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G Model JMSY-538; No. of Pages 9 ARTICLE IN PRESS Journal of Manufacturing Systems xxx (2017) xxx–xxx Contents lists available at ScienceDirect Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys Technical Pap
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  Pleasecitethisarticleinpressas:RenL,etal.Multi-bearingremainingusefullifecollaborativeprediction:Adeeplearningapproach.JManufSyst(2017),http://dx.doi.org/10.1016/j.jmsy.2017.02.013 ARTICLE IN PRESS G Model  JMSY-538;No.ofPages9 JournalofManufacturingSystemsxxx(2017)xxx–xxx Contents lists available at ScienceDirect  Journal   of    Manufacturing   Systems  journal homepage: www.elsevier.com/locate/jmansys Technical   Paper Multi-bearing   remaining   useful   life   collaborative   prediction:   A   deeplearning   approach Lei   Ren a , b , ∗ ,    Jin   Cui a , b ,   Yaqiang   Sun a , b ,   Xuejun   Cheng a , b a SchoolofAutomationScienceandElectricalEngineering,BeihangUniversity,Beijing,China b EngineeringResearchCenterofComplexProductAdvancedManufacturingSystem,MinistryofEducation,China a   r   t   i   c   l   e   i   n   f   o  Articlehistory: Received30September2016Receivedinrevisedform22January2017Accepted21February2017Availableonlinexxx Keywords: CybermanufacturingBearinghealthanalysisRemainingusefullifepredictionDataanalyticsDeeplearningIndustrialbigdata a   b   s   t   r   a   c   t Rolling   bearing   health   analysis   andremaining   useful   life   prediction   have   become   an   increasingly   crucialresearch   area   that   can   promote   reliability   and   efficiency   in   the   modern   manufacturing   industry.   Internet-of-Things   and   cyber   manufacturing   techniques   make   it   convenient   to   collect   large   volumes   of    sensor   datathat   can   provide   powerful   support   for   efficient   data   analytics   such   as   deep   learning.   The   combination   of amassive   amount   of    available   data   and   advanced   machine   learning   models   brings   new   opportunitiesfor   bearing   remaining   useful   life   prediction.   This   paper   proposes   an   integrated   deep   learning   approachfor   multi-bearing   remaining   useful   life   collaborative   prediction   bycombining   both   time   domain   featuresand   frequency   domain   features.   The   method   can   extract   high-quality   degradation   patterns   of    rollingbearing   from   vibration   signals.   Regarding   features   extracted   from   bearing   vibration   signals,   in   additionto   three   conventional   time   domain   features,   anovel   frequency   domain   feature   isadopted   in   the   proposedmethod   aswell.   Based   onthe   extracted   features,   the   deep   neural   network   model   is   introduced   to   predictthe   remaining   useful   life   of    rolling   bearing.   We   evaluate   the   performance   of    the   proposed   method   on   a   realdataset   and   compare   it   with   several   commonly   used   shallow   prediction   methods   Numerical   experimentresults   show   the   effectiveness   and   superiority   of    the   proposed   approach.©   2017   Published   by   Elsevier   Ltd   onbehalf    of    The   Society   of    Manufacturing   Engineers. 