Application of wavelets and neural networks to diagnostic system development, 1, feature extraction

卷积神经网络和一些独立成分分析的外文文献

ComputersandChemicalEngineering23(1999)899–906

Applicationofwaveletsandneuralnetworkstodiagnosticsystem

development,1,featureextraction

B.H.Chen,X.Z.Wang*,S.H.Yang,C.McGreavy

DepartmentofChemicalEngineering,TheUni6ersityofLeeds,LeedsLS29JT,UKReceived14July1997;receivedinrevisedform9March1999;accepted9March1999

Abstract

Anintegratedframeworkforprocessmonitoringanddiagnosisispresentedwhichcombineswaveletsforfeatureextractionfromdynamictransientsignalsandanunsupervisedneuralnetworkforidenti cationofoperationalstates.Multiscalewaveletanalysisisusedtodeterminethesingularitiesoftransientsignalswhichrepresentthefeaturescharacterisingthetransients.Thissimultaneouslyreducesthedimensionalityofthedataandremovesnoisecomponents.Amodi edversionoftheadaptiveresonancetheoryisdeveloped,whichisdesignatedARTnetanduseswaveletfeatureextractionasthesubstituteofthedatapre-processingunit.ARTnetisprovedtobemoreeffectiveindealingwithnoisecontainedinthetransientsignalswhileretainsbeinganunsupervisedandrecursiveclusteringapproach.Theworkisreportedintwoparts.The rstpartisfocusedonfeatureextractionusingwavelets.ThesecondpartdescribesARTnetanditsapplicationtoacasestudyofare nery uidcatalyticcrackingprocess.©1999ElsevierScienceLtd.Allrightsreserved.

1.Introduction

Inmodernprocessplantscontrolledbydistributedcontrolsystems,theroleofoperatorshaschangedfrombeingprimarilyconcernedwithcontroltoabroadersupervisoryresponsibility:analysingoperationaldata,identifyingunusualconditionsastheydevelopandrespondingrapidlyandeffectivelybytakingcorrectiveactions.Thisisachallengingtaskbecauseoftheover-whelmingvolumeofdataoperatorshavetodealwith.Inrecentyearstherehasbeenasigni cantprogressinapplyingintelligentsystemsforprocessmonitoringanddiagnosis.Thisincludestheuseofneuralnetworks,multivariatestatisticalanalysis,expertsystemsaswellasqualitativesimulation.Itisrecognisedthatinprocessmonitoringanddiagnosis,puterbasedprocessingofdynamictrendsignalsisaimedatnoiseremovaland

*Correspondingauthor.Tel.:+44-113-233-2427;fax:+44-113-233-2405.

E-mailaddress:x.z.wang@leeds.ac.uk(X.Z.Wang)

dimensionreductionusingminimumdatapointstocapturethefeaturescharacterisingthetrendsignals.Variousapproacheshavebeenproposedandtheiref-fectivenessdependslargelyonhowtheprocessedinfor-mationistobeused,i.e.byhumanexperts,expertsystemsorneuralnetworks.Inthiswork,anintegratedframework,ARTnetisdevelopedandsubsequentlyap-pliedtoacasestudyofare nery uidcatalyticcrack-ingprocess.ARTnetisamodi edversionoftheadaptiveresonancetheory(ART2)(CarpenterandGrossberg,1987;Whiteley&Davis,1992,1994;White-ley,Davis,Mehrotra,&Ahalt,1996)whichuseswavelettransformsasthesubstituteofthedatapre-processingunitofART2.

Theworkisreportedintwoparts.The rstpartisfocusedonfeatureextractionfromdynamictransientsignalsusingwavelettransformsandthesecondpartisconcernedwiththeintroductionofARTnetanditsapplicationtoacasestudyofare nery uidcatalyticcrackingprocess.The rstpartisorganisedasfollows.InSection2somerepresentativeapproachesforfeatureextractionarebrie yreviewed.Thisnaturallyleadstotheintroductionofwaveletmultiscaleanalysisforfea-tureextractioninSection3.Waveletmultiscaleanalysis ndstheextremaofatransientsignalandanimportant

0098-1354/99/$-seefrontmatter©1999ElsevierScienceLtd.Allrightsreserved.PII:S0098-1354(99)00258-6

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.

