Application of wavelets and neural networks to diagnostic sy(2)

卷积神经网络和一些独立成分分析的外文文献

900B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906

issueishowtoremovetheeffectsofnoisecomponentsandachieveconsistentresultsindifferentscales.ThisisthesubjectofSection4.

2.Previousworkonfeatureextractionofdynamictransients

Thissectionbrie yreviewssomeofthepreviousworkonfeatureextraction.Featureextractionisbasi-callyatransformationofthedatacomposingady-namictrendtoalowerdimensionality.Animportantpropertyofsuchatransformationisthatitisinforma-tionpreserving,thatis,dataisreducedbyremovingredundantcomponentswhilepreserving,insomeopti-malsense,informationwhichiscrucialforpatterndiscrimination.

Someresearchershaveadaptedtheepisoderepresen-tationtechniqueoriginatedbyWilliam(1986)toquali-tativeinterpretationoftransientsignals.JanuszandVenkatasubramanian(1991)developedanepisodeap-proachthatusesnineprimitivestorepresentanyplotsofafunction.Eachprimitiveconsistsofthesignsandthe rstandsecondderivativesofthefunction.There-fore,eachprimitivepossestheinformationaboutwhetherthefunctionispositiveornegative,increasing,decreasing,ornotchangingandtheconcavity.Anepisodeisanintervaldescribedbyonlyoneprimitiveandthetimeintervaltheepisodespans.Atrendisaseriesofepisodesthatwhengroupedtogethercancom-pletelydescribethedynamicfeature.Theapproachautomaticallyconvertson-linesensordatatoqualita-tiveclassi cationtrees.CheungandStephanopoulos(1990)developedaslightlydifferentapproachcalledtriangular-episodethatusesseventrianglecomponentstodescribeadynamictrend.BakshiandStephanopou-los(1994,1996)usedwaveletdecompositionoffunc-tionsindifferentscalesandzero-crossingofwaveletderivativesto ndthein ectionsofdecomposition.Inthisway,episodescanbeidenti edautomaticallybycomputers.Basedonepisodeanalysis,dynamictrendscanbeinterpretedassymbolicrepresentations.Themainideaofdynamictrendinterpretationusingepisodeapproachesistoclassifyatrendsuchasincreasingordecreasingpieces.Thisinterpretationissometimesnotenoughandinadequateinprocessanalysis.Further-more,thereisnonoise lteringinanyoftheepisodebasedapproaches,whichsigni cantlylimitsthetrendrepresentationandidenti cationcapability.

WhiteleyandDavis(1992)appliedback-propagationneuralnetworks(BPNN)toconvertnumericalsensordataintosymbolicabstractions.Themajorlimitationofthisapproachisthatitrequirestrainingdatatotrainthemodel rst.

ThemostwellknowntechniqueforsignalanalysisisprobablytheFouriertransformanditistherefore

necessarytomentionedithere.Fouriertransformusessineandcosineasitsbuildingblockstodecomposeafunctionintoasumoffrequencycomponents.How-ever,Fouriertransformdoesnotshowhowfrequencyvarieswithtime,thereforeitisnotabletodetectwhenaparticulareventtookplace.Itmeansthatthenon-sta-tionaryfeatureofthesignalisnotcaptured.Theshort-timeFouriertransformisabletoovercomethislimitationbyslidingawindowoverthesignalintime.Howeverintime-frequencyanalysisofanon-stationarysignal,therearetwocon ictingrequirements.Thewin-dowwidthmustbelongenoughtogivethedesiredfrequencyresolutionbutmustalsobeshortenoughtolosetrackoftimedependentevents.Whileitispossibletooptimisethedesignofwindowshapestooptimise,ortrade-offtimeandfrequencyresolution,thereisafun-damentallimitationonwhatcanbeachieved,foragiven xedwindowwidth(Dai,Joseph&Motard,1994).

3.Featureextractionusingwavelettransform

Averybriefintroductionofwavelettransformationforsignalprocessingisnowpresented.Thenthemethodemployedinthisstudyforfeatureextractionusingwaveletsisintroducedandillustratedusingexamples.

3.1.Signaltransformationusingwa6elets

Wavelettransformationisdesignedtoaddresstheproblemofnon-stationarysignals.Itinvolvesrepre-sentingatimefunctionintermsofsimple, xedbuild-ingblocks,termedwavelets.Thesebuildingblocksareactuallyafamilyoffunctionswhicharederivedfromasinglegeneratingfunctioncalledthemotherwaveletbytranslationanddilationoperations.Dilation,alsoknownasscaling,compressesorstretchesthemotherwaveletandtranslationshiftsitalongthetimeaxis.&

Themotherwaveletsatis es

+

(t)dt=0(1)

andthetranslationandscalingoperationson (t)createsafamilyoffunctions,

=

1a,b(t) t ba

(2)Theparameteraisascalingfactorandstretches(or

compresses)themotherwavelet.Theparameterbisatranslationalongthetimeaxisandsimplyshiftsawaveletandsodelaysoradvancesthetimeatwhichitisactivated.Mathematicallydelayingafunctionf(t)bytdisrepresentedbyf(t td).Thefactor1/ aisusedtoensuretheenergyofthescaledandtranslatedversionsarethesameasthemotherwavelet.