Using COX Proportional Hazard Model for.docx
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Using COX Proportional Hazard Model for.docx
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UsingCOXProportionalHazardModelfor
UsingCOXProportionalHazardModelfor
MultivariatePrognosticAnalysisofBearings
JinghuaMaandDavidHe
IntelligentSystemsModelingandDevelopmentLaboratory
DepartmentofMechanical&IndustrialEngineering
TheUniversityofIllinoisatChicago
Chicago,IL60607
E-mail:
davidhe@uic.edu
Tel:
312-996-3410
Abstract
TheCoxproportionalhazardmodelhasbeenextensivelyusedinbiomedicalsurvivalanalysis. Itcanprovideastatisticallyrigorouspredictionofsystemfailuresbasedonthetime-to-failuredataobtainedfromexperimentaltests.Inthispaper,theCoxproportionalhazardmodelisusedtoanalyzethemultivariateriskfactorsinfluencingthelifeofthesystemandthentopredictfailurerateofthesystem.Failurerateisakeycriterioninestimatingthereliabilityofasystem.Byemployingthismethod,thefailureriskofsystemwhichisactiveattimeinstantaneoustwithinΔtcanbeobtained,thatis,thefailureprobabilityofthesystembetweentand(t+Δt). PrognosticindexcanbeobtainedfromCoxmodelwhichtakesintoaccountallsignificantinfluencingfactorsincludingoperationalconditionsandcomponentmaterials. Failureratecanbeobtainedfromprognosticindex.TwoparametersWeibulldistributionisusedtorepresentthebaselinehazardfunctionandaNewton-RaphsoniterativemethodofMaximumLikelihoodEstimation(MLE)algorithmisusedtodeterminethecoefficients.Datacollectedfromrealsteelandceramicbearingrun-to-failuretestsunderdifferentoperationalconditionsinanumericalstudyareusedtoevaluatetheeffectivenessoftheproposedapproach. Theresultsshowthattheproposedmethodcanbeeffectiveinpredictingtherealremainingusefullifeofthebearings.
Keywords:
Coxproportionalhazardmodel, multivariateriskfactors, failureprobability.
1.Introduction
Thecurrenttechniquetopredictfailurecanberoughlyclassifiedintotwomethods.Oneisphysicalmodel(design-based)methodinwhichtheinformationofsystemdesignisneeded.Ontheotherhand,data-drivenmethod(dataminingormachinelearning)whichuseshistoricaldatatoautomaticallylearnamodelofsystembehaviortomakeaprognosisdirectlyfromthedata,ratherthanusingahand-builtexperientialmodel.Thispaperdevelopsadata-drivenmethodtoestimatebearingreliability.Theproportionalhazardsmodelisthemostwidelyutilizedmodelforsurvivalanalysisinthemedicalfield.However,theapplicationsinindustrialengineeringarerare.ThatisthepurposeandsignificancewhytheCoxmodelisdiscussedinthispaper[7].Thepurposeofmodelistosimultaneouslyexploretheeffectsofseveralvariablesonthehealthconditionofsystemorcomponent.Coxintroducedconditionallikelihood,latercalledpartiallikelihood,toestimatetheparametersofasemi-parametricproportionalhazardsmodel(PHM),bysupposingthatthebaselinehazardfunctioninthePHMisarbitraryandthecovariatesaretime-independent.Thenwecanlearnwhatkindsofinfluencedifferentfactorshaveontheobjectivesystemorcomponent.Theprobabilityofbeingacertainstatecanbecalculatedbythismethod.
Theframeworkofthepaperisasfollow.InSection2,theCOXproportionalhazardmodelandtheapplicationinthefieldoffailureprognosticarestudied.ThecomparisonbetweenanumericalcasestudyandrealexperimentaldataofbearlifecyclewasdoneinSection3.Finally,someconclusionsaredrawninSection4.
2.ModelstudiesintheFieldofPHM
2.1TheflowchartoffailureprognosticsbyCoxmodel
AdiagramoftheprocedureforpredictionmodelbuildingandtheprognosticsisillustratedinFig.1.
Figure1theflowchartoffailureprognosticsbyCoxmodel
2.2ModelBuilding
Forasystem,havingMarkovpropertymeansthatfuturestatesdependonlyonthepresentstate,andareindependentofpaststates.Thatis,thedescriptionofthepresentstatefullycapturesalltheinformationthatcouldinfluencethefutureevolutionoftheprocess.Markovchainisasequenceofrandomvariables
…
withtheMarkovproperty.
Weconsiderthatthedeteriorationofasystemorcomponentiscausedbyageandoperationconditions.SowecanmodelthedeteriorationprocessbycombiningthebaselinehazardfunctionrelatingtotheagingprocessandCoxPHMformulationwhichincludesthedeteriorationscausedbyutilization.
isthefailurerateattimet.
isthebaselinehazardfunctionwhichrepresentstheagingprocessandistime-dependent.Itcanbeanyfunctionwithoutanylimitationonitsform.Inthispaper,twoparametersWeibulldistributionisusedtorepresent
.Thefailureratesofallsamplesinthetestwillbeidenticalif
isusedtoestimatethefailurerateonlywhatevertheoperationconditions(suchas,rotationspeed,temperatureorlubricatingcondition)are.
isastatefunctionwhichisdependentonlyontheobservationvalue
relatedwithoperationprocess.
isthenumberofinfluencingfactors[14][15].
isthestatevectoroftheinfluencingfactors.
isthevectorofthecoefficients.
isprognosticindex.ThemethodofMLEisalsousedtoestimatethetransitionprobabilitiesoftheMarkovianprocess.
