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AnalysisofUnevenlySpacedTimeSeries
JeffScargleNASAAmesResearchCenter
Commentaryon:Waveletvarianceanalysisofirregularlysampled7meseriesDebashisMondal(UChicago)
Analysisofeventdata Mul?‐taperAnalysis Examples:Time‐Frequency/ScaleDistribu?ons DataAdap?veHistograms
Sta$s$calChallengesinModernAstronomyV
Cross‐andAuto‐Correla?onFunc?onsforunevenlyspaceddata
EdelsonandKrolik:“TheDiscreteCorrela?onFunc?on:aNewMethodforAnalyzingUnevenlySampledVariabilityData”
AstrophysicalJournal333(1988)646
EdelsonandKrolik:TheDiscreteCorrela?onFunc?on:aNewMethodforAnalyzingUnevenlySampledVariabilityData,Ap.J.333,1988,646‐star?ngpointforallelse!
Evenlyspaceddata
Arbitrarilyspaceddata
dt
Height=1/dt n/dt E/dt
dt’=dt×exposure
Area=1/dt’ n/dt’ E/dt’
DataMode
Photonevents Time‐to‐Spill Countsinbins Fluxmeasurements AnyMode/Sampling! Universal
TimeSeriesAnalysisMachine
Auto‐ Correla?onFunc?on FourierPowerSpectrum FourierPhaseSpectrum Waveletscalgram Waveletscaleogram StructureFunc?on Time‐FrequencyDistribu?on Time‐ScaleDistribu?on …
JeffScargle
DataMode
Photonevents Time‐to‐Spill Countsinbins Fluxmeasurements AnyMode/Sampling! Universal
TimeSeriesAnalysisMachine
Cross‐ Correla?onFunc?on FourierPowerSpectrum FourierPhaseSpectrum Waveletscalgram Waveletscaleogram StructureFunc?on Time‐FrequencyDistribu?on Time‐ScaleDistribu?on …
JeffScargle
Photonevents Time‐to‐Spill Countsinbins Fluxmeasurements AnyMode/Sampling!
anycombina$on
Time‐Frequency/Time‐ScaleAnalysisTransformtoanewviewofthe$meseriesinforma$on.
ARealityinjoint?me&frequency(orscale)representa?on Atomicdecomposi?on
Time‐frequencyatoms Over‐completerepresenta?ons Op?malBasisPursuit(Mallat),etc.
UncertaintyPrinciple:T‐Fresolu?ontradeoff Non‐sta?onaryprocesses
Flares TrendsandModula?ons Sta?s?calchange‐points
InstantaneousFrequency Localvs.Globalstructure Interference(cross‐termsinbi‐linearrepresenta?on)Time‐Frequency/Time‐ScaleAnalysis(Temps‐Fréquence)PatrickFlandrinhjp://perso.ens‐lyon.fr/patrick.flandrin/publis.html;AWavelettourofSignalProcessing(UneExplora?ondesSignauxenOndelejes)StéphaneMallat
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Mul?‐taperAnalysis(Thomson1982)
Tapers(windows)reducesidelobeleakage=bias Incompleteuseofdatalossofinforma?on Mul?tapersrecoverthisinforma?on Leakageminimiza?on=eigenvalueproblem
Eigenfunc?ons:efficientwindowfunc?ons Eigenvalues
measureeffec?veness determinehowmanytermstoinclude
SpectralAnalysisforPhysicalApplica3ons:Mul3taperandConven3onalUnivariateTechniques,DonPercivalandAndrewWalden(1993)
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SYNCHROSQUEEZINGAconsistentandinver$ble$me‐frequencyanalysistoolthatcaniden$fyandextractoscilla$ngcomponents(of$me‐varyingfrequencyandamplitude)
E.Brevdo,N.S.Fučkar,G.Thakur,andH‐T.Wu,TheSynchrosqueezingalgorithm:arobustanalysistoolforsignalswith3me‐varyingspectrum,Submijed,2011(arXivid:1105.0010)
I.Daubechies,J.Lu,andH.‐T.Wu,Synchrosqueezedwavelettransforms:Anempiricalmodedecomposi3on‐liketool,AppliedandComputa?onalHarmonicAnalysis,2010.
Matlabtools:hjp://www.math.princeton.edu/~ebrevdo/synsq/
Star?ngvalueforfitrou?ne
Func7on Domain Range Auto‐ Cross‐ PhysicalInterp
Bayesianblk.LightCurve
Time Flux ✔ ✔mul?var.BB
Flares,eventsetc.
ScajerPlot Flux1 Flux2 ✔ Dependency(notjustcor.)
Correla?on Lag <X2><XY> ✔ ✔ Correlatedbehavior/lags
Spectrum Frequency Power ✔ ✔ Periodicity1/fnoise…
Phase ✔ ✔ Shizs,lags
Structure Lag <X2><XY> ✔ ✔ Correlatedbehavior/lags
Scalogram Scale/Time Power ✔ ✔ Dynamicbehavior
Scalegram Scale Power ✔ ✔ 1/fnoiseQPOs
Distribu?on Time/scale/frequency
Power ✔ ✔ Dynamicbehavior
HandbookofSta$s$calAnalysisofEventData
MatLabCode;Documenta?on;Examples;Tutorial…fundedbytheNASAAISRProgram
AdvancesinMachineLearningandDataMiningforAstronomy
Editors:MichaelWay,JeffScargleKamalAli,AshokSrivastava
ChapmanandHall,animprintofCRCPress(adivisionofTaylorandFrancis)
RandiCohen,ComputerScienceAcquisi?onsEditorDataMiningandKnowledgeDiscoveryseries