321 

AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume12|Number4|December2017|pp321–336 Journalhome:apem‐journal.org

https://doi.org/10.14743/apem2017.4.261 Originalscientificpaper

  

An integrated generalized discriminant analysis method and chemical reaction support vector machine model (GDA‐CRSVM) for bearing fault diagnosis 

Nguyen, V.H.a,b,c,*, Cheng, J.S.a,b, Thai, V.T.a,b,c 

aState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, China bCollege of Mechanical and Vehicle Engineering, Hunan University, Changsha, China cMechanical Engineering Department, Hanoi University of Industry, Hanoi, Vietnam   

A B S T R A C T   A R T I C L E   I N F O

Anexperttechniqueinbearingfaultdiagnosisisproposedfortheidentifica‐tion of actual status. A new diagnosismethod based on a two‐stage hybridmodality in integrating generalized discriminant analysis (GDA) with thechemical reaction support vector machine (CRSVM) classification model,namedGDA‐CRSVM,isproposed.TheGDAreduceshigh‐dimensionaldatatoamore compact data,which serves an optimized CRSVM classificationmodelwith input data, inwhich a support vectormachine (SVM) classifiermodelwiththebestparametersareselectedbythemeta‐heuristicchemicalreactionoptimization algorithm (CRO) to build an optimized CRSVM classificationmodel.Theimplementationofthenewproposedmethodisbasedonamulti‐aspectfeature(MAF)setthatpresentsmostoftheactualaspectsofthecom‐plexvibrationsignal.TheMAFsetiscollectedfromthestatisticalfeaturesintime‐domain, frequency‐domain, and time‐frequency domain features areextracted by local characteristic‐scale decomposition (LCD). Experimentshavebeen conductedon twobearing vibrationdatasets by the expert tech‐nique in thebearing faultdiagnosis.Results shown that theeffectivenessofGDA‐CRSVMintermsofclassificationaccuracyandexecutiontime.©2017PEI,UniversityofMaribor.Allrightsreserved.

  Keywords:BearingfaultExpertfaultdiagnosistechniqueChemicalreactionsupportvectormachine(CRSVM)Multi‐aspectfeaturesetGeneralizeddiscriminantanalysis(GDA)

*Correspondingauthor:hung2009haui@gmail.com(Nguyen,V.H.)

Articlehistory:Received8June2017Revised23October2017Accepted26October2017 

  

1. Introduction 

Thebearingisakeycomponentofrotatingmachineryandiscloselyalliedtosystemoperation.Any failureofbearingmaycauseunsafe conditions for theoperatorand inefficientoperation,stoppingworkmayalsoaffectassociatedsystems.Hence,advancedfaultdiagnosismethodsinthemechanicalmaintenance field are a focusof interest tomany researchers.Thesemethodscanbesummarizedintoseveralconsecutivestepsaimedat identifyingpatternsof faultstatus.Thefirststepisacquisitionvibrationdata,whichmayneedpre‐processingsuchasdenoisingorremovingartefacts.Thesecondstep is featureextractionsteptogetthemost important infor‐mation.Then, these featuresare transformed into thepatterndiagnosismodel toclassifypat‐terns.Finally, thepatterndiagnosismodeldeterminesthepatterntypetowhichtheparticularfaultsignalbelongs.

Onethemostimportantactionsforfaultdiagnosistechniqueisfeatureextraction.Aneffec‐tivenessfeaturesetneedstocontainthemostsalientfeatures,beneficialfeaturesoftheclassifi‐cationstage.Thispaperfocusesonamulti‐aspectfeatureextraction(MAF),whichmanyactual

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aspectsofthecomplexvibrationsignal.MAFisbasedonstatisticalfeaturesintime‐domain,fre‐quency‐domain,whichdirectlyrepresenttheoutwardaspectsofsignal,andtime‐frequencydo‐mainfeatures,whichrepresenttheintrinsicaspectsdeeplyhiddeninthevibrationsignal.Fea‐turesinthetime‐frequencydomainareespeciallyextractedbylocalcharacteristic‐scaledecom‐position(LCD),amethodthatbecomessuperiorinrunningtime,restrainingtheendeffectandrelievingmodemixing[1,2].TheMAFsetcanbeextractedfromtheoriginalvibrationsignalasahigh‐dimensionalfeaturevectorincludingsevenfeaturesthatrepresenttheaspectinthetime‐domain,eightfeaturesthatrepresenttheaspectinfrequency‐domain,andfivefeaturesthatrep‐resenttheaspectinthetime‐frequencydomain.TheobtainedMAFcanprovideextremelyade‐quateinformationonvariousbearingconditionstomakeaneffectivediagnosisofperformance.

