Performance evaluation of ensemble empirical mode decomposition

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PERFORMANCEEVALUATIONOFENSEMBLE

EMPIRICALMODEDECOMPOSITION

R.K.NIAZY∗,†,‡,C.F.BECKMANN∗,§,J.M.BRADY†

andS.M.SMITH∗

∗Centre

forFunctionalMagneticResonanceImagingoftheBrain(FMRIB)

UniversityofOxford,Oxford,UK

†Department

ofEngineeringScience

UniversityofOxford,Oxford,UK

‡Cardiff

UniversityBrainResearchImagingCentre(CUBRIC)SchoolofPsychology,CardiffUniversityParkPlace,Cardiff,CFI03AT,UK

§Department

ofClinicalNeurosciences

DivisionofNeuroscienceandMentalHealthSchoolofMedicine,ImperialCollegeLondon,UK

Empiricalmodedecomposition(EMD)isanadaptive,data-drivenalgorithmthatdecom-posesanytimeseriesintoitsintrinsicmodesofoscillation,whichcanthenbeusedinthecalculationoftheinstantaneousphaseandfrequency.EnsembleEMD(EEMD),wherethefinalEMDisestimatedbyaveragingnumerousEMDrunswiththeadditionofnoise,wasanadvancementintroducedbyWuandHuang(2008)totryincreasingtherobustnessofEMDandalleviatesomeofthecommonproblemsofEMDsuchasmodemixing.Inthiswork,wetesttheperformanceofEEMDasopposedtonormalEMD,withemphasisontheeffectofselectingdifferentstoppingcriteriaandnoiselevels.OurresultsindicatethatEEMD,inadditiontoslightlyincreasingtheaccuracyoftheEMDoutput,substantiallyincreasestherobustnessoftheresultsandtheconfidenceinthedecomposition.

Keywords:Ensembleempiricalmodedecomposition;Hilbert–Huangtransform;stoppingcriteria.

1.Introduction

Empiricalmodedecomposition(EMD)1,2isadata-driven,adaptivedataanaly-sismethodthatdecomposesanytimeseriesintoitsintrinsicmodesofoscillation.Eachmode(component),calledintrinsicmodefunction(IMF),isanarrow-band,amplitude-andfrequency-modulated(AMandFM)component,whichallowsthecalculationoftheinstantaneousphaseandfrequencyaspartofHilbert–Huangtransform(HHT).1,2HHTanditsEMDalgorithmhavefoundmanyapplicationsinawiderangeofscientificfieldsincludingmeteorology,3–6financialtime-series

231

232R.K.Niazyetal.

analysis,7structuralfailure,8,9seismology,8imageandtextureanalysis,10,11noisesynthesis,12,13signalfilteringandde-noising,14,15variousbiomedicalapplica-tions16–23andinneuroscience.14,24–26

Despiteitspowerandapplicability,EMDsuffersfromsomeweaknessesthathaveyettobeovercome.Ofinteresthereisthelackofrobustnessintheresultswithregardtofactorssuchasthechoiceofstoppingcriteria,noise,andfre-quencycontent.Thesefactorsaffecttheresultsintheformofmodemixingand“erroneous”low-frequencycomponents.27Inanattempttoaddresstheseissues,WuandHuang27haveintroducedtheconcepttoEnsembleEMD(EEMD)bywhichthefinaldecompositionisachievedbyaveraginganensembleofdecompositions,eachwithsomeGaussiannoiseaddedtoaltertheoutcome.Byaveragingthedifferentdecompositions,thenoiseisaveragedoutandanestimateofthe“true”decomposi-tioniscalculatedwithaconfidenceestimate.Inthiswork,wetesttheperformanceofEEMDagainstconventionalEMDwithregardstorobustnessagainstthelevelofinherentnoiseandvariationinthestoppingcriteria.2.Methods

ThemaingoalofthisworkistomeasuretheperformanceofEEMDcomparedtoconventionalEMDwithrespecttoaccuracyandrobustnessontwofronts.First,theabilitytorecoverasignalburiedinnoise.Second,theeffectofchangesinthestoppingcriteria.

