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1、Onthetrend,detrending,andvariabilityofnonlinearandnonstationarytimeseriesZhaohuaWu*,NordenE.Huang,StevenR.Long,andChung-KangPeng§¶*CenterforOcean-Land-AtmosphereStudies,4041PowderMillRoad,Suite302,Calverton,MD20705;ResearchCenterforAdaptiveDataAnalysis,NationalCentralUniversity,
2、Chungli32054,Taiwan,RepublicofChina;OceanSciencesBranch,Code614.2,NationalAeronauticsandSpaceAdministrationGoddardSpaceFlightCenter,WallopsFlightFacility,WallopsIsland,VA23337;and§DivisionofInterdisciplinaryMedicineandBiotechnology,BethIsraelDeaconessMedicalCenter,HarvardMedical
3、School,Boston,MA02215EditedbyInezY.Fung,UniversityofCalifornia,Berkeley,CA,andapprovedJuly27,2007(receivedforreviewFebruary9,2007)DeterminingtrendandimplementingdetrendingoperationsareInthisarticle,adefinitionfortrendisintroduced,andaimportantstepsindataanalysis.Yetthereisnoprec
4、isedefinitioncorrespondingalgorithmforfindingintrinsicallythetrendandoftrendnoranylogicalalgorithmforextractingit.Asaresult,implementingthedetrendingalsoispresented.Becausethevariousadhocextrinsicmethodshavebeenusedtodeterminedetrendeddatadefineamoremeaningfulvariabilityassociate
5、dtrendandtofacilitateadetrendingoperation.Inthisarticle,awithaparticulartimescaleofthedata,thevariabilityofthedatasimpleandlogicaldefinitionoftrendisgivenforanynonlinearandalsowillbeexamined.Itshouldbenotedherethatthedefinitionnonstationarytimeseriesasanintrinsicallydeterminedmon
6、otonicoftrendandthealgorithmfordetrendinginthisstudyarequitefunctionwithinacertaintemporalspan(mostoftenthatofthegeneralandcanbeappliedtoanydatafromnonstationaryanddataspan),orafunctioninwhichtherecanbeatmostonenonlinearprocesses.Thegoal,however,isnotforpredictionbutextremumwith
7、inthattemporalspan.Beingintrinsic,themethodforanalysis.Theassumptionisthatthepredictivemodelshavetoderivethetrendhastobeadaptive.Thisdefinitionoftrendalsotobeprocess-based,notdata-driven.Theanalysisaspectem-presumestheexistenceofanaturaltimescale.Alltheserequire-phasizesthediscov
8、eryandunderstandingoftheunderlyingmentssuggestt