Normally, a feature set of an original vibration signal can also providemore handy infor‐mation.Increasingtheproblemofusingthesefeaturesefficaciously inawaythatwouldinter‐ferewiththeclassificationstage,suchasthecomputationalburden,theprocessingtimeisslow‐er, theclassificationaccuracyresultsarepoorer.Thispaperaimsatmakingclassificationper‐formancemoreeffective.Thehigh‐dimensional featuredataset thatwasextracted fromvibra‐tionsignalsismappedontoanewfeaturespacetodiscovertheintrinsicstructureinthesenon‐linearhigh‐dimensionaldataandtoobtainamorecompactfeaturedatasetinalowerdimension.Recently,dimensionalityreductionapproacheshavearousedgreatinterestinthefaultdiagnosisresearchfield.Principalcomponentanalysis (PCA) [3,4] isoneof themost traditionallyused,alongwithmulti‐dimensional scaling (MDS) [5] and linear discriminate analysis (LDA) [6, 7].However,while these approaches are remarkably effective on linear data, theymay not ade‐quatelyhandlecomplexnon‐lineardata.Thismaybecauseoflowaccuracyormisjudgementofafaultdiagnosiswithnon‐lineardata.Toexpandthefieldofnon‐lineardataofLDA,thegeneral‐izeddiscriminantanalysis (GDA)methodwasproposedbyBaudatandAnouar(2000)[8].Themainideaistoprojecttheinputspaceintoanadvantageousfeaturespace,wherevariablesarenonlinearlyrelatedtotheinputspace.Accordingtothecurrentliterature,theGDAmethodhasnotbeenpreviouslyappliedforfaultdiagnosis.Inthecaseofmedicine[9]andimaging[10],therearepreviousreferenceworks.AnimportantcontributioninthisstudyistheintroductionoftheGDAmethod in thediscoveryof the intrinsicstructureof thenon‐linearhigh‐dimensional fea‐turedataset.Itscombinationwithaclassifiermodelmakeiteffectiveforclassificationpurposes.

Supportvectormachine(SVM)basedonstatisticallearningtheoryisanewmachinelearningalgorithmproposedbyVapniketal.SVMisapowerfulsupervisedmachinelearningtoolandisusedinanumberofapplicationssuchaspatternrecognition[11],time‐seriesforecasting[12],robotics[13]anddiagnostics[14].WhenSVMisused,oneshouldremarkthattheoptimalpa‐rametersplayaleadingroleforformingaclassificationmodelwithhighclassificationefficiency,thuscreatingsomething thathasaroused thegreat interestof researchers for theselectionofoptimalparameters.Recently, several evolutionarybasedalgorithmssuchas thegeneticalgo‐rithm(GA)[15],particleswarmoptimization(PSO)[16],antcolonyoptimization(ACO)[17],thesimulatedannealingalgorithm(SA) [18]havebeenused tooptimize theSVMparametersandhavealsoshownpromisingabilityaslearningalgorithmsthatcanbeutilizedfordiagnosispur‐poses[19].However,theirperformancemayvaryfromoneobjecttoanotherinfaultdiagnosisandmaynotbe suitable for thedifferent statusesof roller bearings.Besides, the efficiencyoftheseoptimizationalgorithmsischaracterizedbytheprocedureusedforselectiontheparame‐ters,which requires adeepknowledgeof theuseof algorithms.The recent chemical reactionoptimization(CRO)algorithm,whichisanovelcomputationalmethod,isoneoptimizationofthefoundmeta‐heuristicsintroducedin2010[20].CROisanevolutionaryoptimizationtechnique,which iscomprehended fromthenatureofchemical reaction. Itperformsverywell insolvingoptimizationproblemsinaveryshorttime.Inashortperiod,therehavebeenafewapplicationsofCROtotherecognitionfield,datamining,classificationrule[21,22],andefficiencyhasbeendemonstrated. Indeed,CROhasbeenapplied tosolvecomplexproblemssuccessfully,hasout‐performedmanyexistingevolutionaryalgorithms inmostof thetestcases.Aguidelinecanbefoundinthetutorial introducedin[23]tohelpreadersimplementCROforoptimizationprob‐lems.MotivatedbythecapabilityofCRO,animportantcontributioninthisstudyistheauthors’aim touse theCROalgorithm to select thebestparametersof theSVMmodel,which is a fre‐

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quentlyuseddiagnosistechniquecalledCRSVM.Then,theoptimizedCRSVMmodelisusedforbearingfaultdiagnosis.

Finally, inthispaper,atwo‐stagehybridmodalityforintegratingGDAwithCRSVMisintro‐duced,calledGDA‐CRSVM.TheproposedGDA‐CRSVMmethodisbasedonanexperttechniquethataims toexploit thehighest identificationaccuracy in the faultdiagnosisof rollerbearingsbasedonanMAFset.GDAisfirstusedtoreducethehigh‐dimensionalfeaturesetthatactsasthedata pre‐processing for classifier model. Then, the obtained feature set provides the CRSVMmodelwithinputdata.Forexploration,experimentshavebeenconductedontwobearingvibra‐tion datasets with different conditions based on the GDA‐CRSVM method. Moreover, the ac‐quiredvibrationsignalshavebeenanalysedtoextracttheMAFset.Itisremarkablethattheper‐formanceoftheGDA‐CRSVMmethodissignificantlybetterthanthatoftheothermethodsandshowedthemostaccurateresultsforclassificationpurposesalongwithsuperiorexecutiontime.

Thepaper is constructed as follows. Section2presentsmaterials andmethods forbearingfaultdiagnosis.InSection3,theGDA‐CRSVMmethodisproposedbyatwo‐stagehybridmodalitybyintegratingtheGDAmethodwiththeCRSVMclassificationmodelfortheexpertfaultdiagno‐sistechnique.InSection4,wepresentexperimentsforbearingfaultdiagnosis,wherevibrationdataisacquiredforrollerbearings,MAFisusedtoextractvibrationsignals,andtheactualfaultstatusesareidentifiedbyourproposedmethod.Section5istheconclusion.Acknowledgmentsandalistofreferencesfollow.

