A Novel Method for Determining the Nature of Time

1ANovelMethodforDeterminingtheNatureofTimeSeriesTemujinGautama,DaniloP.Mandic,andMarcM.VanHulleT.GautamaandM.M.VanHullearewiththeLaboratoriumvoorNeuro-enPsychofysiologie,K.U.Leuven,Leuven,Belgium.D.P.MandiciswiththeDepartmentofElectricalandElectronicEngineering,ImperialCollegeofScience,TechnologyandMedicine,London,U.K.TheworkofT.GautamawassupportedbyascholarshipfromtheFlemishRegionalMinistryofEducation(GOA2000/11)andaresearchgrantfromtheFundforScientificResearch(G.0248.03).TheworkofD.P.MandicwassupportedbyQLG3-CT-2000-30161andavisitingfellowshipfromtheK.U.Leuven(F/01/079)whileattheLaboratoriumvoorNeuro-&Psychofysiologie,K.U.Leuven,Belgium.TheworkofM.M.VanHullewassupportedbyresearchgrantsreceivedfromtheFundforScientificResearch(G.0185.96N;G.0248.03),theNationalLottery(Belgium)(9.0185.96),theFlemishRegionalMinistryofEducation(Belgium)(GOA95/99-06;2000/11),theFlemishMinistryforScienceandTechnology(VIS/98/012),andtheEuropeanCommission,5thframeworkprogramme(QLG3-CT-2000-30161andIST-2001-32114).Theauthorsaresolelyresponsibleforthecontentsofthisarticle.ItdoesnotrepresenttheopinionoftheCommunity,andtheCommunityisnotresponsibleforanyusethatmightbemadeofdataappearingtherein.August18,2003DRAFT2AbstractTheDelayVectorVariance(DVV)method,whichanalysesthenatureofatimeserieswithrespecttotheprevalenceofdeterministicorstochasticcomponents,isintroduced.DuetothestandardisationwithintheDVVmethod,itispossiblebothtostatisticallytestforthepresenceofnonlinearitiesinatimeseries,andtovisuallyinspecttheresultsinaDVVscatterdiagram.Thisapproachisconvenientforinterpretationasitconveysinformationaboutthelinearornonlinearnature,aswellasabouttheprevalenceofdeterministicorstochasticcomponentsinthetimeseries,thusunifyingtheexistingapproacheswhichdealeitherwithonlydeterministicversusstochastic,orthelinearversusnonlinearaspect.Theresultsonbiomedicaltimeseries,namelyheartratevariability(HRV)andfunctionalMagneticResonanceImaging(fMRI)timeseries,illustratetheapplicabilityoftheproposedDVV-method.IndexTermsnonlinearityanalysis,surrogatedata,fMRI,HRVI.INTRODUCTIONAnalysingthenatureofbiomedicaltimeserieshasreceivedconsiderableattentioninrecentyears,asthepresenceofnonlinearityand/ordeterminisminaphysiologicalsignalcanoftenbeusedasanindicatorofthehealthstatusofapatient(seee.g.,[1]–[4]).Ingeneral,performinganonlinearityanalysisinamodellingorsignalprocessingcontextcanleadtoasignificantimprovementofthequalityoftheresults,sinceitfacilitatestheselectionofappropriateprocessingmethods,suggestedbythedataitself,e.g.,usinglinearornonlinearfilters.Indeed,sincethetrainingofnonlinearmodelsandfiltersismorecomplexandlessconvenientthanthatoftheirlinearcounterparts,thesemodelsshouldbeavoidedwhenthesignalsareactuallylinearinnature.Acomprehensiveaccountoftheimportanceofthisclassofanalysisinengineering,medicineandearthsciences,andanintroductiontobasicmethodsisgiveninRef.[5].Theexistingmethodsfortheanalysisofthenonlinearnatureofatimeseriesaretwofold:inonecase,differentmodels(e.g.,linearandnonlinearones)arefittothetimeseriesandtheiraccuraciesareevaluated[1],[4],[6],whereasintheothercase,certainnonlinearitymeasurescomputedforthesignalunderstudyarecomparedtothoseobtainedforlinearisedversionsofthedata,so-calledsurrogates(foranoverview,seeRef.[7]).Anotheraspectofatimeseries,whichisbasedontheWolddecompositiontheorem[8],highlightstheprevalenceofthedeterministicorstochasticcomponentofatimeseries,apropertywhichcanalsobeexaminedusingthemethodintroducedinRef.[6].August18,2003DRAFT3Thepresenceof(non-)lineardeterministicorstochasticbehaviourinabiomedicalsignalconveysimportantinformation.Achangeinthenatureofamonitoredsignalmightindicateachangeinthehealthcondition.Thisarticlethereforeconsiderstheproblemofdeterminingthenatureofabiomedicalsignal:howwecanjudgeaboutthenatureofatimeseries,giventhatitisrecordedunderanunknownmeasurementconditionandpossiblythroughanonlinearobservationfunction.Theworkpresentedherediffersfrommuchpreviousworkinthatittakesintoconsiderationboththelinearornonlinear,andthedeterministicorstochasticaspectsofatimeseries.Weproposeaunifyingmethodforsequentiallyanalysingthedeterministicorstochasticnature(DelayVectorVarianceor‘DVV’method),andthelinearornonlinearnature(DVVscatterdiagram).Thefirstanalysischaracterisesatimeseriesinastandardisedmanner,whereasthelatteradditionallyemploystheconceptofsurrogatedata.TheproposedDVVmethodisappliedtotwotypesofbiomedicalsignals,namelytoHRVandfMRItimeseries,andtheresultsareinlinewiththoseobtainedusingothermethodsfromtheliterature.Theyconfirmcurrenthypothesesonthepresenceofnonlinearitiesinbiomedicaltimeseries,whiletheproposedmethodfurtherprovidesanaccountofanotheraspectofthetimeseries,namelythedeterministicorstochasticnature.II.TIMESERIESUSEDA.BenchmarkTimeSeriesTheproposedmethodisfirstverifiedonfourbenchmarktimeseriesof1000samples,threeofwhicharesyntheticallygeneratedandtheremainingoneisareal-worldtimeseries.ThefirstsynthetictimeseriesisarealisationoftheH´enonMap,givenbyxk=1¡ax2k¡1+byk¡1yk=xk¡1;wherea=1:4andb=0:3.Next,arealisationoftheMackey-Glassequa