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

12

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)

13

14

15

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