TheoriginalHHTpaper1suggestedstoppingthesiftingoncethestandarddeviationbetweentwosuccessivesiftingsislessthanapresetthreshold.Huangetal.28latersuggestedacriterionbasedonnumberofzero-crossingsandextrema.Oncethenumberofzero-crossings/extremaremainsconstantforSsuccessiveiterations,thesiftingisstopped.ThiscriterionwillbereferredtosimplyasS.AnSof4hasbeenrecommendedasagenerallyacceptablenumberbasedonempiricaltests.Rillingetal.29haveproposedanalternativecriterionwherebytwothresholdsareusedtostopthesifting.Thefirstthreshold,θ1,istheratioofthemeanenvelopetotheabsolutedifferencebetweentheupperandlowerenvelopes.Thesiftingcontinuesaslongasthisratioishigherthanθ1for95%ofthedata.Thesiftingalsocontinuesifmorethan5%ofthedatahasaratioaboveathreshold,θ2,usually=10θ1.Lastly,ithasbeensuggestedbyWuandHuang27thatafixednumberofsiftingsshouldbeused,whichwillbereferredtoasthefixedSstoppingcriterion.Thelattercriterionhasbeenshowntoberobustinproducingconsistentdecompositionsfordatawithsimilarstatisticalproper-tiesandlength.Inthiswork,wewillonlycomparetheSandfixedSstoppingcriteria.

2.1.Testsignalsandrelativeerrormeasurement

Inthefollowingnumericexperiments,theperformanceofconventionalEMDandEEMDistestedfortherecoveryofartificialtestsignalsofdifferentfrequencies.

PerformanceEvaluationofEEMD233

Thetestsignalsaresinusoidswithanormalizedfrequencyrangefrom0.01to0.4.Eachtestsignalhas1001samplepointsandismodulatedbythefollowingenvelope:

m[n]=0.25sin[3πn/fs]+0.75.

(1)

Themodulatorisintroducedtodetectanyerrorsduetooversiftingasthisisknowntoyieldconstantamplitudecomponents.Themodulatoralsoinducesmodemixing.AsampletestsignalandtheamplitudemodulatorareshowninFig.1.

TheperformanceofEMDinthedifferentexperimentswillbemeasuredbytherelativeerrorbetweentherecoveredsignal(theIMFwiththehighestcorrelationtothesignal)andthesimulatedtestsignal.Therelativeerrorwillbecalculatedasfollows:

󰀂󰀁

s[n]−c[n])2n(󰀁2,(2)e=

s[n]nwheres[n]isthesimulatedsignalandc[n]istheIMFcomponentmosthighly

correlatedwiths[n].Finally,allnoiselevelsindicatedarerelativetothestandarddeviationofthesignal,forexampleanoiselevelof0.5hasanamplitude0.5×std(signal).

2.2.EMDperformance

TheaimofthisexperimentistoinvestigatetheperformanceofEMDinseparatingnoisefromsignalatdifferentnoiselevelsandusingdifferentstoppingcriteria.Test

Fig.1.AnexampleofatestsignalusedforEMDperformancetesting;thedashedlineisthemodulatorsignal.

234R.K.Niazyetal.

signalsweregeneratedasdescribedabove,resultinginatotalof40signalsranginglinearlyin(normalized)frequencyfrom0.01to0.4.Foreachsignal(withdifferentfrequency),noisewasaddedwithlevels0.2,0.4,and0.8andanEMDdecompositionwascarriedout.Theprocessofnoiseadditionanddecompositionwasrepeated10timesforeachcombinationofnoiselevel,signalfrequency,andstoppingcriterion.Therelativeerrorwasthencalculatedfromeachdecompositionandaveragedoverthe10runs.Thisexperimentwasrepeatedusingarangeof1–10fortheSstoppingcriterionand1–20forthefixedSstoppingcriterion.2.3.EEMDperformance

Inthisexperiment,theaimistodeterminetheadvantageofusingEEMDovercon-ventionalEMD.Further,weinvestigateddifferentsettingsforperformingEEMDincludingdifferentstoppingcriteriaandperturbation(i.e.,usedforalteringthedecomposition)noiselevels.Theperturbationnoiselevelisthelevelofnoiseusedtoassistinthedecomposition,ratherthanthelevelofnoiseinherentinthesig-nal.Testsignalsforthisexperimentweregeneratedasdescribedinthepreviousexperiment.However,thetestsignalshadnoiselevelsof0.2,0.4,and0.8added.Theamountofperturbationnoisewasvariedusinglevelsof0.2,0.5,and1.0ofthestandarddeviationofthetestsignal(plustheinherentnoise).AteachnoiselevelboththeSandfixedScriteriawereusedwithvaluesofS=2,4,and8,andfixedS=5,10,and15.Foreachcombinationofbackgroundnoise,perturba-tionnoise,andstoppingcriterion,500decompositionswereperformed.Typically,the500decompositionsdidnotallresultinthesamenumberofIMFs.However,therewasusuallyone“winning”numberofIMFs,whichresultsmostfrequently(typically70–80%).However,thismightnotbetheoptimalchoicesinceitmaybearguedthatthemostparsimoniousIMFrepresentationistheonewiththeleasterror.Forthisreason,wealsocomparedtheresultsobtainedusingthewinningandtheminimum(mostparsimonious)numberofIMFs.TherelativeerrorswerecomputedfromtheresultingmeanIMFs.3.ResultsandDiscussion3.1.EMDperformance