卷积神经网络和一些独立成分分析的外文文献

B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906901

Thestretchedandcompressedwaveletsthroughscal-ingoperationareusedtocapturethedifferentfre-quencycomponentsofthefunctionbeinganalysed.Thetranslationoperation,ontheotherhand,involvesshift-ingofthemotherwaveletalongthetimeaxistocapturethetimeinformationofthefunctiontobeanalysedatadifferentposition.Inthisway,afamilyofscaledandtranslatedwaveletscanbecreatedusingscalingandtranslationparametersaandb.Thisallowssignalsoccurringatdifferenttimesandhavingdifferentfre-quenciestobeanalysed.Incontrasttotheshort-timeFouriertransform,whichusesasingleanalysiswindowfunction,thewavelettransformcanuseshortwindowsathighfrequenciesorlongwindowsatlowfrequencies.Thuswavelettransformiscapableofzooming-inonshort-livedhighfrequencyphenomenaandzooming-outonsustainedlowfrequencyphenomena.Thisisthemainadvantageofthewaveletovertheshort-timeFouriertransform.

Wavelettransformcanbecategorisedintocontinu-ousanddiscrete.Continuous,inthecontextofwavelettransform,impliesthatthescalingandtranslationparametersaandbchangecontinuously.However,calculatingwaveletcoef cientsforeverypossiblescalecanrepresentaconsiderableeffortandresultinavastamountofdata.Thereforediscreteparameterwavelettransformisoftenused.Thediscreteparameterwavelettransformusesscaleandpositionvaluesbasedonpow-ersoftwo-so-calleddyadicscalesandpositionsandmakestheanalysismuchmoreef cient,whilstremain-ingaccurate.Todothis,thescaleandtimeparametersarediscretisedasfollows,a=am0,

b=nb0an0

m,nareintegers

(3)

Thefamilyofwavelets{ m,n(t)}isgivenby

m,n(t)=a 0m/2 (a 0

m

t nb0)(4)

resultinginadiscretewavelettransform(DWT)havingtheform

DWTf(m,n)= f, m,n

+

=a

0

m/2&

f(t) (a 0

m

t nb0)(5)

Mallat(1989)developedanapproachforimplement-ingthisusing lters.Formanysignals,thelowfre-quencycontentisthemostimportantpart.Thehigh

frequencycontent,ontheotherhandprovides avourornuance.Inwaveletanalysisthelowfrequencycon-tentiscalledtheapproximationandthehighfrequencycontentiscalledthedetail.The lteringprocessuseslowpassandhighpass lterstodecomposeanoriginalsignalintotheapproximationanddetailparts.Itisnotnecessarytopreservealltheoutputsfromthe lters.Normallytheyaredownsampledandkeeponlytheevencomponentsofthelowpassandhighpass lteroutputs.

Thedecompositioncanbeiterated,withsuccessive

approximationsbeingdecomposedinturn,sothatonesignalisbrokenintomanylower-resolutioncomponents.

Inthecaseofadiscretewavelettransform,recon-structionoftheoriginalsignalisnotguaranteed.Daubechies(1992)developedconditionsunderwhichthe{ m,n}ually,a0=2andb0=1areused,althoughanyvaluescanbeused.Inthiscase,boththetransformandreconstructionarecompletebecausethefamilyofwaveletsformanor-thonormalbasis.

