution for the proposed model is presented laterin an illustrative example.

B. Likelihood Function

A likelihood function is utilized to describe the informationcontained in degradation testing data, condition monitoringdata, and mission operating data. It is constructed based onthe PDF functions of degradation observations or degradationincrements using the models derived above. Due to thesemodels being derived hierarchically for different data types, thelikelihood functions for these data are presented progressivelyin the following part for information integration.Given degradation testing data , suppose units

are tested, and the unit is observed at differenttime points, where . Due to the degra-dation increment of degradation testing being modeled as

with , the likelihoodfunction for the degradation testing data is given as (14)at the bottom of the page, whereand includes all the random effectparameters for the units.Given the ADT data , suppose units are tested

at different levels of accelerated stresses. Under thestress, there are units under test, and the unit is ob-served at different time points, where

, and . Based on thedegradation model for ADT data, which is given as

with , the likelihood functionfor the ADT data is given as (15) at the bottom of thepage, where , andincludes all the random effect parameters for the units.Given the condition monitoring data and mission operating

information , suppose that units are monitored.For the unit, there are degradation observationsobtained from missions. The mission type , themission duration , and the mission intensity arerecorded. The usage environments and accelerated stresses

are also monitored. Based on the degradationincrement for the field observations presented in (10), thelikelihood function for the condition monitoring datawith the mission operating information is given as (16)at the bottom of the page, where

.Based on the likelihood functions derived above, the joint

likelihood function for degradation testing, condition moni-toring, and mission operating information is obtained as (17) atthe bottom of the page.From the joint likelihood function, we can find that all the in-

formation contained in the observed data is concentrated on the

(14)

(15)

(16)

(17)


parameters of the field degradation model, i.e., . A gradualintegrating of information from degradation testing data, ADTdata, and condition monitoring data with mission operating in-formation is implemented through the coupling of model pa-rameters and their likelihood functions.

C. Posterior DistributionBy combing prior distributions of model parameters with the

likelihood function of reliability data through Bayes' theorem,the joint posterior distribution of model parameters is obtained.This joint posterior distribution is a description of the integratedresults of degradation testing, condition monitoring, and mis-sion operating information. Field reliability is then implementedbased on this joint posterior distribution and the correspondingdegradation models.Based on the prior distribution of parameters, and the likeli-

hood function presented in (17), the joint posterior distributionis obtained as (18) at the bottom of the page.There is often no analytical solution to (18). The Markov

chain Monte Carlo (MCMC) method is generally used togenerate posterior samples of model parameters from this jointposterior distribution. These posterior samples are then used toimplement reliability inference and degradation prediction. Inthis paper, the software WinBUGS [35] is used to facilitate theimplementation of MCMC. For specific information about themodeling and calculation through WinBUGS, please refer tothe works [36], [37].

D. Reliability InferenceWhen the parameters of degradation models introduced

above are estimated, it is of interest to investigate the fieldreliability of a product through the degradation models. Thereare generally two kinds of field reliability that manufacturersand customers are concerned about. The first is the overall fieldreliability of the product population, which is related to the de-cision-making of product warranty. This type of field reliabilityis of particular importance for manufacturers of a product. Theother is the particular field reliability of an individual product,which is related to the planning and optimization of preventivemaintenance. This type of field reliability is of critical signif-icance for users of a product. In the following part, both thefield reliability of a product population, and the degradation

prediction and residual life prediction of individual productsare presented.For the field reliability of a product population, if all the prod-

ucts are used under a specific mission type with a missionintensity , the conditional field reliability is given as (19)at the bottom of the page.The mission types and the mission intensity that a

product experienced in the field are generally hard to predictprecisely. This difficulty is because the arrival of different mis-sion types is influenced by various uncertain factors. In addi-tion, the sequence of these mission types and their durations aregenerally unpredictable. As a result, (19) is generally used to es-timate the field reliability of product population in some typicalmission types and mission intensity.Because (19) cannot be solved analytically, simulation

based integration is adopted to facilitate the calculation. It isimplemented by substituting the posterior samples of modelparameters generated fromthrough the MCMC method into the conditional reliabilityfunction . By doing so, eachposterior sample will generate an estimation of fieldreliability. A group of estimations of field reliability are thenobtained using the posterior samples of model parameters.Statistics of these estimations form the estimations of the fieldreliability of the product population, which include kerneldistribution, mean, variance, interval estimation, and so on.For degradation prediction of the product under field

conditions, the degradation prediction for a coming missionis given as follows, when the mission type , mission du-ration , and mission intensity for the comingmission are known. See (20) at the bottom of the next page,where isthe PDF of degradation at the time point

, and.