1.Introduction RapiddevelopmentofInternet-of-Thingsaswellascyberman-ufacturing[1–6]techniquesarechangingmodernmanufacturing industrydramatically.Amassiveamountofindustrialdataaregeneratedfromsensorsincybermanufacturingenvironment,pro-vidingnewpossibilitiesforfurtherimprovementofreliabilityandefficiencyformanufacturingindustry.Inaddition,advancesinAI(ArtificialIntelligence)orientedbigdataanalyticsarecreatingnewresearchopportunitiesforlarge-scalemanufacturingdataprocess-ingandanalysis.Asanindispensableelementofmodernfactorymachines,rollingbearingsplayacriticalroleinindustrialmanufacturingsys-tems,especiallywhererotatingmachineryandequipmentsserveastheessentialcomponents.Bearingfaultsareusuallyconsideredasoneofthemostfrequentcausesofmechanicalfailures[7,8].Bearingreliabilityhasacrucialimpactondependability,durabil-ityandefficiencyoftheequipmentsinmanufacturingindustry. ∗ Correspondingauthorat:SchoolofAutomationScienceandElectricalEngineer-ing,BeihangUniversity,Beijing,China. E-mailaddress: renlei@buaa.edu.cn(L.Ren). Inrecentyears,researchrelatedtobearingdegradationprocessanalysisandservicetimepredictionhasbecomeanincreasinglyimportantarea[9].Thehealthstatusofarollingbearingisinfluencedbyawidevarietyoffactors,i.e.,runningload,operatingtemperature,lubrica-tion,installation,corrosion,materialdefects,andoperationmode,duringthewholeservicelifeofthebearing.Eachofthesefactorshasauniqueeffectonbearinghealthstatus,thusmakingbear-ingremainingusefullifepredictionarathercomplexproblem[10].Infact,multi-bearingcollaborativeanalysisintroducesmorechal-lengingissuesinremainingusefullifeprediction,whichaimsatpredictingremainingusefullifeofabearingbyconsideringagroupofbearingswiththesametypeundersimilaroperatingconditions.Indeed,thereoftenexistdifferencesbetweenrespectivedegrada-tionpatternsinagroupofbearings,somulti-bearingremainingusefullifecollaborativepredictionstillfacesgreatchallenges.Currentexistingbearingremainingusefullifepredictionapproachescanbegenerallyclassifiedintothreecategories,model-basedmethod,knowledge-basedmethod,anddata-drivenmethod[11].Model-basedmethodusuallyrequiresconstructingcontrolequationtodescribetheoperationprinciples[12].Precisionof  amodel-basedmethoddependsontheaccuracyoftheestab-lishedmodel.Sinceitisdifficulttodescribethecomplexprocess http://dx.doi.org/10.1016/j.jmsy.2017.02.0130278-6125/©2017PublishedbyElsevierLtdonbehalfofTheSocietyofManufacturingEngineers.  Pleasecitethisarticleinpressas:RenL,etal.Multi-bearingremainingusefullifecollaborativeprediction:Adeeplearningapproach.JManufSyst(2017),http://dx.doi.org/10.1016/j.jmsy.2017.02.013 ARTICLE IN PRESS G Model  JMSY-538;No.ofPages92 L.Renetal./JournalofManufacturingSystemsxxx(2017)xxx–xxx ofbearingdegradationclearlyandcomprehensively,mechanismmodelforbearingremainingusefullifepredictionisusuallyhardtoconstructandhasdrawbacksfromtheperspectiveofpredic-tionaccuracy.Knowledge-basedmethodmakespredictionbasedonexpertsystemswhichtakeadvantagesofempiricalknowl-edge.Typically,knowledge-basedmethodsaregoodatqualitativeevaluationratherthanquantitativeprediction[13].Therefore, knowledge-basedmethodsaremoresuitabletomakequalitativepredictionsbuthavelimitationsinforecastingbearingremainingusefullifewithhighaccuracy.Data-drivenmethod,featureddataminingandmachinelearning,neednotestablishcomplexcontrolequationsattheinitialanalyticalstage.Inaddition,data-drivenmethodsareabletoprovidequantitativepredictionresultsorprob-abilisticdistributionsofpredictivevariables.Buildingappropriatelearningmodelsformachinelearningalgorithmsplaysanessen-tialroleindata-drivenapproaches[14].Bothquantityandquality ofavailabledatahaveaneffectonthepredictionprecisionofadata-drivenmethod.Inrecentyears,data-drivenmethodshavebecomeincreasinglypopularbyleveragingtheadvantagesofdataanalyticsmethod-ology[15].Thedevelopmentandapplicationofdeeplearning approach[16]haveatremendousimpactonbigdataanalysistech- niqueresearch.Deeplearningapproachfeatureshighprecision[17]andbigdataprocessingcapability[18],allowingthedecreaseof  