Figure2showstheresultsfromthestoppingcriteriaexperiment.Unsurprisingly,asthenoiselevelincreases,theabilityofEMDtoseparatenoisefromsignaldecreases.AsweprogressthroughtherowsofFig.2,therelativeerrorusinganygivenstoppingcriterionincreases.Wenotetheappearanceofpeaksatlowfrequencywheretherelativeerrorincreases.TheappearanceofthesepeaksisnotsurprisingknowingthatEMDbehavesasadyadicfilterwhenoperatingonnoise,30–32especiallythatthepeaklocationsseemtohaveadyadicratio.Thesepeaksaremostlikelyforfrequenciesintheboundarybetweentwodyadicfilterbins,drivenbytheaddedwhitenoise.Thisdoesnotimplythattherearefixedfrequencybinsproducedby

PerformanceEvaluationofEEMD235

(a)(d)

(b)(e)

(c)(f)

Fig.2.TheperformanceofEMDusingdifferentSandfixedSstoppingcriteriaunderdifferentnoiseconditions.Therelativeerrormeasureisthatcomputedbetweentheextractedsignalandthesimulatedsignalbeforeintroducingthenoise.Panels(a)–(c)areresultsfromusingtheSstoppingcriteriaandpanels(d)–(f)arethecorrespondingerrorlevelsusingthefixedScriteriawiththesamenoiselevels.

236R.K.Niazyetal.

theEMD;ratherthesefrequencybinsarespontaneouslydrivenbythepeaksfromtheaddedwhitenoise.Whenthefrequencyofthetestsignalisattheborderoftwoofthesebins,thereismoreerrorassociatedwithitsrecoveryasitismorelikelytosplit.Whatissurprising,though,isthatasthenoiselevelincreases,thelocationofthesepeaksseemstoshifthigherupthefrequencyspectrum.Itcouldbethattheaddedsignalinterfereswiththenormaldecompositionofnoiseinawaythatthesmallerthesignal-to-noiseratio(SNR)is,themoreEMDactsasaspontaneousdyadicfilter,ashasbeenpreviouslyreported.Conversely,thehighertheSNRis,themoreinfluenceithasonthedecomposition.Hence,theshiftinthelocationoftheerrorpeakswiththechangeofSNR.

UsingtheSstoppingcriterion,thereseemstobelittletodistinguishbetweenthevaluesusedapartfromthoseatthebottomoftherange.OneshouldavoidusingS=1,asatthisleveltherelativenoiseismarkedlyhigherthantherest.Also,therearelocalminimaatS=4(alsosuggestedbyHuangetal.28)and8,wherethereseemstobeadipintheerrorvaluescomparedtotherest,butagainnothingsubstantial.SimilarlyforthefixedSthebottomoftherangeshouldbeavoided.Thereseemstobealocalminimumatavalueof4,afterwhichtheerrorpeaksaround8,thencontinuouslydecreasesasthenumberofsiftingsisincreased.Ingeneral,theuseoftheSstoppingcriterionresultedinslightlylowererrorvaluescomparedwiththefixedScriterion,atallnoiselevels.3.2.EEMDperformance

Figure3illustratesthemodemixingproblemandthepotentialofEEMDtodealwithit.Inthe“signal”panel,thetestsignalplusnoiseisshowninblueandthetestsignalonitsownisshowninred.ThefirstfourIMFsfromconventionalandEEMDarealsoshown.AreaswheremodemixingwaspresentintheoriginaldecompositionhaveeffectivelybeencorrectedusingEEMD.Thiscanbeseenaroundtime-points80,150,205,and230.

Finally,Figs.4and5showtheresultsoftheEEMDexperimentatabackgroundnoiselevelof0.4,comparedwithconventionalEMD.Regardingaccuracy,thefind-ingfromtheseresultsisthedecreaseinrelativeerrorregardlessoftheperturbationnoiselevelorthestoppingcriteria:almostallcombinationsworkedequallywellwithlittletochoosebetweenthem.Theuseofthe“winning”numberofIMFs,however,providesmuchbettererrorvaluesthroughoutthefrequencyrange.Moreimpor-tantly,asshownintheboxplotsofFig.5,thedistributionaroundtheresultsaremuchtighterwhenusingEEMD;thatis,thereisamarkedincreaseintherobust-nessoftheoutcomeusingEEMDcomparedtoconventionalEMDasindicatedbythetighterwhiskersoftheboxplots.Thiswasaconsistentfindingregardlessofthestoppingcriteriaorperturbationnoiselevel.Also,usinglowerlevelsofperturbationnoiseseemstoresultinslightlylowererrorvalues.