2. Materials and methods 

2.1 Generalized discriminant analysis (GDA) method 

Dimensionality reduction can be done by feature transformation to a low‐dimensional dataspaceoncefeatureshavebeenextractedfromthevibrationsignals.Thepurposeofthedimen‐sionalityreductionistoselectthemostsuperiorfeaturesoftheoriginalfeatureset,whichcanprovide dominant actuality‐related information. Irrelevant or redundant factorsmust be dis‐cardedtoimproveclassificationperformance,toavoidproblemswithdimensionality.Therefore,theGDAmethodispresentedandusedtoselectthesuperiorfeaturesfromtheoriginalfeatureset[8].

TheobjectiveofGDA is tofindmapping for the input feature set into a lowerdimensionalspace/newspace.Theratioofcentre‐classpartner towithin‐classpartner canbemaxim‐ized[8].AsetofinputpatternsSoftrainingfeatures‐setcanbegivenas:

1,2, . . , ; 1,2, . . , (1)

ThisisaC‐classproblem, isthenumberofsamplesinclass .Themapping : ⟶ isnon‐linearfortrainingpatternsinthenewspace,thus → , 1,2, . . , ; 1,2, . . , isrepresented.

Thecenter‐classpartner towithin‐classpartner ofthetrainingfeaturesetcanbecalcu‐latedasbelow:

1 1 (2)

1 (3)

Wehavetocalculatetheeigenvaluesandeigenvectors , ∈ ∖ 0 tosatisfytheequation:

(4)

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′′

(5)

The eigenvectors are combinations of elements and the existing coefficients, 1,2, . . , ; 1,2, . . , suchthat:

(6)

Tosimplify,wecanwritethecoefficientvectorasbelow:

, ,..,, ,..,

(7)

Further,letusconsiderthisvector.Weusedthekerneltechniqueinthenewspace.Usingthedot product of a sample from class and another sample from class , the dot product

givesthefollowing:

⋅ (8)

First,let bea matrixdefinedintermsoftheclasselementsby 1,2,..,1,2,..,

.Inthe

newspace,the matrixisrepresentedasbelow:

, ,..,, ,..,

(9)

where isa matrix:

, ,..,, ,..,

(10)

Then,a matrix isintroduced, isdefinedas:

, ,.., (11)

where isa matrixwithalltermsequalto1 .Finally, fromtheEqs.2,3,6and4,we foundthe innerproductwithvector onboth

sides.

(12)

There, representsacolumnvectorwithvalues , 1,2, . . , ; 1,2, . . , .ThesolutionofEq.(12)issatisfactorywhentheeigenvectorsofmatrix arecal‐

culated.

2.2 Optimal classification model 

ThissectionemphasizesthesuperiorityoftheCROalgorithm,whichisthenappliedtoselectthebestparametersoftheSVM.TheseparametersplayaleadingroleforbuildinganoptimalCRSVMclassificationmodel.TheobtainedCRSVMmodelcanbeusedforfaultdiagnosisofbearingcom‐ponents,combiningtheinputfeaturesettobecometheexpertclassifiermodelwithhighclassi‐ficationaccuracy,stabilityandeffectivenessofperformance.Fig.1depictstheflowchartforus‐ingtheCROalgorithmtoselectparametersoftheSVMmodel.

PrincipleofSVM

SupportVectorMachine(SVM)wereintroducedbyVapnik[24].TheSVMclassifierisdesignedforclassificationtaskswithtwo‐classdatasets.Thedataareseparatedbyahyperplaneinordertomaximizedistance.Theseparatinghyperplaneisdefinedbytheclosestpointsofthetrainingdataset,whicharecalledsupportvectors.ThedetailsofSVMarepresentedin[14].Theparame‐terpair , intheRBFkernelfunction(thepenaltyparameterof andwidthparameterof )playsanimportantpartintheclassificationpurpose.Theparameterpairvaluescoverabroad

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rangeandcontrolsthegeneralizationcapabilityofSVM.ThebestselectionofparameterpairisimportantandnecessarytotrainingtheSVMclassifier.Inthiswork,valuesof , areselectedusingtheCROalgorithmforthebestperformanceforaccuratebearingfaultdiagnosis.

TheoptimizedSVMmodelbasedonCRO

Tofulfil theaimtobuildanoptimalclassificationmodel,CRSVM,theCROalgorithmisusedtoexploitthebestparameterpair , oftheSVM.Theobtainedclassificationmodelisemployedtoidentifythebearingconditions.

Fig.1ThearchitectureofCRSVMclassificationmodel

Chemicalreactionoptimizationalgorithm

TheCROalgorithm is introduced in2010 [20] in the findingmeta‐heuristicsof computationalmethod which is the efficient optimization technique that enjoys the advantages of previousgeneticalgorithmsandsimulatedannealing.Thisalgorithmisnotonlyinspiredbytheelemen‐tarychemical reactions, that isdifferent fromevolutionaryalgorithmsmotivatedbybiologicalevolution,butalsoeasilyconstructedbydefiningtheagentsandtheenergydirectionalscheme.Consequently, algorithm has been deployed for different problems and has been successfullyused to counteract complicated problems, outperforming other prevailing evolutionary algo‐rithmsintestconditions.

Furthermore,CROalgorithmcarriedoutparallelofsub‐reactionstepsintheoptimalprocesswhich benefits the minimizing for accomplishing time. Algorithm accomplishes local, globalsearchwithelementaryreactions.Inthese,fourtypesofelementaryreactionsareincluded:(1)on‐wallineffectivecollision,(2)decomposition,(3)inter‐molecularineffectivecollision,and(4)synthesis.Infact,eachreactionistheinteraction(thecombinationandvariation)ofmoleculesatahighenergyleveltobecomenewproductswithalowenergylevel,inastablestatus.Thede‐tailsoftheCROalgorithmcanbeseenin[20,23].MotivatedbythesuperiorcapabilityofCRO,theauthorsappliedselectparametersoftheSVMmodel.