Similarly, the field reliability of the product duringthe mission time of the coming mission is given as fol-lows, when the mission type , mission duration ,and mission intensity for the coming mission areknown. See (21) at the bottom of the next page, where

is the reliabilityof the product during the mission duration of mission with

(18)

(19)


, and which is alower incomplete gamma function.Both (20) and (21) are indices derived for the coming mis-

sion of the product, which are derived on the condition thatthe mission type , mission duration , and mission inten-sity of the coming mission are known. Due to the unpre-dictability of the mission sequence that the product will endurein the remaining life, it is impractical to derive an estimation ofRUL of this product. However, a preventive maintenance and anoperation planning still can be made based on the prediction ofthe degradation and reliability of the product in the coming mis-sion. This ability exists because there is a time interval betweenthe present observation time point and the predicted observa-tion time point . A decision can be drawn based on the com-parison of the degradation level between the predicted degrada-tion and the degradation threshold. It is of practical meaning forthe management of the engineering systems or products that canfulfill multiple missions, yet the information of these missionsis only available for the coming one or two missions. The fieldanalysis of machine tools introduced in the introduction sessionis a good example. A further demonstration is also presented inthe later example section.The calculations of (20) and (21) are implemented

through the incorporation of the simulation based integra-tion, which are the same as the one used for (19). Theposterior samples of model parameters generated from

using the MCMCmethod are substituted into the conditional PDF function

, and the conditional re-liability function . Both of thesetwo functions have analytical expressions as presented in (20),and (21). By doing so, a group of calculated samples of theconditional PDF and conditional reliability are then obtainedusing the posterior samples of model parameters. Degradationprediction and reliability inference for individual units duringthe coming mission are then simulated and summarized basedon these calculated samples.

V. EXAMPLES

Take the spindle system of a machine tool as an example. Thespindle system transmits the required energy, and rotates the tool(grinding, milling and drilling) precisely to implement high-pre-cision machining. It exerts a great effect on the material removalrate and the final quality of machined parts. The spindle systemis expected to possess high reliability and availability. Degra-dation testing and condition monitoring are implemented on thespindle system. Themeasurement and monitoring of the amountof debris in the lubricating oil is a possible way to monitor thedeterioration of bearings and gears in the spindle system. In thissection, degradation analysis of oil debris is used to illustratethe proposed method.To avoid proprietary issues, degradation data are simulated

using the estimated parameters from the original degradationdata, and the units of values are omitted. Largely, however, thecharacteristics of the degradation observations and the applica-tion of the proposed methods are the same as with the originaldata.

A. Degradation Data

The degradation data of oil debris include the testing dataand field observations. In detail, five spindle systems are testedby the manufacturers during the design and manufacturing pro-cesses. Testing data are recorded from the degradation tests,which include general degradation test data and ADT data. Thedegradation data collected by manufacturers are presented inFig. 1. These tests are designed and implemented by profes-sional reliability test engineers. We are not able to discuss moredetail about the reliability tests. However, the accelerated degra-dationmodel is presented in the following part. The specific datafor the normal degradation test, and ADT tests are respectivelygiven in Tables I, and II.These five systems are also monitored by the users during the

usage processes. Field observations are obtained from condition

(20)

(21)


TABLE IDEGRADATION DATA OBTAINED FROM NORMAL DEGRADATION TEST

Fig. 1. Normal degradation test data, and ADT data.

monitoring and mission tracking, which include field degrada-tion observations, and mission operating records. Degradationdata and operating information collected by users are presentedin Fig. 2.The degradation observations are observed during the idle

time of the machine tools when they finish specific missions.As presented in Fig. 2, there are five types of missions that

Fig. 2. Field degradation data, and mission type information.

the machine tools fulfilled. The mission types, mission inten-sity, and mission durations were recorded by users of machinetools based on the operation sheets of these missions. In addi-tion, the accelerated variables experienced by the machine toolsunder typical use conditions in the factories, and the environ-ment information, were also recorded. The effect of these vari-ables and the specific model are presented in the following part.The specific data for mission operating information, condition


TABLE IIADT DATA WITH ACCELERATED VARIABLE PRESENTED IN THE BRACKET FOR EACH SAMPLE

Fig. 3. Prediction of field degradations at three leave-one-out cross-validationpoints.

monitoring degradation data, and the values of accelerated vari-ables and environment variables for machine tools are given inTable III.