modelingcomplexity[19].Therefore,deeplearningbaseddata- drivenapproachhasthepotentialtoprovidenewopportunitiesformulti-bearingremainingusefullifeprediction.Inaddition,thedevelopmentofhigh-performancecomputingalsosupportstheimplementationofcomplexmachinelearningalgorithmstopro-cessthelarge-scaledataefficiently.Toaddresstheissuesofmulti-bearingremainingusefullifepre-diction,thispaperproposesanintegrateddeeplearningapproachbasedoncollaborativeanalysisofmonitoringdatafrommulti-bearingvibrationsignalsbycombiningbothtimedomainfeaturesandfrequencydomainfeatures.Thefeatureparameterssystemconsistsofthreefeaturesoftimedomainandonefeatureoffre-quencydomain.Thetimedomainfeaturesincluderootmeansquare(RMS),crestfactorandkurtosis.Allthesethreefeaturesderivefromclassicaltimedomainfeaturesforbearingvibrationsignalsanalysis.Thefrequencydomainfeatureisanewlydefinedfeature,namedasfrequencyspectrumpartitionsummation(FSPS),whichisrepresentedasasix-dimensionalvectorinthispaper.Thefeaturesextractedfromthebearingvibrationsignalsalmostcoverawholeprocessofbearingdegradation.Thefrequencydomainparameterissensitivetotheearlierstageandthelaterstage,whilethetimedomainfeatureshaveadvantagesinrepresentingthemid-dlestageofbearingdegradationprocess.Thefullyconnecteddeepneuralnetworkisadoptedintheproposedmulti-bearingremain-ingusefullifepredictionmodel.Andthedeepneuralnetworkrelatedparametersaredeterminedaccordingtoaseriesofgridsearchexperiments.Inordertovalidatetheeffectivenessoftheproposedmethod,numericalexperimentsareimplementedonarealdataset,whichisprovidedbyAS2MdepartmentofFEMTO-STInstitute,forperfor-mancecomparisonwiththegradientboostingdecisiontree(GBDT)method,thesupportvectormachine(SVM)method,BPneuralnet-work,GaussianregressionmethodandBayesianRidgemethod.Experimentalresultsshowtheeffectivenessandsuperiorityoftheproposedapproach.Theremainingpaperisorganizedasfollows.Section2inves- tigatestherelatedwork.Section3presentsdescriptionsoftheproblemandtheproposedmethodology.NumericalexperimentdetailsandresultanalysisaregiveninSection4.Section5concludes thepaper. 2.Relatedwork Ingeneral,solutionsforrollingbearingremainingusefullifepredictioncouldbeclassifiedintothreecategories,model-basedpredictionapproach,knowledge-basedpredictiontechnique,anddata-drivenpredictionmethod.Systemmechanismmodelisthefoundationofmodel-basedpredictionmethod.Systemstateequationsorcontrolequationsforrollingbearingdegradationprocessareestablishedbyanalyz-ingthecomplexrelationshipsamongbearinghealthstatusandthemainaffectfactors.Basedonthestate/controlequationsandthecurrentstateofbearing,whichcanbeusedforinitializationparametersextraction,theremainingusefullifeofrollingbearingisderivedandcalculated.Consideringthenoiseinbearingstatemon-itoringdata,filteringalgorithmslikeKalmanfilter[20]andparticle filter[21]arecommonlyadoptedforsignalpreprocessinginorder toimprovetheperformanceofthemodel-basedpredictionmethod.Theoretically,model-basedpredictionmethodisabletoreflectthenatureandthelawofasystemadequately.Theestablishedsystemmodelisexpectedtofullydescribethemechanismandcharacteris-ticsofbearingdegradationprocess,thusmodel-basedmethodhasthepotentialtoproduceareasonableforecastingresultwithhighaccuracy.However,sincethereexistsvariousaffectfactorsforbear-ingdegradationprocessandsomeeffectmechanismsareunclearorunpredictable,soitisusuallyquitedifficultorevenimpossibletoestablishapreciseandreliablemechanismmodelforbearingremainingusefullifepredictionproblem.Itisforthisreasonthatmodel-basedpredictionmethodcanonlybeusedinverylimitedapplicationfields.Byleveragingtheadvantagesofaccumulatedtechnicalexperi-enceorknowledgeontherelevantissue,knowledge-basedmethodmakespredictionorjudgmentwithouttheneedofprecisesystemmechanismmodel.Accumulationofdomainknowledgeandrea-sonablejudgmentofapplicationsituationarethecrucialaspectsoftheknowledge-basedpredictionmethod.Expertsystem[22]andfuzzylogic[23]arecommonlyusedclassicknowledge-based