Thesolution tooptimize theCROalgorithm involvesusing thenaturalchemical reactionofreactantstosolveproblems.Thebeginningofthealgorithmestablishesinitialreactants,whichplayan importantrole inasolution.Then, thereactantsreactandproduce four typesofreac‐tions. The algorithm is stoppedwhen the termination criterion reaches final status,when no

ParametersofSVM:Initialvalues( , )

CRO

TrainingSVMmodel

EvaluatefitnessbydeteriorationparameterEq.(13)

Updateparameters( , )

TrainedSVMmodelwithbestparame‐ters ,

Start

Terminationcriterial

Trainingset

End

Y

N

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morereactionscantakeplace.Inthiswork,theparameterpair , ofSVMissetasreactantsinfourtypesofreactions.Accordingtothis,theCROalgorithmconsistsofthefollowingsteps[21]:

Step1:Initializetheparameters.Step2:Settheinitialreactantsandevaluateenthalpy.Step3:Applychemicalreactionstoreactants.Step4:Updateandselectreactants.Step5:Gotostep3ifterminationcriterionnotsatisfied.Step6:Outputreactantwithbestenthalpy.

ClassificationmodelCRSVM

The optimization parameter pair , of the SVM can be obtained using the CRO algorithm.ThisCROalgorithmconductsstochasticsearchesusingapopulationofmolecules,eachofwhichrepresentsapossiblesolutiontoaproblem.Apopulationincludesafinitenumberofmolecules,witheachmoleculedefinedbyanevaluatingmechanismtoobtainitspotentialenergy.

TheprincipledtrainingphaseoftheCRSVMmodelincludessevenmainsteps,whichareim‐plementedasfollows:

Step1:Trainingandtestingdatasetsarepreparedafter featureextractionfromoriginalvibra‐tionsignals.

Step2:Thisisinitializationstep.Theinitial , parametersarerandomforSVM.Setthemaxi‐mum iterationnumber . Set the iterative variable: 0 andperform the trainingprocessforthenextsteps.Theparametersforthisoptimizationalgorithmareiteration

50,populationsize 5,upperbound 2 andlowerbound 2 .Step3:Increasetheiterationvariablebyset 1Step4:Deteriorationevaluation.Thedeteriorationfunctionisemployedtoevaluatethequality

ofeveryelement.Eq.(13)showstheclassificationaccuracyofanSVMclassifier:

% 100 % (13)

where isfalseclassifiedsamples, istotalsamplesinthetestingprocess.Thedesirablevalueissmallforhighclassificationaccuracy.

Step5:Stopcriteriachecking.IfthedeteriorationfunctionsatisfiesEq.(13)oriterationismax‐imal,gotostep7.Ifnot,gotothenextstep.

Step6:Updatethenew , parametersbasedonconditions.Gotostep3.Step7:Endofthetrainingprocedure.Fittingparametersareoptimaloutputvalues.

The efficient search capability of the chemical reaction algorithm is incorporatedwith thegeneralizationcapabilityofSVMtobringoutsynergiesoftheclassificationaccuracy.Thearchi‐tectureforCRSVMispresentedinFig.1.Eachreactantrepresentsthecandidatesolutionforthemodel,whichincludestheparameters , .

3. An expert technique based on the proposed GDA‐CRSVM method 

Inthissection,theauthorsproposeanewdiagnosismethodbasedonatwo‐stagehybridmodal‐ityforintegratingGDAwiththeCRSVM,calledtheGDA‐CRSVM.Thistakesspecialconsiderationof improved computational time, reduction of calculation memory and enhanced recognitionaccuracyof faultdata.Themethodologyofdimensionalityreduction(GDA) isclose,whichcanobtain the total intrinsic emergent information of the original high‐dimensional feature set.Combined,theoptimizedCRSVMmodelcanobtaineffectiveclassificationperformance.

Theprocessoftheproposedmethodconsistsoftwoparts:dimensionalityreductionandpat‐ternrecognition.First, theauthorsused theGDAmethod toreduce thehigh‐dimensional faultfeaturedatasetbytakingoutthemostresponsivefeaturestoproducealow‐dimensionalfeatureset.Theobtainedfeaturesetincreasedtheoverallreliabilityofthefaultdiagnosistechniqueaswell as the accuracy of diagnosis of an actual fault condition. Second, the reduced feature is

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served as input to theoptimized classificationmode,whichwas elaborately optimizedby theCROalgorithmbasedontheSVM,namelyCRSVM.

Overall,theexpertbearingfaultdiagnosistechniquebasedontheGDA‐CRSVMhybridmeth‐od aims to further improve fault diagnosis performance and ensure diagnosis reliability. It ispresentedinFig.2.Fromthisfigure,thetechniqueincludesfourmainstepsasfollows:Step1isvibrationsignalacquisition,Step2isMAFextraction,Step3isdimensionalityreduction,Step4ispatternclassification.Theimplementationprocessisdescribedbelow:

Step1:Inthefirststep,theoriginalvibrationsignalsareacquiredfromaccelerationsensors.Step2:Featureextraction isanurgentstepof thediagnosisprocess.Theextracted featureset

represents important information about the actual bearing conditions, which governsthefinalresultsofthediagnosisprocess.Thisfeaturesetcontainsthetime‐domain,fre‐quency‐domainfeatures,andtime‐frequencydomainfeaturesareextractedbytheLCDmethod,whichisusedtoformtheMAFsetoftheoriginalvibrationsignal.