B. Degradation ModelsBased on the degradation models introduced in Section III,

the models for degradation testing data and the model for con-dition monitoring information are chosen. These models are

Fig. 4. Boxplot of relative errors of degradation predictions.

chosen by considering both the characteristic of degradationdata presented in Fig. 1 and the subjective information of ex-perts in that domain.A gamma process with random effects is used as the degrada-

tionmodel for normal degradation tests. As the oil debris is oftencharacterized as an increasing degradation rate, a power-law


TABLE IIIDEGRADATION OBSERVATION AND MISSION OPERATING INFORMATION

function is used as the shape function in the degradation modelas follows.

(22)where .The degradation model for ADT data is formulated by in-

corporating the accelerated variables into the shape function.For the ADT of the oil debris, only one accelerated variableis used, and a linear accelerated model is adopted in the de-sign of ADT tests. Here, we incorporate this accelerated modelinto the normal degradation test model. The model for ADT is

then given as (23) at the bottom of the page, where, and is the accelerated variable.

The degradation model for condition monitoring dataand mission operating information are formulated by sepa-rately incorporating covariates into the shape function andscale parameters of the ADT model presented above. It isgiven as shown in (24) at the bottom of the page, where

is the accelerated vari-able observed under typical use conditions in the factories,

is a nominal mission intensity designed for themachine tool, and is the environment variable.As discussed in Section III, the effect of the mission in-

tensity is considered by modifying the shape functioninto . It is implemented by

introducing an exponential function

(23)

(24)


TABLE IVSTATISTICAL SUMMARY OF POSTERIOR SAMPLES OF MODEL PARAMETERS

into the shape function to directly affect the lifetime . Theeffect of the mission intensity is presented through itscontrast with a nominal mission intensity designed forthe machine tool. In addition, the environment variable isconsidered by modifying the scale parameter into ,which aims to adjust the variance of the degradation model.

C. Parameter Estimation and Reliability InferenceFollowing the procedure presented in Section IV, prior dis-

tributions for model parameters are derived. In this ex-ample, non-informative prior distributions are used for theseparameters. In this way, the results of estimation and inferencegenerated by the proposed method are mainly based on the inte-gration of information presented above. In detail, according tothe principle of indifference [38], -uniform distributions withlarge intervals are assigned for the model parameters as priordistributions:

, and. denotes a -uniform distribution with

an interval . The intervals of these prior distributions arechosen large enough to make these priors act as non-informativepriors in the Bayesian analysis.By combining the degradation models presented in

(22), (23), and (24) with the general likelihood functionspresented in Section IV, the joint likelihood functionfor the degradation data given above is obtained shownin (25) at the bottom of the page, where

are the data sets given above; and

, and

.According to Bayes' theorem, the posterior distribution for

model parameters is obtained by combining the prior distribu-tions given above with the joint likelihood function given in(25). It is given as (26) at the bottom of the page.

(25)

(26)


Fig. 5. Estimations of population reliability with different mission types.

Based on the joint posterior distribution, posterior samples ofmodel parameters are simulated by implementing the MCMCthrough WinBUGS. In this example, 20,000 posterior samplesare generated. A statistical summary of these posterior samplesis presented in Table IV.To verify the effectiveness of the proposed method, a leave-

one-out cross-validation is implemented. We retain three latestdegradation points of the field observations given in Table III,and carry out parameter estimation using the remaining data.The posterior samples generated by these data sets are then usedto predict the degradations at these three leave-one-out timepoints. A pictorial description of the predicted degradations andthe observed degradations at the three leave-one-out time pointsis presented in Fig. 3. The relative errors of these degradationpredictions for each sample are also obtained as presented inFig. 4. Both boxplots of the degradation predictions and the rel-ative errors demonstrate high precision of the proposed methodfor field degradation prediction. We then proceed to the reli-ability estimation, degradation inference, and RUL predictionusing the results of information integration.By utilizing the simulation based method presented in

Section IV, the reliability assessment of the machine tool'spopulation is obtained based on the posterior samples of modelparameters from (26). The degradation prediction, and theRUL prediction for individual machine tools are obtained aswell based on these posterior samples of model parameters.For the manufacturers of machine tools, the conditional fieldreliability of the product population is calculated by assumingthat the products are working under a specific mission typewith a mission intensity . Fig. 5 presents the estimationsof population reliability with different mission types. The

corresponding mission intensity are 1800, 2200, 2600, 3000,and 3400 for mission types 1, 2, 3, 4, and 5.From the estimations of population reliability, we can find

out that the field reliability of machine tools can be greatly af-fected by the mission type and mission intensity fulfilled by themachine tools. A mission type with a large mission intensity,such as the mission type 5 with mission intensity 3400 shownin Fig. 5, generally leads to poor field reliability of the machinetool. Manufacturers can utilize these estimation results to fa-cilitate decision making for the machine tools. A more specificestimation of field population reliability can be obtained basedon the field degradation and posterior samples derived above, ifmore information about the process of mission types that a ma-chine tool will experience is available.For the user of a machine tool, if missions are assigned for

