An adaptive-binning method for generating constant-uncertainty/constant-significance light curves with Fermi-LAT data

0.57 MB·English

by  B. Lott

Astronomy&Astrophysicsmanuscriptno.adaptive˙binning˙one (cid:13)c ESO2012
January24,2012
An adaptive-binning method for generating
constant-uncertainty/constant-significance light curves with
Fermi-LAT data
B.Lott1,2,L.Escande1,2,S.Larsson3,4,5,andJ.Ballet6
1 Univ.Bordeaux,CENBG,UMR5797,F-33170Gradignan,France
2 CNRS,IN2P3,CENBG,UMR5797,F-33170Gradignan,France
2
3 DepartmentofPhysics,StockholmUniversity,AlbaNova,SE-10691Stockholm,Sweden
1
4 TheOskarKleinCentreforCosmoparticlePhysics,AlbaNova,SE-10691Stockholm,Sweden
0
5 DepartmentofAstronomy,StockholmUniversity,SE-10691Stockholm,Sweden
2
6 LaboratoireAIM,CEA-IRFU/CNRS/Universite´ParisDiderot,Serviced’Astrophysique,CEASaclay,91191GifsurYvette,France

n
a Preprintonlineversion:January24,2012
J

3 ABSTRACT
2
Aims.Wepresentamethodenablingthecreationofconstant-uncertainty/constant-significancelightcurveswiththedataoftheFermi-
] LargeAreaTelescope(LAT).Theadaptive-binningmethodenablesmoreinformationtobeencapsulatedwithinthelightcurvethan
E
withthefixed-binningmethod.Althoughprimarilydevelopedforblazarstudies,itcanbeappliedtoanysources.
H Methods.Thismethodallowsthestartingandendingtimesofeachintervaltobecalculatedinasimpleandquickwayduringafirst
. step.Thereportedmeanfluxandspectralindex(assumingthespectrumisapower-lawdistribution)intheintervalarecalculatedvia
h
thestandardLATanalysisduringasecondstep.
p
Results.The absence of major caveats associated with this method has been established by means of Monte-Carlo simulations.
-
Wepresent the performance of thismethod in determining duty cycles aswell as power-density spectra relativeto the traditional
o
fixed-binningmethod.
r
t
s Keywords.gammarays:analysis
a
[