decision-makingtechniques.Knowledgeandexperiencearefullyusedtosupportthedecision-makingprocess.Knowledge-basedpredictionmethodiswidelyadoptedinvariousapplicationfields,especiallyinsomequalitativedecisionmakingscenarios.How-ever,knowledge-basedmethodhaslimitationsinsolvinghighprecisequantitativepredictionproblems.Thus,knowledge-basedpredictionapproachesmay   havedifficultyinprovidingsatisfactoryforecastingresultsofbearingremainingusefullife.Data-drivenpredictionmethodlearnsthelatentassociationrelationshipsamongbearingdegradationprocessanditsaffectfactorsautomaticallyfromthesensormonitoringdataofrollingbearingbyutilizingmachinelearningalgorithmsorotherintelli-gentdataanalysistechniques.Thenthewell-traineddata-drivenpredictionmodelcanbeusedforbearingremainingusefullifepre-diction.Theforecastingprecisedependsonthequalityandquantityoftheavailabletrainingdataaswellastheeffectivenessofthelearningalgorithm.Data-drivenpredictionmethodhasadvantagesinbothmodelingaspectandquantitativepredictionaspect,butitrequiresthesupportofnumeroushigh-qualityavailablelearningdataaswellaseffectivedataanalysistechniques.Rapiddevel-opmentandwidelyapplicationofInternet-of-Thingsandcybermanufacturingtechniquesaswellasadvancesinintelligentdataanalysisandhigh-performancecomputingtechniquescreatenewopportunitiesforresearchofdata-drivenpredictionmethodandpromoteitsdevelopmentaswell.Astheselectedsignalfeaturesandlearningmodelvaries,thereexistsvariousdata-drivenpredictionmodelsforrollingbearingremainingusefullifeforecastingproblem.However,onlyiffea-turesofhighrepresentationalcapabilityandsuitablehighefficient  Pleasecitethisarticleinpressas:RenL,etal.Multi-bearingremainingusefullifecollaborativeprediction:Adeeplearningapproach.JManufSyst(2017),http://dx.doi.org/10.1016/j.jmsy.2017.02.013 ARTICLE IN PRESS G Model  JMSY-538;No.ofPages9 L.Renetal./JournalofManufacturingSystemsxxx(2017)xxx–xxx 3 learningmodelareintegratedproperlywillthepredictionmethodbeabletomakeprecisepredictions.Thefeaturesthatextractedfrombearingvibrationsignalandusedforbearingremainingusefullifepredictioncanbegroupedintothreecategories,timedomainfeature,frequencydomainfea-ture,andtime-frequencydomainfeature.Inwhich,commonlyusedtimedomainfeaturesincludingmean,variance,RMS,crestfactorandkurtosis[24].Timedomaincharacteristicindexesare abletoreflectthegeneraltendencyofrollingbearingdegrada-tionprocessinanintuitivewaybutinsensitivetosmallchanges.Besides,noisesignalsmay   influencethepredictionresultseriouslyastimedomainfeaturesaresensitivetonoise.Frequencyspec-trumvarianceandfrequencyspectrumRMS   aretwo   commonlyusedfrequencydomainfeaturesinbearingvibrationsignalanal-ysis.Frequencydomainfeaturesaresuitableforstationarysignalprocessingandwidelyusedinbearingfailurediagnosisresearch.Inaddition,somestudiesalsoapplytime-frequencydomainfea-turestobearingvibrationsignalanalysisandbearingremainingusefullifeprediction[25].Typicaltime-frequencydomainfeature extractiontechniquesincludingwaveletpackettransform[26]and empiricalmodedecomposition.Time-frequencydomainfeaturesarecapableofrepresentingweaksignalcharacteristicsandsuit-ablefornonlinearsignalprocessing.However,thesefeatureshavedrawbacksininformationredundantandusuallyneedcomplexmathematicalderivationforfeatureextraction.Therefore,asetof