Simple Denoising Algorithm Using Wavelet Transform

arXiv:nlin/0002028v1[nlin.CD]18Feb2000SimpledenoisingalgorithmusingwavelettransformManojitRoy,V.RaviKumar1,B.D.KulkarniChemicalEngineeringDivision,NationalChemicalLaboratory,Pune411008,IndiaJohnSanderson,MartinRhodesDepartmentofChemicalEngineering,MonashUniversity,Clayton,Victoria,3168,AustraliaMichelvanderStappenUnileverResearch,Vlaardingen,Postbox114,3130,ACVlaardingen,TheNetherlandsFebruary6,2008keywords:Noise,Discretewavelettransform,Chaos,DifferentiationApplicationofwaveletsandmultiresolutionanalysistoreactionengineeringsystemsfromthepointofviewofprocessmonitoring,faultdetection,systemsanalysisetc.isanimportanttopicandofcurrentresearchinterest(see,BakshiandStephanopoulos,1994;Safaviet.al.,1997;Luoet.al.,1998;CarrierandStephanopoulos,1998).Inthepresentpaperwefocusononesuchimportantapplication,whereweproposeanewandsimplealgorithmforthereductionofnoisefromascalartimeseriesdata.Presenceofnoiseinatime–varyingsignalrestrictsone’sabilitytoobtainmeaningfulinformationfromthesig-nal.Measurementofcorrelationdimensioncangetaffectedbyanoiselevelassmallas1%ofsignal,makingestimationofinvariantpropertiesofadynamical1e-mailaddressforcorrespondence:[e-mail]1system,suchasthedimensionoftheattractorandLyapunovexponents,al-mostimpossible(KostelichandYorke,1988).Noiseinexperimentaldatacanalsocausemisleadingconclusions(Grassbergeret.al.,1991).Ahostoflit-eratureexistsonvarioustechniquesfornoisereduction(KostelichandYorke,1988;H¨ardle,1990;FarmerandSidorowich,1991;Sauer,1992;CawleyandHsu,1992;Cohen,1995;DonohoandJohnstone,1995;KantzandSchreiber,1997).Forinstance,FastFourierTransform(FFT)reducesnoiseeffectivelyinthosecaseswherethefrequencydistributionofnoiseisknown(KostelichandYorke,1988;Cohen,1995;KantzandSchreiber,1997);singularvalueanaly-sismethods(CawleyandHsu,1992)projecttheoriginaltime–seriesontoanoptimalsubspace,wherebynoisecomponentsareleftbehindintheremainingorthogonaldirections,etc.Intheexistingwavelet–baseddenoisingmethods(DonohoandJohnstone,1995)twotypesofdenoisingareintroduced:lineardenoisingandnonlineardenoising.Inlineardenoising,noiseisassumedtobeconcentratedonlyonthefinescalesandallthewaveletcoefficientsbelowthesescalesarecutoff.Nonlineardenoising,ontheotherhand,treatsnoisereductionbyeithercuttingoffallcoefficientsbelowacertainthreshold(socalled‘hard–thresholding’),orreducingallcoefficientsbythisthreshold(socalled‘soft–thresholding’).Thethresholdvaluesareobtainedbystatisticalcalculationsandhasbeenseentodependonthestandarddeviationofthenoise(Nason,1994).Thenoisereductionalgorithmthatweproposeheremakesuseofthewavelettransform(WT)whichinmanywayscomplementsthewellknownFourierTransform(FT)procedure.Weapplyourmethod,firstly,tothreemodelflowsystems,viz.Lorenz,Autocatalator,andR¨osslersystems,allex-hibitingchaoticdynamics.Thereasonsforchoosingthesesystemsarethefollowing:Firstly,allofthemaresimplifiedmodelsofwell–studiedexperimen-talsystems.Forinstance,Lorenzisasimplerealizationofconvectivesystems(Lorenz,1963),whiletheAutocatalatorandR¨osslerhavetheirmorecompli-catedanalogsinchemicalmulticomponentreactions(R¨ossler,1976;Lynch,1992).Secondly,chaoticdynamicsisextremelynonlinear,highlysensitive,possessesonlyshort–timecorrelationsandisassociatedwithabroadrange2offrequencies(GuckenheimerandHolmes,1983;Strogatz,1994).BecauseofthesepropertiesitiswellknownthatFTmethodsarenotapplicableinastraightforwardwaytochaoticdynamicalsystems(Abarbanel,1993).Ontheotherhand,WTmethodsareparticularlysuitedtohandlenotonlynonlin-earbutnonstationarysignals(StrangandNguyen,1996).Thisisbecausethepropertiesofthedataarestudiedatvaryingscaleswithsuperiortimelocaliza-tionanalysiswhencomparedtoFTtechnique.Ournoisereductionalgorithmisadvantageous,because,asshallbeshown,thethresholdlevelfornoiseisidentifiedautomatically.Inthisstudy,wehaveusedthediscreteanalogofthewavelettransform(DWT)whichinvolvestransformingagivensignalwithorthogonalwaveletbasisfunctionsbydilatingandtranslatingindiscretesteps(Daubechies,1990;Holschneider,1995).Forstudypurposeswecorruptonevariablex(t)foreachofthesesystemswithnoiseofzeromean,andthenapplyouralgorithmfordenoising.Weanalyzetheperformanceofthismethodinallthethreesystemsforawiderangeofnoisestrengths,andshowitseffectiveness.Importantly,wethenvalidatetheapplicabilityofthemethodtoexperimentaldataobtainedfromtwochemicalsystems.Inonesystemthetimeseriesdatawasobtainedfrompressurefluctuationmeasurementsofthehydrodynamicsinafluidizedbed.Intheothertheconductivitymeasurementsinaliquidsurfactantmanufacturingexperimentwereanalyzed.MethodologyThenoisereductionalgorithmbasedonDWTconsistsofthefollowingfivesteps:Step1:Infirststep,wedifferentiatethenoisysignalx(t)toobtainthedataxd(t),usingthemethodofcentralfinitedifferenceswithfourthordercorrectiontominimizeerror(Constantinides,1987),i.e.,xd(t)=dx(t)dt.(1)3Step2:WethentakeDWTofthedataxd(t)andobtainwaveletcoefficientsWj,katvariousdyadicscalesjanddisplacementsk.Adyadicscaleisthescalewhosenumericalmagnitudeisequalto2(two)raisedtoanintegerexponent,andislabeledbytheexponent.Thus,thedyadicscalejreferstoascaleofmagnit