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Time Series Analysis of Data With Gaps
Jeff Scargle
Space Science Division NASA Ames Research Center
TheFirstYearofMAXI:MonitoringvariableX‐raysources
SpecialthankstoTatehiroMihara‐san 1
PracAcalTimeSeriesMethods
DataIssues:SamplingIntervalsandGaps LightCurveRepresentaAons(DataCells) ScaKerPlots CorrelaAonFuncAons(EdelsonandKrolikalgorithm) Spectralanalysis:
Amplitude(Power) Phase WaveletTransform(Scalogram) WaveletPower(Scalegram) StructureFuncAons Time‐Scale/Time‐FrequencyAnalysis
CauAons:“staAonarity”,“nonlinearity”,“correlaAons”,…
BayesianBlocksrepresentaAon
EdelsonandKrolik:TheDiscreteCorrelaAonFuncAon:aNewMethodforAnalyzingUnevenlySampledVariabilityData,Ap.J.333,1988,646‐starAngpointforallelse!
Evenlyspaceddata
Arbitrarilyspaceddata
Time‐Frequency/Time‐ScaleAnalysisTransformtoanewviewofthe5meseriesinforma5on.
ARealityinjointAme&frequency(orscale)representaAon AtomicdecomposiAon
Time‐frequencyatoms Over‐completerepresentaAons OpAmalBasisPursuit(Mallat),etc.
UncertaintyPrinciple:T‐FresoluAontradeoff Non‐staAonaryprocesses
Flares TrendsandModulaAons StaAsAcalchange‐points
InstantaneousFrequency Localvs.Globalstructure Interference(cross‐termsinbi‐linearrepresentaAon)Time‐Frequency/Time‐ScaleAnalysis(Temps‐Fréquence)PatrickFlandrinhKp://perso.ens‐lyon.fr/patrick.flandrin/publis.html;AWavelettourofSignalProcessing(UneExploraAondesSignauxenOndeleKes)StéphaneMallat
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MulA‐taperAnalysis(Thomson1982)
Tapers(windows)reducesidelobeleakage=bias IncompleteuseofdatalossofinformaAon MulAtapersrecoverthisinformaAon LeakageminimizaAon=eigenvalueproblem
EigenfuncAons:efficientwindowfuncAons Eigenvalues
measureeffecAveness determinehowmanytermstoinclude
SpectralAnalysisforPhysicalApplica3ons:Mul3taperandConven3onalUnivariateTechniques,DonPercivalandAndrewWalden(1993)
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Func4on Domain Range Auto‐ Cross‐ PhysicalInterp
Bayesianblk.LightCurve
Time Flux ✔ ✔mulAvar.BB
Flares,eventsetc.
ScaKerPlot Flux1 Flux2 ✔ Dependency(notjustcor.)
CorrelaAon Lag <X2><XY> ✔ ✔ Correlatedbehavior/lags
Spectrum Frequency Power ✔ ✔ Periodicity1/fnoise…
Phase ✔ ✔ Shirs,lags
Structure Lag <X2><XY> ✔ ✔ Correlatedbehavior/lags
Scalogram Scale/Time Power ✔ ✔ Dynamicbehavior
Scalegram Scale Power ✔ ✔ 1/fnoiseQPOs
DistribuAon Time/scale/frequency
Power ✔ ✔ Dynamicbehavior
PracAcalSuggesAons(somewhatexaggerated)
StudydistribuAonofsampleintervalsdtn=tn+1‐tn NeversubtractmeanofAmeseries EdelsonandKrolikCFisthesourceofallotheranalysis UseselftermsinE&KCFtoassessobservaAonalerrors Don’tconfuse:sourcerandomness/observaAonalnoise H0:AGNsareidenAcalstochasAcdynamicalsystems StaAonarityisalocalproperty AnystaAonaryrandomprocessisexactlyshotnoise (randompulses;theWoldDecomposiAonTheorem) Linearityisaphysicalproperty,notoneofAmeseries Donotbindata
VariableSource
PropagaAonToObserver
PhotonDetecAon
Luminosity:randomordeterminisAc PhotonEmissionIndependentRandomProcess(Poisson)
RandomDetecAonofPhotons(Poisson)
CorrelaAonsinsourceluminositydonotimplycorrelaAonsinAmeseriesdata!
RandomScinAllaAon,Dispersion,etc.?
Allofthiswillbeinthe
HandbookofSta3s3calAnalysisofEventData
…fundedbytheNASAAISRProgram
MatLabCodeDocumentaAonExamplesTutorial
ContribuAonswelcome!
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dt
Height=1/dt n/dt E/dt
dt’=dt×exposure
Area=1/dt’ n/dt’ E/dt’
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Preliminary
!"#$%&'(
)*
+,-./0
.12+,345
WaveletKurtosis
NewSta4s4ctoDetectandCharacterizeIntermiEency
DanielEngavatov,EllioEBloom,JS;SLACPhDThesis
!!
"#$%&'($)*)+
StaAonarityvs.Non‐StaAonarity FormaldefiniAonrequiresinfiniteamountofdata
LocalstaAonaritydependsonscale
ConstructstaAonaritymeasureS[x(t)] E.g.varianceofTFdistribuAonvs.Amemarginal AnysuchmeasurehasstaAsAcalfluctuaAons Simulatesurrogatedata:scrambleFourierphase
ConstructdistribuAonofS(surrogatedata)
Tes3ngSta3onaritywithTime‐FrequencySurrogates,JunXiao,PierreBorgnat,andPatrickFlandrin
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From:Flandrin&Borgnat“RevisiAngandtesAngstaAonarity,”2008
…interpretedas“staAonary”or“nonstaAonary”dependingontheobservaAonscale…
TL:nonstaAonary
TR:staAonary(periodic)
BL:nonstaAonary
BR:staAonary(homogeneoustexture)
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