3.2.Singularitydetectionusingwa6eletsforfeatureextraction

Singularitiesoftencarrythemostimportantinforma-tioninsignals.Singularitiesofasignalcanbeusedasthecompactrepresentation,i.e.thefeaturesoftheoriginalsignal.Mathematically,thelocalsingularityofafunctionismeasuredbyLipschitzexponents(Mallat&Hwang,1992).MallatandHwang(1992)provedthatthelocalmaximaofthewavelettransformmodulusdetectsthelocationsofirregularstructuresandpro-videsnumericalproceduresforcomputingtheLipschitzexponents.Withintheframeworkofscale-space lter-ing,in exionpointsoff(t)appearasextremafor(f(t)/(tandzerocrossingfor(2f(t)/(t2,soMallatandZhong(1992)suggestsusingawaveletwhichisthe rstderivativeofascalingfunctionF(t), (t)=

d (t)dt

withacubicspinebeingusedforthescalingfunction.BakshiandStephanopoulos(1996)usedthein exionpointsastheconnectionpointsofepisodesegmentsofasignal.

Thewaveletmodulusmaximaandzero-crossingrep-resentationsweredevelopedfromunderlyingcontinu-ous-timetheory.Forcomputerimplementation,thishastobecastindiscrete-timedomain.BermanandBaras(1993)provedthatwavelettransformextrema/zero-crossingprovidestablerepresentationsof nitelengthdiscrete-timesignals.Amorecompletediscrete-timeframeworkfortherepresentationofthewavelettransformwasdevelopedbyCvetkovicandVetterli(1995)andthereforeisusedinthisstudy.Theyde-signedanon-subsampledmulti-resolutionanalysis ingthis lterbank,thewaveletfunctioncanbeselectedfromawiderrangethantheB-splineinMallat’smethod.Non-subsampledmulti-resolutionanalysiswasusedtodeterminesingularitiesofasignal.Anoctavebandnon-subsampled lterbankwithanalysis ltersH0(z)andH1(z)isshowninFig.1.Inthismethod,awavelettransformreferstotheboundedlinearoperators

卷积神经网络和一些独立成分分析的外文文献

902B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906

Wj:l2(Z) l2(Z);j=1,2,…j+1.TheoperatorsWj,aretheconvolutionoperatorswiththeimpulsere-sponsesofthe lters:V1(z)=H1(z)V2(z)=H0(z)H1(z2)Vj(z)=H0(z)···H1

0(z2

j 2

)H1(z2

j )Vj+1(z)=H 1

0(z)···H0(z2

j 2

)H0(z2

j)

ThemultiresolutionproceduredepictedinFig.1canbedescribedlessrigorously.Fig.1showsfoursteps,orfourscales.Inthe rstscale,theoriginalsignalissplitintoapproximationAx1anddetailDx1.ThedetailDx1issupposedtobemainlythenoisecomponentsoftheoriginalsignal.Ax1isfurtherdecomposedintoapprox-imationAx2anddetailDx2,Ax2toAx3andDx3andAx3toAx4andDx4.Ineachsteptheextremaofthedetailarefound.Apparently,inthe rstfewsteps,theextremaarebothasaresultofthenoiseandthetrendofthenoise-freesignal.Withscalesbeingincreased,thenoiseextremawillgraduallyberemovedwhiletheextremaofthenoise-freesignalremain.Inthisway,throughmulti-scaleanalysisandextremadetermina-tion,theextremaofthenoise-freesignalcanbefound,whichareregardedasthefeaturesofthesignal.

Fortherepresentationofextrema,itisconvenienttousea niteimpulseresponse(FIR)wavelet lter.TheFIRisa lterwiththesequence{ak:k Z}andhasonlyKnon-zeroterms.AtypicalexampleistheHaarwavelet,whichhasonlytwonon-zerocoef cients.Daubechies’wavelets(Daubechies,1992)arealsoFIR ltersandsmootherthantheHaarwavelet.Daubechies’waveletshavingmorecoef cientsaresmootherandhavehighervanishingmoments.Theyalsorequirelesscomputationaleffortastheyarecon-structedby lterconvolution.

Fig.1.Anoctavebandnon-subsampled lterbank.

TheDaubechies’scaleandwaveletfunctionsareexpressedas

(t)=%h(k) (2t k)

(6)k

(t)=%g(k) (2t k)

(7)

k

where{h(k)}isthelow-pass ltercoef cientsand{g(k)}theband-pass ltercoef cients.