Step3:TheGDAmethodisusedtodiscovertheintrinsicstructureoftheMAFset.Thereducedfeaturesetasalow‐dimensionalfeaturevectorhasmoreeffectiveclassificationperfor‐mance,suchasreducedcalculationmemory,computationaltime,andthebestclassifica‐tionaccuracyresults.

Step4:Thereducedfeaturesetisdividedintotrainingsetandtestingset.Eachlow‐dimensionaltrainingsampleinitsrespectiveclasslabelledasthetrainingsetisusedtodiscoverthebestparameterpair , of theSVMbyCRO.TheobtainedCRSVMclassifiermodel isthenused to recognize the samples in the testing set.For reliablediagnosis capability,thediagnosistechniquebasedonthisproposedGDA‐CRSVMmethodisappliedforbear‐ingfaultdiagnosis.

Fig.2Structdiagramofexpertfaultdiagnosistechnique

Dataacquisition

Vibrationsignalcollection

Multi‐aspectfeature

extraction

7featuresintime‐domain

8featuresinfrequency‐domain

5featuresintime‐frequencydomain D

imensionalreduction

UsingGDAonthehigh‐

dimensionalfeatureset

Reducedfeatureset

Trainingset

Testingset

TrainingSVMmodel

withbestparam

etersto

gettingoptimalmodel

CRO

TrainedCRSVMmodel

Diagnosisresults

OptimizedCRSVMclassificationmodel

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Fig.5Theschematicdrawingoftestrig

4. Results and discussion for bearing fault diagnosis performance case study 

Inthissection,twodatasetsofbearingconditionswerecollected.Theyareusedfortheexperi‐mentsbasedontheproposalofproposedexperttechniqueinactualstatusidentification.

4.1 Data acquisition 

The first dataset is vibration signals of bearing component fault cross from the Bearing DataCenteratCaseWesternReserveUniversity(Loparo,2013).Fig.3showstheexperimentalsetupmodel,whichconsistsofatwo‐HPrelianceelectricmotor,atorquetransducerandadynamom‐eter.The testbearingswere installedon themotorshaft,whichwas loadedbydynamometer.TheaccelerometerdataatDEwereusedasoriginalsignalsforthedetectionoffourbearingcon‐ditions: healthy bearing (HB), inner race (IR) fault, outer race (OR) fault, and rolling element(RE) fault.Adefectwas testedon the IR,OR,REof testbearingusingdefectsizes0.5334mm(0.021 inches) in diameter with a depth of 0.2794mm (0.011 inches) generated by electro‐discharge machining. Four vibration signal datasets were acquired from the bearings with asamplingfrequencyof12kHz,testedundermotorloadoftwo‐HPataspeedof1750RPM.Fig.4presentsthevibrationsignalsintime‐domain,infrequency‐domainfromfoursignalsamplesofthebearingconditions.Ineachfaultpattern,25sampleswereacquiredfromvibrationsignals.Each sample includes 4096 continuous data points in the time‐domain. The results obtainedgroupswith100vibrationsignalsatvariousbearingconditions.

Tofurthertesttheefficacyofthefaultdetectiontechnique,theseconddatasetwasacquiredfromtestrig,asshowninFig.5[25].ThebearingfaultswereintroducedbylasercuttingintheIRorREwithslotwidthof0.15mmanddepthof0.13mm,respectively.Thethreeexperimentalconditionstestedwerehealthybearing(HB),bearingwithIRfaultandbearingwithREfault.Atotalof15accelerationmeasuringsignalswithsamplingfrequencyof4096Hzwereacquiredforeachbearingcondition.

Fig.3Schematicoftheexperimental Fig.4 Thebearingconditionsinthetimedomainandfrequencydomain

‐a,b)Normalbearing ‐c,d)Innerbearingfault‐e,f)Outerbearingfault‐g,h)Rollerelementfault

Time‐domain Frequency‐domain

a) b)

d)c)

e)

g)

f)

h)

Amplitude

Magnitude

Time(s) Frequency(Hz)

Drivermotor

CouplingBearing Rotor

Shaft

Worktable

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Fig.6Thetime‐domainfeaturesofabearingsampleindifferentconditions

4.2 Multi‐aspect feature extraction 

TheMAFsetextracted fromoriginalvibrationsignalsplaysan importantrole for theachieve‐ment of the diagnosismodel. TheMAF includes time‐domain features, frequency‐domain fea‐turesandfeaturesinthetime‐frequencydomain,whichareconsideredahigh‐dimensionalfea‐turevector.

Time‐domainfeatures

The signalwas analysed to extract seven time‐domain statistical features . Table 1showsthesevenfeaturedefinitions.Inthistable,thefirstfivedimensions reflectthevibrationamplitudeandenergy in the time‐domain, the last twodimensions , are thecrestfactorandclearancefactor,whichrepresentthetime‐seriesdistributionofthesignals.Fig.6describesthetime‐domainfeaturesofthebearingsamplesinthedifferentconditions.

