Application of wavelets and neural networks to diagnostic sy(3)

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