Daubechies’waveletshaveamaximumnumberofvanishingmomentsforthesupportspace.Thevanish-ingmomentsofthewaveletsalsohaveadifferentnumberofcoef ingwaveletswithmorevan-ishingmomentshastheadvantageofbeingabletomeasuretheLipschitzregularityuptoahigherorder,whichishelpfulin lteringnoise,butitalsoincreasesthenumberofmaximalines.Thenumberofmaximaforagivenscaleoftenincreaseslinearlywiththenum-berofmomentsofthewavelet.Inordertominimisecomputationaleffort,itisnecessarytohaveaminimumnumberofmaximatodetectthesigni cantirregularbehaviourofasignal.ThismeanschoosingawaveletwithasfewvanishingmomentsaspossiblebutwithenoughmomentstodetecttheLipschitzexponentsofthehighestordercomponentsofinterest.

Inthisstudy,aneightcoef cient‘least-asymmetric’Daubechies’waveletisusedasa lter.Thescaleandwaveletfunctionforthis lterareillustratedinFig.2.Asignalf(t)=sin(t)anditsextremaofwaveletanalysisusingnon-subsampled lterbankwithDaubechies’eightcoef cientsleastasymmetrywaveletisillustratedinFig.3,whichshowsthatextremaofwaveletanalysiscorrespondtothesingularitiesofthesignal.InFig.3b,thewaveletisusedas lterandthe rstsingularityofthesignalinFig.3acorrespondstominimumofwaveletanalysis.InFig.4itisamaximumbecauseadifferentwaveletisemployed.Theformerisusedhere.

3.3.Noiseextremaremo6al

Theextremaobtainedfromwaveletmulti-resolutionanalysiscorrespondtothesingularitiesofthesignal,whichmayalsoincludethoseproducedbynoise,de-pendingontheanalysisscales.Therefore,infeatureextractionitisnecessarytofurtheridentifyand lteroutnoiseextremafromwavelettransform.Themostclassicaltechniqueofremovingnoisefromasignalisto lterit.Partofthenoiseisremovedbutitmayalsosmooththesignalsingularitiesatthesametime.MallatandHwang(1992)andMallatandZhong(1992)devel-opedatechniqueforevaluatingnoiseextremainwaveletanalysis.Theyfoundthatsomenoisemaximaincreaseonaveragewhenthescaledecreasesordon’tpropagatetolargerscales.Thesearethemodulusmax-imawhicharemostlyin uencedbynoise uctuations.

卷积神经网络和一些独立成分分析的外文文献

B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906903

Fig.2.The‘least-Asymmetric’scalefunctionandwaveletfunction.

Fig.3.Signal(a)anditsextrema(b)ofwaveletanalysisusingDaubechies’eightcoef cientswavelet.

Fig.5andFig.6illustratethisidea.InFig.5,threedifferentnoisefrequenciesarestudied.Thewaveletmulti-resolutionanalysisisshownontheleftandex-tremaofwaveletanalysisareontheright.Clearly,theextremawilldecreaseandthendisappearasthescaleincreases.

Fig.6showsasignalwhichisbasicallythesineinFig.3acorruptedbywhitenoiseaswellasthewaveletmulti-scaleextremaanalysis.Noisecomponentsarere-ducedandthendisappearasthescaleincreases.Theresultsforscales-4and-5aresimilartothatofFig.3bwhichisnoise-free.Thisshowsthattheextremaofthetrendsignalareretainedwhilenoiseextremaare ltered.

theextremarepresentationinscale-4isavectorofdimension70,

Scale-4=(…x5…x23…x37…x53…)

wherex5standsforanon-zerodatumincolumn5.Whileinscale-5,itbecomes

3.4.Piece-wiseprocessing

Twoobservationsaremadeabouttheabovediscus-sions.Firstly,extremaanalysisusingwaveletmultireso-lutionanalysisremainssteadywiththeincreaseofscales,sotherepresentationissteady.Forexample,inFig.6whenthescaleisincreasedfromfourto ve,thefourextremaremain.Secondly,thelocationofextremamayslightlyshiftwithtimeasscaleincreases.InFig.6,

Fig.4.ExtremaofwaveletanalysisusingDaubechies’tencoef cientswavelet.