Table1Thefeaturedefinitionequationsintime‐domain

No.TimeDomain(TD) Remark

Feature Equation

1 Mean1

isavibrationsignalintimedomain 1,2, . . , , isthenumberofdatapoints.2 Standarddeviation

11

3 RootMeanSquare1

4 Skewness1

1

5 Kurtosis1

1

6 Crestfactor| |

7 ClearanceFactor1

| |

Frequency‐domainfeatures

Some signal information is also described in the frequency‐domain and reveals informationaboutthedemodulationspectrum,amplitudefrequencyordistribution,whichcannotbefoundinthetime‐domain.Toextractthesefeatures,theHilberttransformwasfirstusedtotransformthevibrationsignals.Eightfrequency‐domainfeatures werethenextractedfromthe

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frequencyspectrumofvibrationsignals,asshowninTable2.Theobtainedfeatures describe theconvergenceordivergenceof thespectrumpower, , indicateachange inposition frequency. The spectrumpower energy degree can centralize or de‐centralize as de‐scribedbyparameter canquantitativelymeasuredisorderinthesystem.Thefre‐quency‐domainfeaturesofthebearingconditionsinaninnerracefault,outerracefault,rollerelementfaultandhealthyfaultaredepictedinFig.7.Thefeaturesoftheconditionalouterracefaultandinnerracefaultareshownespeciallyclearly.

Time‐frequencydomainfeatures

LCD is a self‐adaptivemethodused indatadecomposition. LCDhasbeen successfullyused toanalyse non‐linear, non‐station signals, especially fault signals [2]. Obviously, LCD can extractthedeeplyhiddenfeatures inbearingfaultdata,asthesefeaturesareveryhardtodistinguishonly from the time‐domain and frequency‐domain statistical characteristics. In this study, theauthorsinvestigatedtheenergycorrelationcoefficientsbetweenthefirstseveralintrinsicscalecomponents (ISC),whichweredecomposedby theLCDmethod.These coefficients can revealtheoriginalvibrationsignalinthetime‐frequencyamplitudeanddistributionview,whichiswellandgoodforaccuratediagnosisofabearingfault.Further,anycomplexvibrationsignal isdecomposedintoISCandtheresiduebyLCD,asintheequationbelow[1]:

(14)

where isthe ICSoforiginalsignalobtainedbyLCD,theresidueis .

Table2Thefeaturedefinitionequationsinfrequency‐domain

No.Frequencydomain(FD)

RemarkFeature Equation

1 Meanfrequency1

istheenergyprobability

distributiondefinedas:| |

∑ | |

where isthespectrumofvibrationsignal,1,2, . . , ,Misthenumberof

spectrumlines.isthefrequencyvalueofthespectrumline.

2Standarddeviationfre‐quency

11

3 Skewnessfrequency1 ∑

11∑

4 Kurtosisfrequency1 ∑

11∑

5 Frequencycentre∑

∑

6Rootmeansquarefre‐quency

∑∑

7 Rootvariancefrequency ∑

8 ShannonEntropy log

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ThefirstseveralISCscontainalmostallvalidfaultinformationthatcharacterizestheoriginalsignal.There,ISCshavehigherenergythantherest.Inthiswork,thefirstfiveISCswereusedtocalculatethecorrelationofenergywiththeoriginalvibrationsignal.Thesefiveenergycorrela‐tion coefficients formed a feature subset representing the time‐frequency domain features ofbearingfaultstatus.Thesefollowingstepsweretaken:

Step1:Theoriginalvibrationsignalswerecollectedfromfaultsamplesoftherollerbearing.Step2:TheLCDmethoddecomposedthevibrationsignalsintoISCs.ThefirsthISCswerechosen.Step3:Theenergy ofthefirst ISCswascalculated,asrelatedinEq.15andEq.16:

(15)

where 1,2, . . , , isdatalengthof ISC, istheamplitudeofpoint inthe compo‐nent,and istheobtainedamplitudeofthe ISCbyHilberttransform:

(16)

Step4:Afeaturevector wasconstructedwiththeenergycorrelationcoefficientaselement:

, , . . , (17)

where , 1,2, . . , aretheenergycorrelationcoefficients.

∑ (18)

Fig.8alsoshowstime‐frequencydomainfeaturesofthebearingconditions.Thedeeplyhid‐denfeaturesinthebearingfaultsignalwereextractedbyLCD.ThefeaturevaluesshowthattheenergylevelofISCsgraduallydecreased.

Theobtainedtime‐frequencydomain featureswereadded into the featuresetandthus thecompleteMAFwasformedcontaining20features.TheobtainedMAFrepresentsabearingfaultconditionasahigh‐dimensional featurevectorthatservestheGDA‐CRSVMmethodwith inputdata.

 4.3 Diagnosis analysis based on GDA‐CRSVM 

Thehigh‐dimensionalMAFdiscoverednon‐linearcharacteristicsbytheGDAmethodasdimen‐sionality reduction, which can be given as a low‐dimensional feature set. The obtained low‐dimensionalfeaturesetwasthenrandomlydividedintoatraining‐testingpartition,70%:30%.Finally,asmentionedinSection3,thetrainingsetwasusedtotraintheoptimalCRSVMdiagno‐sismodelinactualbearingconditions.Theobtainedoptimaldiagnosismodelwasemployedtoclassifythesamplesinthetestingset.

Fig.7Thefrequency‐domainfeaturesofabearingsampleindifferentconditions

Fig.8 Time‐frequencydomainfeatures

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332  Advances in Production Engineering & Management 12(4) 2017

Feature‐dimensionalreductionbyGDAmethod

Practically, the high‐dimensional feature dataset involves too much memory for parameters,whichresultsinacomplicatedandinefficientforclassificationmodel.Forthisgoal,theextractedMAF of original vibration signalswas reduced to three features using the GDAmethodmen‐tionedinSection2.TodemonstratethesuperiorityoftheintroducedGDAdimensionalityreduc‐tionmethod,whenGDAisusedintheprocessofthetrainingpatternlabelledintoCclasses,Scnumberofsamplesineachclass,Cissetto4,Scissetto25.

