Application of wavelets and neural networks to diagnostic sy(4)

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.