TheincreaseinEMDestimationrobustness,andassociatedcharacterizationofestimationuncertainty,throughtheapplicationofEEMD,canbethoughtofas

PerformanceEvaluationofEEMD237

Fig.3.ModemixinginEMDandEEMD.Thetoppanelshowsinbluethesignalbeingdecom-posed(signal+noise).Overlaidinredisthenoise-freetestsignal.ThefirstfourIMFsareshownusingboththeconventionalEMD(blue)andEEMD(green).Modemixingcanbeseenaroundtimepoints80,150,205,and230usingtheconventionalEMD.Inthiscase,theEEMDseemstoremedytheproblemofmodemixing.

somewhatanalogoustotheuseofMonteCarlosamplingmethods,usedtoachieverobustmodelfittinganderrorcharacterization;inthecaseofEMD,whichprovidesanentirelydata-drivendescription/decompositionoftheoriginaldata,itisclearlynotpossibletotakemultiplesamplesof‘modelparameters’,butinsteadweareusingmultiple‘samples’ofthedataitself.

Finally,theerrorpeaksseenpreviouslyinFig.2alsoappearinFig.4usingEEMD.However,inthecaseofEEMD,therealsoseemstobeamarkedshiftintheerrorpeaksasafunctionoftheamountofsifting!Again,theerrorpeaksresultingfromanygivencriterionseemtohaveadyadicratiowhereeachpeakoccursathalfthefrequencyofthenext.ThereasonwhytheerrorpeaksinEEMDshowa

238R.K.Niazyetal.

Minimum number of IMFs

S

1Perturbation Noise level=0.20.80.60.40.20

S=2

S=4

S=8

Winning number of IMFsNormal EMD

S=5

fixed S

S=10

S=15

10.80.60.40.20

00.10.20.30.400.10.20.30.4

1Perturbation Noise level=0.50.80.60.40.20

10.80.60.40.20

Relative Error00.10.20.30.400.10.20.30.4

1Perturbation Noise level=1.00.80.60.40.20

10.80.60.40.2000.10.20.30.400.10.20.30.4Normalised Frequency

Fig.4.EEMDperformanceatnoiselevel0.4,usingdifferentperturbationnoiselevelsandstop-pingcriteria.Eachrowoffiguresisadifferentperturbationnoiselevel.TheleftandrightcolumnscontaintheresultsusingtheSandfixedSstoppingcriteria,respectively.ResultsshownindashedlinesareobtainedbyaveragingEMDdecompositionswiththeleastnumberofIMFs,whilesolidlinesrepresentresultsobtainedbyaveragingthedecompositionswiththemostfrequentnumberofIMFs(winningnumberofIMFs).IngrayaretheresultsfromtheconventionalEMDwiththesamenoiselevel.

PerformanceEvaluationofEEMD239

Fig.5.BoxplotsofEEMDrelativeerrorcomparedtonormalEMDatanoiselevelof0.4.Theboxplotdistributionsareacrossthefrequencyrange.Eachrowisatadifferentperturbationnoiselevel.TheleftandrightcolumnsareusingtheSandfixedSstoppingcriteria,respectively.Thecrossesindicatepossibleoutliers.TheresultsfromnormalEMDarethesameineachcolumn,asthenoiseperturbationleveldoesnotapply.EEMDperformanceisclearlysuperiortoconventionalEMDasindicatedbythelowererrorvalues(accuracy)andtighterwhiskers(robustness).

240R.K.Niazyetal.

dependenceontheamountofsifting,whilstthisisnotthecaseforconventionalEMD,isnotclear.Similarresultsandconclusionswereobtainedfromthe0.2and0.8noiselevelexperiments(resultsnotshown).Eventhoughhypothesescanbemadeaboutthebehavioroftheerrorpeaksandtheirdependenceonnoiselevelandtheamountofsifting(inthecaseofEEMD),thesewouldremainspeculativeandspecificnumericexperimentsneedtobecarriedouttoinvestigatethesephenomenafurther.4.Conclusion

Insimulationtests,theEEMDwasshowntoimprovetheaccuracyofthedecom-position,andsignificantlyincreasetherobustnesswithregardtoSNRandchangesinthestoppingcriteria.Moreover,intestingtherecoveryofsignalsburiedinnoise,testsshowedarelationshipbetweenthefrequencyofthesignalandtherelativeerror,wheretheerrorpeakedatsomefrequencies.Thelocationoftheseerrorpeaksseemtofollowadyadicpattern.Inaddition,inthecaseofEEMD,thelocationoftheerrorpeaksshiftedhigherupthefrequencyspectrumatmorestringentstoppingconditionsthatarelikelytocausemoresifting.References

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