Anexperimentwas conductedon the feature set, theauthors explored this toevaluate theGDAmethod’sdimensionalityreductionperformanceonthesample featureset.WecomparedGDAwithPCAandLDAas representativemethods.The experimental results of thePCA, LDAandGDAmethods are shown in Fig. 9 to Fig. 11, respectively,which show that PCA andLDAhavedimpatternclassificationperformance,withthreeclassesofoverlap.Comparedwiththese,GDAcanobtainaclearerseparationonthemapping.Therefore,GDAcanaccuratelyseparatethebearingfaultstatusfortheextractedMAFset.Infact,thisisbecausetheGDAhasgreaterabilitytodiscoverthemaximalratioofcentre‐classpartnertowithin‐classpartnerinthemulti‐aspectdatabyemploying class label information.Overall, thebestGDAmethod isused for thehigh‐dimensionalMAFdatasettoobtainthelow‐dimensionalfeaturedatasetwithprominentfeaturesasadimensionalityreductiontask.

Fig.11ScatterplotforthereducedfeaturesetbyGDA

Fig.9ScatterplotforthereducedfeaturesetbyPCA Fig.10 ScatterplotforthereducedfeaturesetbyLDA

An integrated generalized discriminant analysis method and chemical reaction support vector machine model… 

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CRSVMtraining

Inthisstudy,CRSVMclassifiermodelsweredesignedtoidentifyvariousbearingconditions.Infact,CRSVM1wasdesignedtoidentifynormalbearingconditions,withnormalbearingconditiondataassignedto 1,otherdataassignedto 1.CRSVM2wasdesignedtoidentifyinnerrace faults of bearings,with inner race fault data assigned to 1, other data assigned to

1.CRSVM3wasdesignedtoidentifyouterracefaults,withouterracefaultdataassignedto 1,otherdataassignedto 1.Inthesamework,CRSVM4wasdesignedtoidentifyrollerelementfaultsofbearings.ToevaluatetheperformanceofthediagnosistechniquebasedontheproposedGDA‐CRSVMmethod,theSVMwasadoptedtoperformbearingconditiondiag‐nosis. This is a traditional model, with the parameter set selected to follow experience at

200, 1.TheseclassifiersweretrainedonboththereducedfeaturesetandtheoriginalMAFsettoevaluatetherecognitionaccuracyresultsandtimeofdiagnosis.

Table3showstheMAF‐GDA‐CRSVMdiagnosistechniques.Theyweredesignedtoidentifythedifferent bearing conditions of the first dataset based onMAF and the proposedGDA‐CRSVMmethod.Intheexperimentationinthisdataset,theseclassifiersweretrainedwiththereducedfeatureset,with18samplesperclassselectedrandomlyasthetrainingset,meaning72sampleswere collected as the training set. Theywere used to calculate the deterioration function Eq.(13)andconstructtheoptimizedclassifiers.Thesevenrestsamplesperclasswereusedtotesttheobtainedclassifierwiththebestparameters.Thearchiveddiagnosisresultsforbearingcon‐ditionsareshowninTables4and5.

TodemonstratetheeffectivenessoftheMAF‐GDA‐CRSVMdiagnosistechnique,wedesigneddiagnosismodelsbasedonGDA‐CRSVMtoidentifybearingconditionsof theseconddataset inTable6.Similartotheprocessforthefirstdataset,thebearingfaultdiagnosisresultsarelistedinTables7and8.

Table3TheMAF‐GDA‐CRSVMdiagnosistechniqueofthefirstdataset.

Bearingcondition

AsamplefeaturevectorDiagnosistechnique

MAF‐ GDA‐CRSVM1

MAF‐ GDA‐CRSVM2

MAF‐GDA‐CRSVM3

MAF‐GDA‐CRSVM4

HB ‐53.2173 0.2071 0.1407 (+1) (‐1) (‐1) (‐1)IRfault 49.6764 8.0919 ‐0.2715 (‐1) (+1) (‐1) (‐1)ORfault 35.4594 ‐8.7378 0.8780 (‐1) (‐1) (+1) (‐1)REfault ‐31.0524 2.6843 ‐0.4135 (‐1) (‐1) (‐1) (+1)

Table4Thediagnosisaccuracyresult(%)offirstdatasetwithvariousfeaturesets

BearingconditionSamples Originalfeatureset Reductionfeatureset

Training Test SVM CRSVM GDA‐SVM GDA‐CRSVMHB 72 28 75 99.35 98.70 100IRfault 72 28 75 97.08 92.85 99.03ORfault 72 28 75 96.42 85.71 99.67REfault 72 28 75 88.63 96.42 98.70

Table5Thetimecost(s)offirstdatasetwithvariousfeaturesets

BearingconditionOriginalfeatureset Reductionfeatureset

SVM CRSVM GDA‐SVM GDA‐CRSVMHB 1.3249 1.4095 1.2473 1.3035IRfault 1.1137 1.4454 1.0277 1.0883ORfault 0.9838 1.4490 0.9833 1.2997REfault 0.8671 1.5266 0.8869 1.5039

