1-C-C算法(延迟时间确定)

PhysicaD127(1999)48–60Nonlineardynamics,delaytimes,andembeddingwindowsH.S.Kim1;a,R.Eykholtb;,J.D.SalascaDepartmentofCivilEngineering,ColoradoStateUniversity,FortCollins,CO80523,USAbDepartmentofPhysics,ColoradoStateUniversity,FortCollins,CO80523,USAcHydrologicScienceandEngineeringProgram,DepartmentofCivilEngineering,ColoradoStateUniversity,FortCollins,CO80523,USAReceived11September1996;receivedinrevisedform10January1998;accepted12August1998CommunicatedbyA.M.AlbanoAbstractInordertoconstructanembeddingofanonlineartimeseries,onemustchooseanappropriatedelaytimed.Often,disestimatedusingtheautocorrelationfunction;however,thisdoesnottreatthenonlinearityappropriately,anditmayyieldanincorrectvalueford.Ontheotherhand,thecorrectvalueofdcanbefoundfromthemutualinformation,butthisprocessisrathercumbersomecomputationally.Here,wesuggestasimplermethodforestimatingdusingthecorrelationintegral.WecallthistheC–Cmethod,andwetestitonseveralnonlineartimeseries,obtainingestimatesofdinagreementwiththoseobtainedusingthemutualinformation.Furthermore,someresearchershavesuggestedthatoneshouldnotchooseafixeddelaytimed,independentoftheembeddingdimensionm,but,rather,oneshouldchooseanappropriatevalueforthedelaytimewindowwD.m−1/,whichisthetotaltimespannedbythecomponentsofeachembeddedpoint.Unfortunately,wcannotbeestimatedusingtheautocorrelationfunctionorthemutualinformation,andnostandardprocedureforestimatingwhasemerged.However,weshowthattheC–Cmethodcanalsobeusedtoestimatew.Basicallywistheoptimaltimeforindependenceofthedata,whiledisthefirstlocallyoptimaltime.Astests,weapplytheC–CmethodtotheLorenzsystem,athree-dimensionalirrationaltorus,theRosslersystem,andtheRabinovich–Fabrikantsystem.Wealsodemonstratetherobustnessofthismethodtothepresenceofnoise.c1999ElsevierScienceB.V.Allrightsreserved.Keywords:Delaytime;Correlationintegral;Embedding;TimeseriesPACS:05:45:Cb;47:52:Cj1.IntroductionAnalysisofchaotictimeseriesiscommoninmanyfieldsofscienceandengineering,andthemethodofdelayshasbecomepopularforattractorreconstructionfromscalartimeseries.Fromtheattractordynamics,onecanestimatethecorrelationdimensionandotherquantitiestoseewhetherthescalartimeseriesischaoticorstochastic.Therefore,attractorreconstructionisthefirststageinchaotictimeseriesanalyses.SincethechoiceofthedelayCorrespondingauthorTel.:[phone];fax:[phone];e-mail:[e-mail]1PresentAddress:DepartmentofConstructionEngineering,SunMoonUniversity,Korea0167-2789/99/$–seefrontmatterc1999ElsevierScienceB.V.Allrightsreserved.PII:S0167-2789(98)00240-1H.S.Kimetal./PhysicaD127(1999)48–6049timedforattractorreconstructionusingthemethodofdelayshasnotbeenfullydeveloped,manyresearchersusetheautocorrelationfunction,whichiscomputationallyconvenientanddoesnotrequirelargedatasets.However,ithasbeenpointedout[1]thattheautocorrelationfunctionisnotappropriatefornonlinearsystems,and,instead,dshouldbechosenasthefirstlocalminimumofthemutualinformation.Unfortunately,thisapproachiscumbersomecomputationallyandrequireslargedatasets[2].AccordingtoPackardetal.[3]andTakens[4],themethodofdelayscanbeusedtoembedascalartimeseriesfxig;iD1;2;:::;intoanm-dimensionalspaceasfollows:xxxiD.xi;xiC1;:::;xiC.m−1/t/;xxxi2Rm;(1)wheretistheindexlag.Ifthesamplingtimeiss,thedelaytimeisdDts.Takens’theoremassumesthatwehaveaninfinitenoise-freedataset,inwhichcase,wecanchoosethedelaytimealmostarbitrarily.However,sincerealdatasetsarefiniteandnoisy,thechoiceofthedelaytimeplaysanimportantroleinthereconstructionoftheattractorfromthescalartimeseries.Ifdistoosmall,thereconstructedattractoriscompressedalongtheidentityline,andthisiscalledredundance.Ifdistoolarge,theattractordynamicsmaybecomecausallydisconnected,andthisiscalledirrelevance[5].Incommonpractice,thedelaytimedischosensoastoensurethatthecomponentsofxxxiareindependent,andthesamedelaytimeisusedforallembeddingdimensionsm.However,inrecentyears,ithasbeensuggestedthatthedelaytimewindowwD.m−1/,whichistheentiretimespannedbythecomponentsofxxxi,shouldbeindependentofminstead[6–11].Inthiscase,thedelaytimevarieswiththeembeddingdimensionm.Rosensteinetal.[6]comparedseveralapproachesforestimatingdandwandindicatedthecriticaldisadvantagesofeach,suchasinconsistenciesandlongcomputationtimes.Thegeometricalconceptsofredundanceandirrelevancewereusedtoevaluatethequalityofanattractor’sreconstruction.Ithasbeensuggested[11]thatthedelaytimewindowmaybesettowp,wherepisthemeanorbitalperiod,whichcanbeapproximatedbyexaminingtheoscillationsofthetimeseries.Martinerieetal.[7]examinedthedelaytimewindowandcompareditwiththedelaytimesestimatedusingtheautocorrelationfunctionandthemutualinformation,buttheyconcludedthatwcannotbeestimaedusingeitherofthesemethods.Basically,wistheoptimaltimeforindependenceofthedata,butthesemethodsestimateonlythefirstlocallyoptimaltime,whichisd.Inthiswork,wedevelopatechniqueforchoosingeitherthedelaytimedorthedelaytimewindowwusingthecorrelationintegral.Althoughwmaybeamoreusefulquantityfortheestimationofthedimensionthand,manyresearcherscontinuetousedfo