this machine tool, degradation prediction can be obtained forthe time points when the missions are completed. In addition,a prediction of RUL for individual machine tools under a spe-cific mission type and mission intensity can be obtained as well.Fig. 6 presents degradation predictions for a machine tool underdifferent mission types. This machine tool is the sample 1 pre-sented in Fig. 2. The predictions of RUL under different missiontypes are also presented in Fig. 6.The degradation predictions and the RULs vary with the mis-

sion types that the machine tool endured. The RULs changedfrom 110 to 23 with the mission type changing from missiontype 1 to mission type 5. As a result, different strategies aboutoperation management and preventive maintenance are goingto be implemented on the machine tool if different missiontypes are carried out by the machine tool. In addition, basedon the degradation model and the inference procedure de-scribed above, more specific predictions of field reliability forindividual machine tools can be implemented if more specific


Fig. 6. Prediction of degradation and RUL of a machine tool with different mission types.

information about their respective missions in the future isavailable.

VI. CONCLUSIONThis paper investigates field reliability analysis by lever-

aging degradation testing and condition monitoring data.Time-varying mission operating information is incorporatedto facilitate the degradation analysis with degradation testingdata and condition monitoring observations. The models fornormal degradation testing data, ADT data, and degradationobservations are derived progressively by separately incor-porating random effects, dynamic covariates, and markerprocesses, into a baseline degradation model. The effectsof inherent unit-to-unit variation, external condition hetero-geneity, and time-varying missions are modelled coherentlywith these degradation models. Estimations of populationreliability, and predictions of individual degradations andRULs, are implemented using a Bayesian information fusionstrategy. Degradation analysis of machine tools' spindle sys-tems is used to demonstrate the proposed method, where thedegradations of oil debris in the lubrication systems of themachine tools' spindle systems are used. Within this example,the degradation inference capability of the proposed methodis verified through leave-one-out cross-validation. In addition,by studying the degradation under different mission types, theeffectiveness of incorporating mission operating information isdemonstrated for both the reliability estimation of product pop-ulation and the degradation prediction of individual products.Time-varying missions can exert significant influence in thedegradation of the product under field conditions. Accordingly,the decision-making concerning product warranty for productmanufacturers, and strategy-designing about preventive main-tenance for system users, should consider the effect of timevarying missions.

There are several points that deserve further investigation.One is the studying of a real-time estimation method for the pro-posedmodels and the integration ofmulti-source information. Astudy of degradation test planning and ADT planning by consid-ering the effect of field condition heterogeneity and time-varyingmissions is also of interest for further investigation.

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Weiwen Peng is currently a Ph.D. candidate in mechanical engineering at theUniversity of Electronic Science and Technology of China.His research interests include degradation modeling, Bayesian reliability, and

the reliability of complex systems.

Yan-Feng Li received his Ph.D. in mechatronics engineering from the Univer-sity of Electronic Science and Technology of China in 2013.His research interests include the reliability analysis and evaluation of com-

plex systems, dynamic fault tree modelling, Bayesian networks modelling, andprobabilistic inference.

Yuan-Jian Yang is currently a Ph.D. candidate in mechanical engineering atthe University of Electronic Science and Technology of China.His research interests include degradation modeling, and reliability

assessment.

Jinhua Mi is currently a Ph.D. candidate in mechanical engineering at the Uni-versity of Electronic Science and Technology of China.Her research interests include reliability analysis, the evaluation of complex

systems, and Bayesian networks modelling.

Hong-Zhong Huang (M’06) received his Ph.D. in reliability engineering fromShanghai Jiaotong University, China.He is a Professor of the School of Mechanical, Electronic, and Industrial En-

gineering at the University of Electronic Science and Technology of China. Hehas held visiting appointments at several universities in the USA, Canada, andAsia. He has published 200 journal papers and 5 books in the fields of relia-bility engineering, optimization design, fuzzy sets theory, and product devel-opment. His current research interests include system reliability analysis, war-ranty, maintenance planning and optimization, and computational intelligencein product design.Prof. Huang is an ISEAM Fellow, a technical committee member of ESRA,

a Co-Editor-in-Chief of the International Journal of Reliability and Applica-tions, and an Editorial Board Member of several international journals. He re-ceived the William A. J. Golomski Award from the Institute of Industrial Engi-neers in 2006, and the Best Paper Award of the ICFDM2008, ICMR2011, andQR2MSE2013.

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