Accordingtotheseresults,theMAFsetextractedfromoriginalvibrationsignalscancopewellwithboththeSVMandCRSVMdiagnosismodels forall fourbearingconditions.Theuseofallinputfeaturesdoesnotensureanimprovementintheclassificationaccuracyresultsforthevar‐iousclassifiers.Infact,thediagnosisaccuracyresultsinTables4,7shownthatevaluationontheconventionalSVMmodelwiththehigh‐dimensionalfeaturesetobtainedtheverypooraccuracy,the accuracy is only 75% in every bearing condition. Thus, this diagnosis technique usuallytends to produce a rejection and not reuse themodel. It should be emphasized that the low‐

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334  Advances in Production Engineering & Management 12(4) 2017

dimensionalfeaturesetgainedfromGDAgeneratedthebetterresultsincomparisonwithhigh‐dimensionalfeaturesetbythesameSVMmodels,theaccuracygotmaximumof98.70%forthefirstdatasetandgettingmaximumof90%fortheseconddataset.

Additionally, in this work, we explored the CRSVM model to exploit accuracy results forclassification purpose. The CRSVMmodel has achieved good diagnosis accuracy result in thehealthybearing(HB)andinnerrace(IR)faultconditionsevenwithusingthehigh‐dimensionalfeatureset.Thus,thisCRSVMmodelismoreappropriatewithdiagnosistechniqueforindividualconditionofbearing. Inparticular,Table4and7alsoshowed that theproposedGDA‐CRSVM‐based diagnosis technique provides the best results formost bearing conditions. ItsmeaningthatGDAmethodgeneratedthecompactfeaturesetthatinputtedtheoptimizedCRSVMclassifi‐cationmodelinintegratingtoproduceeffectiveness.Consequently,thisdiagnosistechniquecanbemoreused in themechanicalengineeringenvironment tosatisfywith theexpectedresults.Fig.12,13presentedtheresultsincomparisontheproposedmethodwiththeothermethods.

Moreover,theexecutiontimeoftheCRSVMclassifierforbearingfaultdiagnosisisfasterthanotherclassifiersforboththeoriginalfeaturesetandthereducedfeatureset,astheresultsshowinTables5and8.Theconsiderableusefulnessofreducingtheoriginalinputfeaturespacede‐finedby theGDAwascombinedwith theoptimal classifierCRSVMtobuildanexpertbearingfaultdiagnosistechnique.

Table6TheMAF‐GDA‐CRSVMdiagnosistechniqueoftheseconddataset

Bearingcondition AsamplefeaturevectorDiagnosistechnique

MAF‐ GDA‐CRSVM1

MAF‐GDA‐CRSVM2

MAF‐ GDA‐CRSVM4

HB ‐52.6982 0.2977 ‐5.8266 (+1) (‐1) (‐1)IRfault 15.0308 ‐2.0533 ‐1.4911 (‐1) (+1) (‐1)REfault ‐25.3478 0.9476 9.9225 (‐1) (‐1) (+1)

Table7Thediagnosisaccuracyresult(%)ofseconddatasetwithvariousfeaturesets 

BearingconditionSamples Originalfeatureset Reductionfeatureset

Training Test SVM CRSVM GDA‐SVM GDA‐CRSVMHB 30 15 75 99.09 90 100IRfault 30 15 75 95.45 86.81 96.82REfault 30 15 75 93.18 87.27 95

Table8Thetimecost(s)ofseconddatasetwithvariousfeaturesets 

BearingconditionOriginalfeatureset Reductionfeatureset

SVM CRSVM GDA‐SVM GDA‐CRSVMHB 1.0190 1.0456 0.8793 1.0240IRfault 1.0800 1.1376 0.9420 1.1522REfault 1.0495 1.1607 0.8901 0.9278

   

Fig.12Theclassificationaccuracyofdifferentdiagnosistechniquesforthefirstdataset

Fig.13Theclassificationaccuracyofdifferentdiagnosistechniquesfortheseconddataset

An integrated generalized discriminant analysis method and chemical reaction support vector machine model…

5. Conclusion In this paper, a GDA-CRSVM-based expert fault diagnosis technique is proposed. The GDA-CRSVM method is a two-stage hybrid method that integrates GDA with CRSVM for an expert di-agnosis technique. The original vibration dataset is firstly extracted the high-dimensional fea-ture set by the MAF extraction. This feature set then provides GDA-CRSVM, in which the GDA method exploits to produce a reduced feature set which serves as input to the CRSVM classifica-tion model. The most of reduced feature set is used for training the optimized CRSVM classifier and the rest use for evaluation. The experimental results demonstrate the high efficiency of the proposed method and its expertness in bearing fault diagnosis.

In fact, the MAF extraction produces features in the different domain to represent the bearing status which can restrain the effect of proposed method. Furthermore, the proposed method capacity can be restricted due to the operating of GDA method depends on the classes label which reveals the non-objective condition in supervised feature learning. Thus, a feature reduc-tion method is very useful and necessary for un-supervised feature learning in the future. Never-theless, we have unshaken confidence that the GDA-CRSVM method can help to improve the fault classification performance of any diagnosis technique with different subjects. The practical applications of GDA-CRSVM is enquired and attached the most of features corresponding to the real subject status.

Acknowledgements This research is supported by the National Key Research and Development Program of China (2016YFF0203400), and the National Natural Science Foundation of China (51575168 and 51375152). The authors would also like to thank the Collaborative Innovation Center of Intelli-gent New Energy Vehicles and the Hunan Collaborative Innovation Center for Green Car for sup-port.

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