the frequency domain approach of a time series

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1、ChapterTheFrequencyDomainApproachofaTime4SeriesTheprecedingsectionsfocussedontheanalysisofatimeseriesinthetimedomain,mainlybymodellingandttinganARMA(p;q)-processtostationarysequencesofobservations.Anotherapproachtowardsthemodellingandanalysisoftimeseriesisviathefrequencydomain:Aseriesisoft

2、enthesumofawholevarietyofcycliccomponents,fromwhichwehadalreadyaddedtoourmodel(1.2)alongtermcycliconeorashorttermseasonalone.Inthefollowingweshowthatatimeseriescanbecompletelydecomposedintocycliccomponents.Suchcycliccomponentscanbedescribedbytheirperiodsandfrequencies.Theperiodistheinterval

3、oftimerequiredforonecycletocomplete.Thefrequencyofacycleisitsnumberofoccurrencesduringaxedtimeunit;inelectronicmedia,forexample,frequenciesarecommonlymeasuredinhertz,whichisthenumberofcyclespersecond,abbre-viatedbyHz.Theanalysisofatimeseriesinthefrequencydomainaimsatthedetectionofsuchcycle

4、sandthecomputationoftheirfrequencies.Notethatinthischaptertheresultsareformulatedforanydatay1;:::;yn,whichneedformathematicalreasonsnottobegeneratedbyastationaryprocess.Neverthelessitisreasonabletoapplytheresultsonlytorealizationsofstationaryprocesses,sincetheempiri-calautocovariancefunctio

5、noccurringbelowhasnointerpretationfornon-stationaryprocesses,seeExercise1.21.136TheFrequencyDomainApproachofaTimeSeries4.1LeastSquaresApproachwithKnownFrequenciesAfunctionf:R!RissaidtobeperiodicwithperiodP>0iff(t+P)=f(t)foranyt2R.Asmallestperiodiscalledafundamentalone.Thereciprocalvalue=1

6、=Pofafundamentalperiodisthefundamentalfrequency.Anarbitrary(time)intervaloflengthLconsequentlyshowsLcyclesofaperiodicfunctionfwithfundamentalfrequency.Popularexamplesofperiodicfunctionsaresineandcosine,whichbothhavethefundamentalperiodP=2.Theirfundamentalfrequency,therefore,is=1=(2).Th

7、epredominantfamilyofperiodicfunctionswithintimeseriesanalysisaretheharmoniccomponentsm(t):=Acos(2t)+Bsin(2t);A;B2R;>0;whichhaveperiod1=andfrequency.AlinearcombinationofharmoniccomponentsXr
g(t):=+Akcos(2kt)+Bksin(2kt);2R;k=1willbenamedaharm