reasonableandefficientvibrationsignalcharacteristicsindexesiscrucialtothepredictionresultofbearingremainingusefullife.Theselectionofsignalfeaturesneedsacomprehensiveconsider-ationofdegradationpatternrepresentationalability,informationredundancy,andcomplexityofsignalfeature.Classiclearningmodelsaccountforthegreatmajorityoflearn-ingmodelsoralgorithmsappliedinbearingremainingusefullifeprediction.Amongthem,neuralnetworkmodelplaysanimportantroleandmakesexcellentpredictionsinsomescenariosbecauseoftheoutstandingcapabilityfortheneuralnetworktofittinganonlinearsystem[27].However,fromanotherperspective,how toavoidover-fittingproblem[28]andimproveforecastingeffect aretwoknottyproblemsforusingneuralnetworkmodel.Withthedevelopmentofdataanalysistechniquesandadvancesinefficientintelligentlearningmodelresearch,thedeeplearningmethodisbecomingpopulargradually[16].Deeplearningmethodmakesup theshortcomingsanddeficienciesofneuralnetworkmodel,espe-ciallyintermsoferrorpropagation.Deeplearningmethodchangestheerrorpropagationmechanismoftheneuralnetwork,prevent-ingtheerrordiffusionlayerbylayer[29],whichgreatlyimproves thepredictionaccuracy. 3.Problemandmethodology   3.1.Multi-bearingremainingusefullifecollaborativeprediction Rollingbearingremainingusefullifepredictionmeanspre-dictingtheremainingusefullifeofrollingbearingaccordingtoconditionmonitoringdataacquiredintheiroperatingprocess.Commonlyusedconditionmonitoringdataofrollingbearingincludevibrationsignal,accelerationsignal,andtemperaturesig-nal.Inthispaper,vibrationsignalisappliedinrollingbearingremainingusefullifeforecastingandhealthevaluationanalysis.Existingstudiesonrollingbearingremainingusefullifepredic-tionmainlyfocusonsinglebearinganalysis.However,performancedeclinefeaturesofdifferentbearingsinsameoperatingconditionshowsomesimilarity.Alltheperformancedeclinefeaturesanddelicaterelationsamongthemareimpliedinthevibrationdata.Machinelearningalgorithmsprovideaneffectivetoolforobserva-tiondataanalysis.Themulti-bearingremainingusefullifecollaborativepredictionreferstotheproblemofremainingusefullifepredictionformultibearingsbasedontheconditionmonitoringdataacquiredinsameoperatingcondition.Multi-bearingremainingusefullifecollabo-rativepredictionaimedatforecastingremainingusefullifeofabearingaccordingtotheavailablemonitoringdataofthebearingitselfaswellasmonitoringdataofotherbearingsofthesametypeandoperatingconditions.  3.2.Featureextraction 3.2.1.Timedomainfeature Timedomainsignalfeaturesareeffectivetoreflecttherunningconditionandthefaultinformationofrollingbearing.Timedomainfeaturesonlydependontheprobabilitydensityfunctionsofsig-nalamplitudeandaresensitivetothefaultsanddefectsofrollingbearing.Asaresultofbalancebetweeninformationrepresentationabilityandinformationdimensionality,threeclassicaltimedomainfeatures,rootmeansquare  X  RMS  ,crestfactor  X  Crest  ,andkurtosis  X  Kurtosis ,areusedinthispaper.Formulasforthesethreefeaturesarepresentedasfollows:  X  RMS   =   ni = 1 (  x ( i )) 2 n  (1)  X  Crest   = max   x ( i )   X  RMS  (2)  X  Kurtosis  =  ni = 1 (  x ( i ) − ¯  x ) 4 n (  X  RMS  ) 4  (3)where  x ( i )isaseriesofvibrationsignaland n referstothenumberofvibrationsignaldatapoints.  3.2.2.Frequencydomainfeature Rollingbearingisofdifferenthealthconditionwhenindiffer-entdegradationstage.Thedegradationfeatureschangeovertimethroughoutthewholelife-cycleofrollingbearing.However,timedomainfeaturesarenotsensitivetosignalfrequency.Therefore,timedomainfeaturesarenotsufficientenoughtoreflectthebear-ingdegradationprocess.Inordertoextractmorecomprehensivehealthconditioninformationofrollingbearingfromthemonitoringvibrationdata,anewfrequencydomainfeaturenamedFrequencySpectrumPartitionSummation(FSPS)isdefined.Givenaseriesofvibrationsignal  x ( i )for i =1,2, ... ,   n , s (  j )referstoitsfrequencyspectrumwhichobtainedthroughFouriertrans-formationand  j =1,2, ... ,   m .   