卷积神经网络和一些独立成分分析的外文文献

904B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906

Fig.5.Noisesignal,itswavelettransformationandtheextremaofwaveletanalysis.

Scale-5=(…x7…x22…x38…x54…)

Itisobviousthatnon-zerodatumintheposition5ofscale-4isshiftedtotheposition7ofscale-5.Thisinconsistencyshouldbeavoided.Forinstance(2,0…0,3)and(2,0…3,0)shouldbeconsidereddif-ferent.Thisisnecessaryespeciallywhenthetrendsofavariableatdifferentoperationsconditionsareconsidered.

Theextremarepresentationcanbehighlysparsevec-tors.Thisistrueforprocessdynamicresponseswhichare

slowinfrequency.Themethodweusediscalledpiece-wiseprocessing.Theideaistomapahighlysparsevectortoadenservectorbydimensionreduction.Forexample,withscale-4andscale-5discussedabove,ifthepiece-wisesub-regionis xedasfourdatapoints,thenscale-4andscale-5willbetransformedtovectorsofdimension18.Scale-4%=(…x2…x6…x10…x13…)Scale-5%=(…x2…x6…x10…x13…)

Itisclearthatafterpiece-wiseprocessing,thedimen-

卷积神经网络和一些独立成分分析的外文文献

B.H.Chenetal./ComputersandChemicalEngineering23(1999)899–906905

sionisreducedandscale-4%andscale-5%areconsistent.Thereforeusingpiece-wiseprocessingtechnique,itcanachieveconsistentfeatureextractionaswellasdimen-sionreduction.

4.FinalRemarks

Featuresofaprocessdynamictransientsignalareidenti edasthesingularitiesandirregularitiesbecausetheycontainthemostimportantinformationcorre-spondingtochangesofoperationalstates.Theap-

proachdevelopedbyMallatandHwang(1992)andCvetkovicandVetterli(1995)fordeterminingsingulari-tiesandirregularitiesisintroducedforfeatureextrac-tionofdynamictransientsignalsofprocessoperations,whicharetheextremaofwaveletanalysis.Anapproachfornoiseextremaremovalandpiece-wisedimensionreductionarealsodiscussed.Inthesecondpartofthepaper,theuseoftheapproachtoreplacethedatapre-processingpartoftheadaptiveresonancetheorytodevelopamoreef cientunsupervisedandrecursivelearningsystemARTnetanddescribeitsapplicationtoare nery uidcatalyticcrackingprocessisreported.

Fig.6.Noisesignalanditsmulti-resolutionanalysis.Axi,approximationofmultiresolutionanalysis;Dxi,detail.

卷积神经网络和一些独立成分分析的外文文献

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

5.NotationAapproximationinwaveletmultiresolutionanalysis

awaveletdilationparameter

a0,b0discretewavelettransformparametersbwavelettranslationparameter

D

detailinwaveletmultiresolutionanalysisDWTfdiscretewaveletcoef cientf(t)afunctioninthetimedomaing(k)thekthwaveletsynthesis lterHwaveletanalysis lter

h(k)thekthwaveletanalysis lter

m,ndiscretewavelettransformparameterssscalettime

GreekaLipchitzexponent waveletfunction

(t)

waveletscalefunctionororthogonalfunction

Acknowledgements

Theauthorsareindebtedtotheanonymousrefereesfortheirconstructivecommentswhichledtoanim-provedmanuscript.Theauthorsacknowledgethepartial nancialsupportoftheAigisSystems,USA.The rstauthorthanksthe nancialsupportoftheSino-BritishFriendshipScholarshipScheme(SBFSS).

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