2.3RegressioncoefficientestimationofCOXmodel
Lispartiallikelihoodfunction.AccordingtoBayesfunction
Thedenominatoristhesummationofallhazardforallobjectswhicharewordingnormallyattestingtime
.
isthenumberofobservations(i.e.thenumberofsamples).
inthedenominatorrepresentsthesummationofallobjects.Thenumeratoristhesummationofallobjectsforwhichfailurehappenedattime
.
Tomakecomputationeasy,thevalueofvector
iscalculatedbymaximizethelogarithmoftheequationabove.
So,
2.4Parametersestimationofbaselinehazardfunction
Theshapeparameter
inthebaselinehazardfunctioncanbeobtainedbythefollowingequation[16]
Thescaleparametercanberepresentedby[16]
and
canbeobtainedbysolvingtheequation(5)and(6).
3.Casestudies
3.1FailuremodedefinitionandinfluencingparametersSelection[14][15]
Bearingsarethecriticalcomponentsontherotatingmachine.Thatisimportanttomakeapredictionofthefailurebeforethefailurehappens.Fatigueisthemostimportantfailuremechanismofbearingswhichistheresultofshearstressescyclicallyappearingimmediatelybelowtheloadcarryingsurface.Afteratimethesestressescausecrackswhichgraduallyextenduptothesurface.
Acasewithreal-worlddataisstudiedtoprovethevalidityofthismethod.Changesinacousticnoise,vibration,andLubricatingtemperaturecanbeusedtodetectmechanicaldegradationofbearings.Inthiscase,wedefinefailurehappenwhen%&*%*&%*
Table1Commonfailuremodeofbearingandfailurecauses[13]
Commonfailuremodesofbearing
Causes
Wear
causedbyabrasiveparticles
inadequatelubrication
vibration
Indentation
Overloadingpressureorpressureappliedtothewrongplace
Pollutionofforeignparticles
Smearing
Slidingunderheavyaxialloadingandwithinadequatelubrication
Ringrotationrelativetoshaftorhousing
Diagonalsmearstreaksintheraceways
SurfaceDistress
Inadequateorimproperlubrication
Corrosion
water,moistureorcorrosivesubstances
Fittooloose.
Flutingorcraterscausedbyelectriccurrent
Electriccurrentthroughrotatingbearingornon-rotatingbearing.
Flaking(spawling)
excessivepreload
Temperaturedifferentialbetweeninnerandouterringstoogreat
Incorrectmounting
Unsuitablymatchedconfigurationbetweenparts
Cracks
Slidingunderheavyaxialloadingandwithinadequatelubrication
water,moistureorcorrosivesubstances
Theselectionofinfluencingfactorsisthesecondstepinthediagnosticsprocess.Theinfluencingfactors[relatingtothereliability]aretheinternalandexternalpartsofanitemwhichactonitsreliability,forexamplebycausingfailureratechanges[8].
Table2Classificationofinfluencingfactors
Designfactors
workingprinciple,sizes,materials,
componentqualityetc.
Installationandactivationfactors
Operationfactors
Solicitationfrequencyandload,electricalload(voltage,intensity),environment(mechanicalconstraints,temperature,humidity,pollution),performanceorreliabilityrequirementsetc.
Maintenancefactors
thequantityandthequalityofpreventiveandcorrectiveactionsetc.
Humanandorganizationalfactors
Inthiscase,weselecthumidity,temperature,lubricating,rotationspeed,materialparametersanddimensionasinfluencingfactors.Thevaluesofthemarepresentedinthetable2.Forquantitativefactors,thevaluesofmeasurementsareusedassuch,forexample,humidity,temperature,rotationspeed,materialparametersanddimension.Forqualitativefactors,thevaluesofmeasurementsshouldbecodedas“classes”,forexample,wedefinebearingwithgoodlubricatingconditionas“0”,drybearingas“1”.
3.2ModelBuilding
Thefailurerateattimetis:
Table3theselectedsignificantinfluencingfactors
Humidity
Lubricating
temperature
RotationSpeed
Material(Ceramic=1,Metal=0)
x1
x2
x3
x4
x5
3.3ParameterEstimation
Estimationofthemodel’sparametersistheprerequisitetoimplementinganymodelinCBM.MaximumLikelihoodEstimationmethodisusedtoestimatethemodel’sparameters(boththeparametersofbaselinehazardfunctionandtheCoxmodel).
MLEisawellknownstatisticalmethodusedtoestimatetheparametersofanunderlyingprobabilitydistributionthatrepresentsagivendataset.
Newton-Raphsonalgorithmisusedtosolvethelikelihoodequationswhichariseinmaximumlikelihoodfactoranalysis.
Table4C
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