Then,theFSPSindex  X  FSPS   ( k ) canbecalculatedasfollows:  X  FSPS   ( k ) =  mkK   j = K  + m ( k − 1) K  s (  j ) (4)where k =1,2, ... ,   K  .Itshouldbenotethatthenewdefinedfre-quencydomainfeature  X  FSPS   ( k )isaone-dimensionalvectorwhichbeconsistofKelements.Kisanempiricalparameterandisgener-allydeterminedbytheconcretedomainproblem.  3.3.Deepneuralnetwork Deepneuralnetworkreferstoaneuralnetworkthathastwoormorehiddenprocessinglayersbetweentheinputandoutputlay-ers[16].Deepneuralnetworkisadeeplearningmodeldeveloped basedonthebasicneuralnetwork.Comparedwithshallowneuralnetworkwithasinglehiddenlayer,deepneuralnetworkhasthepotentialtosolvemorecomplexlearningproblems.Adeepneuralnetworkconsistsofaninputlayerandanoutputlayerseparatedbytwoormorehiddenlayers.Areasonablenumberofthehiddenunits  Pleasecitethisarticleinpressas:RenL,etal.Multi-bearingremainingusefullifecollaborativeprediction:Adeeplearningapproach.JManufSyst(2017),http://dx.doi.org/10.1016/j.jmsy.2017.02.013 ARTICLE IN PRESS G Model  JMSY-538;No.ofPages94 L.Renetal./JournalofManufacturingSystemsxxx(2017)xxx–xxx canleadtohigherpredictionaccuracy.Weightsandbiasvalues,aswellasactivationfunctionsareprimaryeffectfactorsoftheper-formanceofadeepneuralnetwork.Adeepneuralnetworkmodelcanbetraineddiscriminativelybyusingthestandardbackpropaga-tionalgorithm.Tocombattheoverfittingproblem,regularizationmethodscanbeappliedduringthetrainingprocess.Currently,withtherapiddevelopmentandsuccessfulappli-cationofdeeplearningtechniques,manyopensourcedeeplearningframeworksandlibrariesaredevelopedtofacilitaterelatedresearch.Amongthem,Kerasisoneofthedeeplearninglibraries.Kerasisapythonwrittenmodularityframeworkdevel-opedtoenablingfastdeeplearningexperimentation.ModularityandextensibilitymakesKerasapowerfultoolforrapiddeeplearn-ingexperimentimplementation.  3.4.Performanceindex Twocommonlyusedperformanceindicators,themeanabsoluteerror  X  MAE   andtherootmeansquareerror  X  RMSE  ,areemployedtoevaluatetheperformanceoftheproposedmodel.Mathematicaldescriptionsoftheindexesaregivenasfollows:  X  MAE   =  ni = 1 |  f  i − ˆ  f  i | n (5)  X  RMSE   =   ni = 1   f  i −  ˆ  f  i  2 n  (6)where  f  i  istherealremainingusefullifeand ˆ  f  i  isthepredictedremainingusefullife. 4.Experimentandanalysis 4.1.Datadescription Inordertovalidatetheeffectivenessofthenewlydefinedsignalfrequencydomainfeatureandthedeeparchitecturecollaborativepredictionmodel,themulti-bearingremainingusefullifecollabo-rativepredictionmodelwasappliedtoadataset,providedbyAS2MdepartmentofFEMTO-STInstitute,asanumericalexperiment.Thevibrationdataofrollingbearingweregeneratedfromrun-to-failuretestsperformedonthePRONOSTIAexperimentalplatform[30].OverviewillustrationofthetestrigisshowninFig.1.Therollingbearingwasinstalledontheexperimentalplatformandrun-to-failureexperimentwasperformedunderconstantloadconditionandconstantrotatingspeed.Forthebearingsusedinthispaper,therotatingspeedwasmaintainedat1650rpmandthepay-loadweightis4200N.Twovibrationsensors,placedonverticalaxisandhorizontalaxisseparately,areadoptedtoacquirethebearingvibrationdatawithafrequencyof25.6kHz.Andthemeasureisacquiredatafrequencyequalto100Hz.Thebearingisregardedasfailureaslongastheamplitudeofvibrationsignaloverpassed20g.Adatasetwithfourgroupsofbearingtestingvibrationsignalswasusedinthefollowingmulti-bearingremainingusefullifecollabo-rativepredictionexperiment.Fig.2depictedavibrationrawsignal acquiredfromanexperiment. 