Time Series Analysis of Unequally Spaced Data Intercomparison Between Estimators of the Pow

TimeSeriesAnalysisofUnequallySpacedData:IntercomparisonBetweenEstimatorsofthePowerSpectrum

V.V.Vityazev

AstronomyDepartment,St.PetersburgUniversity,Bibliotechnayapl.2,Petrodvorets,St.Petersburg,198904,Russia.

Abstract.ItisshownthatthelikenessoftheperiodogramandtheLS-spectrum(bothestimatorsofthepowerspectrumarewidelyusedinthespectralanalysisoftimeseries),dependsonthepropertiesofthespectralwindowW(ω)correspondingtothedistributionoftimepoints.Themainresultsare:a)alltheestimatorsevaluatedatfrequencyωareidenticalifW(2ω)=0;b)theSchusterperiodogramdiffersfromtheLS-spectraatthefrequenciesω=ωˆk/2,whereωˆkarethefrequenciesatwhichthespectralwindowhaslargesidepeaksduetoirregulardistributionoftimepoints.Twoexamplesforsituationstypicalinastronomyillustratetheseconclusions.

1.Introduction

Invariousbranchesofastronomy,wefacetheproblemoffindingperiodicitieshiddeninobservations.Ifthedataareregularlyspacedintime,theSchusterperiodogramisthebasictoolforevaluatingthepowerspectra(Marple1987;Terebizh1992).Unfortunately,astronomicalobservationsareirregularforvari-ousreasons:day-timechanges,weatherconditions,positionsoftheobjectunderobservations,etc.Presentdaytheoryandpracticeofthespectralanalysisofun-equallyspacedtimeseriesarebasedontwoapproaches.ThefirstoneemploystheSchusterperiodogram(Deeming1975;Robertsetal.1987).Thesecondoneusestheprocedureoftheleastsquaresfittingofasinusoidtothedata(Barning1962;Lomb1976;Scargle1982)withresultingestimatorsknownastheLS-spectra.ThemostvaluablefeatureoftheLS-spectraiswelldefinedstatisticalbehavior.Atthesametime,theLS-spectraloseveryimportantproperties:descriptionintermsofthespectralwindow,connectionwiththecorrelationfunction,etc.Ontheotherhand,theSchusterperiodogramofagappedtimeseriessatisfiesallthefundamentalrelationsoftheclassicalspectralanalysis,butitsstatisticalpropertiesarecomplicatedascomparedtothecaseofregu-lardata.Itisworthmentioningthatdespitedifferenttheoreticalfoundations,theSchusterperiodogramandtheLS-spectrafrequentlyturnouttobealmostidentical.Thissimilarityrequiresanexplanation,andwearetryingtofindsituationswhentheSchusterperiodogramandtheLS-spectraareveryclosetoeachotherordiffergreatly.Theultimategoalofthisstudyistoclarifythepropertiesofvarioustechniqueswhichareusedtoderivetheperiodicitiesfromtheunequallydistributeddata.

166

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TimeSeriesAnalysisofUnequallySpacedData

2.

TwoEstimatorsofthePowerSpectrum

167

ForasetofNobservationsxk=x(tk),k=0,1,...,N−1withzeromeanobtainedatarbitrarytimestk,wecansetupthemodel

f(t)=

where

2󰀃i=1

aiφi(t),(1)

φ1(t)=cosωt,φ2(t)=sinωt.(2)

Usingthefollowingnotation

(p,q)=

1

N2

−iωtk

|

N−1󰀃k=0

xke−iωtk|2.

(4)

Ifthesignalcontainsasinefunctionoffrequencyω0,thentheproduct

xkemakesalargecontributiontoSprovidedthatω=ω0.Inotherwords,theSchusterperiodogram,tothelimitofnormalizingfactor,isasquareofthecorrelationcoefficientbetweenthedataandaharmonicfunction.

