The-empirical-mode-decomposition-and-the-Hilbert-s

TheempiricalmodedecompositionandtheHilbertspectrumfornonlinearandnon-stationarytimeseriesanalysisByNordenE.Huang1,ZhengShen2,StevenR.Long3,ManliC.Wu4,HsingH.Shih5,QuananZheng6,Nai-ChyuanYen7,ChiChaoTung8andHenryH.Liu91LaboratoryforHydrosphericProcesses/OceansandIceBranch,NASAGoddardSpaceFlightCenter,Greenbelt,MD20771,USA2DepartmentofEarthandPlanetarySciences,TheJohnHopkinsUniversity,Baltimore,MD21218,USA3LaboratoryforHydrosphericProcesses/ObservationalScienceBranch,NASAWallopsFlightFacility,WallopsIsland,VA23337,USA4LaboratoryforAtmospheres,NASAGoddardSpaceFlightCenter,Greenbelt,MD20771,USA5NOAANationalOceanService,SilverSpring,MD20910,USA6CollegeofMarineStudies,UniversityofDelaware,DE19716,USA7NavalResearchLaboratory,Washington,DC,20375-5000,USA8DepartmentofCivilEngineering,NorthCarolinaStateUniversity,Raleigh,NC27695-7908,USA9NavalSurfaceWarfareCenter,CarderockDivision,Bethesda,MD20084-5000,USAReceived3June1996;accepted4November1996Contentspage1.Introduction9042.Reviewofnon-stationarydataprocessingmethods907(a)Thespectrogram907(b)Thewaveletanalysis907(c)TheWigner{Villedistribution908(d)Evolutionaryspectrum909(e)Theempiricalorthogonalfunctionexpansion(EOF)909(f)Othermiscellaneousmethods9103.Instantaneousfrequency9114.Intrinsicmodefunctions9155.Theempiricalmodedecompositionmethod:thesiftingprocess9176.Completenessandorthogonality9237.TheHilbertspectrum9288.ValidationandcalibrationoftheHilbertspectrum9339.Applications948(a)Numericalresultsfromclassicnonlinearsystems949(b)Observationaldatafromlaboratoryandeldexperiments962Proc.R.Soc.Lond.A(1998)454,903{995c1998TheRoyalSocietyPrintedinGreatBritain903TEXPaper904N.E.Huangandothers10.Discussion98711.Conclusions991References993Anewmethodforanalysingnonlinearandnon-stationarydatahasbeendevel-oped.Thekeypartofthemethodisthe`empiricalmodedecomposition'methodwithwhichanycomplicateddatasetcanbedecomposedintoaniteandoftensmallnumberof`intrinsicmodefunctions'thatadmitwell-behavedHilberttrans-forms.Thisdecompositionmethodisadaptive,and,therefore,highlyecient.Sincethedecompositionisbasedonthelocalcharacteristictimescaleofthedata,itisapplicabletononlinearandnon-stationaryprocesses.WiththeHilberttransform,the`instrinicmodefunctions'yieldinstantaneousfrequenciesasfunctionsoftimethatgivesharpidenticationsofimbeddedstructures.Thenalpresentationoftheresultsisanenergy{frequency{timedistribution,designatedastheHilbertspectrum.Inthismethod,themainconceptualinnovationsaretheintroductionof`intrinsicmodefunctions'basedonlocalpropertiesofthesignal,whichmakestheinstanta-neousfrequencymeaningful;andtheintroductionoftheinstantaneousfrequenciesforcomplicateddatasets,whicheliminatetheneedforspuriousharmonicstorep-resentnonlinearandnon-stationarysignals.Examplesfromthenumericalresultsoftheclassicalnonlinearequationsystemsanddatarepresentingnaturalphenomenaaregiventodemonstratethepowerofthisnewmethod.Classicalnonlinearsystemdataareespeciallyinteresting,fortheyservetoillustratetherolesplayedbythenonlinearandnon-stationaryeectsintheenergy{frequency{timedistribution.Keywords:non-stationarytimeseries;nonlineardierentialequations;frequency{timespectrum;Hilbertspectralanalysis;intrinsictimescale;empiricalmodedecomposition1.IntroductionDataanalysisisanecessarypartinpureresearchandpracticalapplications.Imper-fectassomedatamightbe,theyrepresenttherealitysensedbyus;consequently,dataanalysisservestwopurposes:todeterminetheparametersneededtoconstructthenecessarymodel,andtoconrmthemodelweconstructedtorepresentthephe-nomenon.Unfortunately,thedata,whetherfromphysicalmeasurementsornumericalmodelling,mostlikelywillhaveoneormoreofthefollowingproblems:(a)thetotaldataspanistooshort;(b)thedataarenon-stationary;and(c)thedatarepresentnonlinearprocesses.Althougheachoftheaboveproblemscanberealbyitself,thersttwoarerelated,foradatasectionshorterthanthelongesttimescaleofasta-tionaryprocesscanappeartobenon-stationary.Facingsuchdata,wehavelimitedoptionstouseintheanalysis.Historically,Fourierspectralanalysishasprovidedageneralmethodforexamin-ingtheglobalenergy{frequencydistributions.Asaresult,theterm`spectrum'hasbecomealmostsynonymouswiththeFouriertransformofthedata.Partiallybecauseofitsprowessandpartiallybecauseofitssimplicity,Fourieranalysishasdominatedthedataanalysiseortssincesoonafteritsintroduction,andhasbeenappliedtoallkindsofdata.AlthoughtheFouriertransformisvalidunderextremelygeneralconditions(see,forexample,Titchmarsh1948),therearesomecrucialrestrictionsofProc.R.Soc.Lond.A(1998)Nonlinearandnon-stationarytimeseriesanalysis905theFourierspectralanalysis:thesystemmustbelinear;andthedatamustbestrict-lyperiodicorstationary;otherwise,theresultingspectrumwillmakelittlephysicalsense.ThestationarityrequirementisnotparticulartotheFourierspectralanalysis;itisageneraloneformostoftheavailabledataanalysismethods.Therefore,itbehovesustoreviewthedenitionsofstationarityhere.Accordingtothetraditionaldenition,atimeseries,X(t),isstationaryinthewidesense,if,forallt,E(jX(t)2j)1;E(X(t))=m;C(X(t1);X(t2))=C(X(t1+);X(t2+))=C(