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1、TimeSeriesAnalysisOutline1Timeseriesinastronomy2Frequencydomainmethods3Timedomainmethods4ReferencesTimeseriesinastronomyPeriodicphenomena:binaryorbits(stars,extrasolarplanets);stellarrotation(radiopulsars);pulsation(helioseismology,Cepheids)Stochasticphenomena:accretion(C
2、Vs,X-raybinaries,Seyfertgals,quasars);scintillation(interplanetary&interstellarmedia);jetvariations(blazars)Explosivephenomena:thermonuclear(novae,X-raybursts),magneticreconnection(solar/stellarflares),stardeath(supernovae,gamma-raybursts)Difficultiesinastronomicaltimeseries
3、Gappeddatastreams:Diurnal&monthlycycles;satelliteorbitalcycles;telescopeallocationsHeteroscedasticmeasurementerrors:Signal-to-noiseratiodiffersfrompointtopointPoissonprocesses:Individualphoton/particleeventsinhigh-energyastronomyImportantFourierFunctionsDiscreteFourierTran
4、sformXn−1/2d(ωj)=nxtexp(−2πitωj)t=1XnXn−1/2−1/2d(ωj)=nxtcos(2πiωjt)−inxtsin(2πiωjt)t=1t=1Classical(Schuster)Periodogram2I(ωj)=
5、d(ωj)
6、SpectralDensityhX=∞f(ω)=exp(−2πiωh)γ(h)h=−∞Fourieranalysisrevealsnothingoftheevolutionintime,butratherrevealsthevarianceofthesignalatdiffere
7、ntfrequencies.Itcanbeprovedthattheclassicalperiodogramisanestimatorofthespectraldensity,theFouriertransformoftheautocovariancefunction.Formally,theprobabilityofaperiodicsignalinGaussiannoiseisP∝ed(ωj)/σ2.GingaobservationsofX-raybinaryGX5-1GX5-1isabinarystarsystemwithgasfr
8、omanormalcompanionaccretingontoaneutronstar.HighlyvariableX-raysareproducedintheinneraccretiondisk.XRBtimeseriesoftenshow‘rednoise’and‘quasi-periodicoscillations’,probablyfrominhomogeneitiesinthedisk.Weplotbelowthefirst5000of65,536countratesfromGingasatelliteobservationsdu
9、ringthe1980s.gx=scan(”˜/Desktop/CASt/SumSch/TSA/GX.dat”)t=1:5000plot(t,gx[1:5000],pch=20)FastFourierTransformoftheGX5-1timeseriesrevealsthe‘rednoise’(highspectralamplitudeatsmallfrequencies),theQPO(broadenedspectralpeakaround0.35),andwhitenoise.f=0:32768/65536I=(4/65536)*
10、abs(fft(gx)/sqrt(65536))ˆ2plot(f[2:60000],I[2:60000],type=”l”,xlab=”Frequency”)Limitationsofthesp