ThealternativeestimatorofthepowerspectrumbasedontheleastsquaresfittingofthesinefunctiontothedatawasproposedbyLomb(1976)andScargle(1982).Theirapproachisbasedontheintroductionofthenewtimepoints

1

tˆk=tk−

󰀂

kcos2ωtk

,(5)

wherethetimeshiftprovidestheorthogonalityofthefunctions

ˆ1(t)=cosωtˆφk,

ˆ2(t)=sinωtˆφk.

(6)

UnderthisassumptiontheLS-spectrumlooksasfollows:

L(ω)=

1

ˆ1󰀅2󰀅φ

+ˆ2)2(x,φ

168Vityazev

wherethespectralwindowW(ω)is

W(ω)=

1

sin2(mω∆T/2)

n2sin2(ω∆t/2)

m∆T

j,j=1,2,...(11)

satisfyEq.(8),providedthatj=m/2,m,...ifmisevenandj=m,2m,...,ifmisodd.3.2.

ObservationswithaLongGap

Consideredhereisasituationwheretwosetsofobservations(eachoneconsistingofnsuccessivepoints)areseparatedbypmissingpointsformingthegap.Asearlier,allthepointsaresupposedtoberegularlyspacedoverthetimeinterval∆t=const.Now,forthespectralwindowwehave(Vityazev1994)

W(ω)=

sin2(nω∆t/2)

(n+p)∆t

(j+

1

TimeSeriesAnalysisofUnequallySpacedData

4.

Conclusions

169

TheLS-spectragainedpopularityduetothefactthattheyretaintheexponentialdistributionoftheiraccountswhenthetimeseriesisassumedtobewhitenoise.NowweseethatatfrequenciesthatsatisfyEq.(8)theSchusterperiodogramretainstheexponentialdistributiontoo.

TheSchusterperiodogramdiffersfromtheLS-spectraonlyatthefrequen-ciesthatsatisfythecondition1−W(2ω)≪1.ItmeansthatthediscrepanciesbetweentheSchusterperiodogramandtheLS-spectraarelargewhenthetimeseriescontainaharmonicofthefrequency,thedoublevalueofwhichcoincideswiththefrequencyatwhichthespectralwindowhasalargesidepeak.Inthecaseofperiodicalgapsithappenswhentheperiodofasignalhiddeninthedataisonehalftheperiodofthegaps.Inthissituation(Vityazev1997a),thespectralestimationfacesunrealisticintensitiesofthespectralpeaksandthestrongdependenceoftheheightsofpeaksonthephaseofthesignal.Itisveryimportanttoemphasizethattheseproblemscomenotfromthechoiceofthetooltoevaluatethepowerspectrum;theyoriginatefrommixingtwosourcesoftheperiodicities:oneisthephysicalprocessthatweobserveandanotheroneisaperiodicalinterruptionofobservations.Inastronomy,therotationandrevolutionoftheEarthimposediurnalandannualcyclesontheEarth-basedobservations.TheperiodshiddeninobservationsoftheSun,stars,quasars,etc.,arehardlyconnectedphysicallywiththeperiodsspecifictotheEarth.Fortheseobservations,theprobabilityofmixingperiodicitiesisnegligible.Onthecontrary,ifwestudytheEarthfromtheEarth(suchisthecasewithastrometricobservationsoftheEarth’srotationparameters),thenthesemi-annualperiodintheEarth’srotationinterfereswiththeannualgapsinobservations.

ForfurtherdetailsthereaderisreferredtoVityazev(1997a,1997b).References

Barning,F.J.M.1962,B.A.N.,17,N1,22Deeming,T.J.1975,Ap&SS,36,137Lomb,N.R.1976,Ap&SS,39,447

Marple,Jr.,S.L.1987,DigitalSpectralAnalysiswithApplications(Englewood

Cliffs,NJ:Prentice-Hall)

Roberts,D.H.,Lehar,J.,&Dreher,J.W.1987,AJ,93,968Scargle,J.D.1982,ApJ,263,835

Terebish,V.Yu.1992.TimeSeriesAnalysisinAstrophysics(Moscow:Nauka)Vityazev,V.V.1994,Astron.andAstrophys.Tr.,5,177Vityazev,V.V.1997a,A&A,inpressVityazev,V.V.1997b,A&A,inpress