4.2.Experimentimplementation4.2.1.Identifythefailuretimeoftestingrollingbearings Theveryfirststepofthenumericalexperimentisdataprepro-cessing.Asabearingisconsideredasfailureifitsvibrationsignalamplitudeexceeds20g,thefailuretimeoftestingrollingbearingaredeterminedbyexaminingbothverticalvibrationsignalandhor-izontalvibrationsignal.Forsimplicity,dimensionlessvariabletimestepisadoptedinsteadofaccuratetimewithitsunit.Measurefre-quencyofPRONOSTIAexperimentalplatformequalto100Hz,thus,0.1sisdefinedasonetimestepinthenumericalexperiment.Theremainingusefullifeofarollingbearingisdefinedastheavailableservicetimeformthepresentmomenttothemomentitfailed.Forexample,forabearingatsamplingpoint t  1 ,itfailedatsamplingpoint t  2 ,thenremainingusefullifeofthebearingiscalculatedby t  2 − t  1 . 4.2.2.Extractbothtimedomainandfrequencydomainfeatures fromtheobservedvibrationsignaldata Highqualitysignalfeaturesareabletorepresenttheuse-fulinformationofthedatasetmoreintensivelyandtofiltertheunrelatedinformationeffectively.Asdescribedinprevioussec-tion,threetimedomainfeatures,rootmeansquare,crestfactor,andkurtosisaswellasafrequencydomainfeature,FSPSindex,areemployedtoextractdegradationinformationfromthevibra-tiondataacquiredfromrun-to-failureexperiment.Consideringtheindividualizationcharacteristicofmulti-bearingremainingusefullifecollaborativepredictionproblem,theempiricalparameterKforfrequencydomainfeatureFSPSissetas6.Figs.3and4showtheRMS   curveandthecrestfactorcurveofonetestingbearing,respectively. 4.2.3.Featuredatanormalizationprocessinganddatasplitting  Differentfeatureshavedifferentrangesofvalues.Toeliminatethenegativeeffectcausedbydifferentrangesofvalues,min-maxnormalizationmethodwasappliedtotransformtherangeofvaluesas[0,1]forallextractedfeatures.We   adoptedthenine-dimensionalvectortorepresentthefeaturesofbearingvibrationsignal,includ-ingthreetimedomainfeaturesandonefrequencydomainfeaturethatrepresentedasasix-dimensionalvector.Asbearingvibrationsignaldatainthedatasetwas   collectedfromtwodirections,ver-ticaldirectionandhorizontaldirection,sothedimensionalityof themodelinputdatais18.Besides,thecorrespondingnormalizedremainingusefullifeisusedasthelabelforeachdatarecord.Inaddition,toperformthemulti-bearingremainingusefullifecollaborativepredictionnumericalexperiment,thepreprocessedvibrationdatasetisseparatedastrainingdataandtestingdata.Acertainpercentageofdatapointsareselectedastestingdataran-domlyandtherestareusedastrainingdata.Inourexperiment,sixtestingdatasets,accountfordifferentpercentagesofthewholedataset(5%,10%,15%,20%,25%,30%),aresetforcomparison. 4.2.4.Deepneuralnetworkmodelconstructionandtraining  Inthispaper,weconstructedafullyconnecteddeepneuralnet-workformulti-bearingremainingusefullifeprediction.ThewholeexperimentisdevelopedandimplementedbasedontheKerasdeeplearningframework.Inputlayerofthedeepneuralnetworkmodelisfullyconnectedtothefirsthiddenlayer,andbythesamerule,thesecondhiddenlayerisfullyconnectedtothefirsthiddenlayerandsoonuptotheoutputlayer.Numberoflayersandamountofneuronsforeachlayeraredeterminedaccordingtoaseriesof gridsearchexperiments.Thedeepneuralnetworkisconsistingof8hiddenlayerswithdifferentamountofneurons(300,200,150,100,80,50,30,1).Kerasframeworkprovidesvariousproba-bilisticdistributionbasedparameterinitializationmethods.Inthispaper,uniformdistributionisemployedforweightparameterini-tialization.Becauseoftheirexcellentperformanceinthenumericalexperiments,ReLUfunctionisappliedasactivationfunctionformiddlelayersandactivationfunctionforthelastlayerisasigmoidfunction.Thesigmoidactivationfunctionmatchesthenormalizedremainingusefullifevalueswellsincetheoutputofasigmoidfunctionrangesfrom0to1.Asforthetrainingsettinginitializa-tion,meansquarederrorandRMSpropalgorithmareselectedastheoptimizationobjectiveandtheoptimizer,respectively.After
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