1 1. Introduction
v
1 Variabilitystudiesofblazarsprovideawealthofinformationonthedynamicalprocessesatworkintheseobjectsandyieldimportant
5 constraints on their physical parameters. Thanks to its all-sky coverage in 3 hours and large sensitivity, the Fermi-Large Area
8 Telescope(LAT),isenablingalong-termviewofvariabilityintheenergyband0.1–300GeVforalargesampleofγ-rayblazars.
4 Aboutone thousandblazars are included in the Second LAT Active Galaxy Nuclei Catalog (Ackermannetal. 2011). Due to the
1. largevariationsinfluxshownbyblazarsoverdifferenttimescales,producinglightcurvesthatpreservetheinformationprovidedby
0 thedataisnotaneasytask.Witharegular(fixed)timebinning,usinglongbinswillsmoothoutthefastvariationsassessableduring
2 brightflares.Conversely,usingshortbinsmightleadtohard-to-handleupperlimitsduringlow-activityperiods.Herewepresenta
1 simplemethodwherethebinwidthisadjustedbyrequiringaconstantrelativefluxuncertainty,σ =∆ ,(alternativelyaconstant
lnF 0
: significance)ineachbin.Thismethodallowsformoreinformationtobeencapsulatedwithinthelightcurvethancanpossiblybe
v
achievedfor a fixed-binninglightcurve,withoutfavoringanyaprioriarbitrarytimescale andwhile avoidingupperlimits. These
i
X advantagescomeattheexpenseofaniterativeprocedurerequiredtofindtheproperbinwidthssatisfyingthechosencriterion.The
r elaborate,fairlycomputing-intensivemaximum-likelihoodprocedurenecessarytoanalyzeLATdatadoesnotlenditselfverywell
a
tosuchaniterativeprocedure.Whilelightcurveswithconstantsignal-to-noiseratiosarecommonlyusedatotherwavelengthssuch
astheX-rays,theabovedifficultyhassofarhamperedthegenerationofcorrespondinglightcurvesintheLATdomain.Thepurpose
ofthemethodpresentedinthispaper(referredtoastheadaptive-binningmethod)istoprovideasolutiontothisproblem.
The adaptive-binning method bears some resemblance to the Bayesian-Blocks (BB) method (Scargle 1998). Both methods
proposea time binningwhichadaptsitself to thedata insteadof anaprioriarbitraryregularbinning.The maindifferenceis that
theBBmethodgivesthemostprobablesegmentationoftheobservationperiodintotimeintervalsduringwhichthephotonarrival
rateisperceptiblyconstant,i.e.,hasnostatisticallysignificantvariations.Abinthendefinestheperiodoverwhichaconstantfluxis
observed.Intheadaptive-binningmethod,thesegmentationcriterionisbasedonagivenvalueofσ (orsignificance)regardless
lnF
of whether the flux varies within the bin. The adaptive-binningmethod is expected to be more sensitive (i.e. allows the study of
faintersources)thantheBBmethodsinceitmakesuseofboththephotonspatialandenergyinformationandproperlyaccountsfor
thediffusebackgroundswhiletheBBmethoddoesnot.Theflareriseanddecaytimescanalsobedeterminedmoreaccuratelywith
theadaptive-binningmethod.
This paper presents the method and explores its possible inherent biases/caveats. For the sake of clarity, only light curves
obtainedwith a constantrelative flux uncertaintywill be discussed, butchangingthe criterionto a constantsignificance, defined
1
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
fromtheTestStatistic1 (TS),isstraightforward.ThemethodisdescribedinSection2.Validationswithsimulationsarepresented
in Section3.Section 4presentsresultsontiminganalysesusingthe adaptive-binninglightcurvesregardingthe determinationof
thedutycycleandthepowerdensityspectraoftheγ-raysource.AsummaryisgiveninSection5.
2. Descriptionofthemethod
2.1.Optimizationofthetimebins
ThegoalofthemethodistocomputelightcurvesforanintegralfluxaboveagivenenergyE ,withconstant∆ inalltimebins.
min 0
Thesourceenergyspectrumisassumedtobeapower-lawdistributionwithphotonspectralindexΓ.Althoughthemethodallows
anyE tobeused,fluxesarereportedinthefollowingabovetheoptimumenergy(E )forwhichtheaccumulationtimesneeded
min 1
tofulfilltheabovecondition(i.e.,binwidths)aretheshortestrelativetootherchoicesofE (seetheAppendixforthederivation
min
of E ). Theenergy E dependsonthe signal/backgroundratiobutisindependentofexposure.Itis calculatedhereusingtheflux
1 1
andΓdeterminedoverthewholetimerangespannedbythedata.
ThemethodconsistsinsolvingthefollowingequationforT (endingtimeoftheinterval)givenastartingtimeT :
1 0
σ (T ,T ,F¯,Γ¯)=∆ (1)
lnF 0 1 0
whereF¯ andΓ¯ aretheaveragefluxandphotonspectralindexovertheinterval[T ,T ]respectively.Inordertoavoidthepenaltyof
0 1
excessivecomputingtime requiredbythe standardmaximum-likelihoodanalysis,σ is estimatedfromthetime-orderedlistof
lnF
photonscontainedintheregionofinterest(ROI),asdescribedintheAppendix(seeEq.A.8).
Sincethesourceispotentiallyvariable,F¯ mustbeevaluatedconsistentlyineachtimebin.F¯ isestimatedthroughaprocedure
maximizing the likelihood (see Eq. A.4), this procedure being similar to a simplified version of the standard LAT analysis. In
principleΓ¯ shouldbeevaluatedsimilarly,butinpracticeΓ¯ canbeleftfixed2tospeedupcomputation.TakingaconstantΓislargely
justifiedsincespectralvariationswithtimehavebeenfoundtobeverymoderateinLAT-detectedblazars3(see Abdoetal.2010b).
Inpractice,duetothediscretenatureofthedata,Eq.1canonlybesolvedinanapproximateway.Theprocedureisthefollowing:
– ForagivenT ,F¯ (initiallyestimatedoverthewholeperiodforthefirstbin)andΓ¯,thefunctionσ (T ,T ,F¯,Γ¯)ismonoton-
0 lnF 0 1
ically decreasing with increasing T (see Eq. A.8). The detection time of theearliestphoton leading to the fulfillment of the
1
conditionσ <∆ ishencetakenasT .
lnF 0 1
– F¯ isthenreestimatedover[T ,T ]andσ isreevaluatedusingthisnewF¯ value.Then
0 1 lnF
•ifσ isequalto∆ withinapredefinedtolerance,convergenceisachievedandT isreplacedbyT ;
lnF 0 0 1
•otherwise,T isreevaluatedwiththeupdatedvalueofF¯.Abisectionmethod,makinguseofthesetofT estimatesobtained
1 1
inpreviousiterations,complementstheproceduretospeedupconvergence.
– Thewholeprocedureisrepeateduntilconvergenceisachieved.
The timesT forthedifferentintervalsarecalculatedsequentiallyuntiltheconditionσ < ∆ cannotbefulfilledusingthe
1 lnF 0
remainingset of photons(these photonsare left unused).Once the whole set of time intervalshasbeen determined(this stage is
referredtoas“Step1” inthe following),F¯, Γ¯ andσ are recalculatedforeachintervalwiththe standardpylikelihoodanalysis
lnF
(“Step 2”). The consistency between these values of σ and ∆ is then checked. Using Monte-Carlo simulations, an excellent
lnF 0
agreementhasbeenobservedinmostcases,makingfurtheriterationsunnecessary.
Beingdrivenbythenumberofaccumulatedsourcephotons,thebinwidthsprovidedbythismethoddependonboththesource
flux and the exposure of the instrument for the source location. The modulation of the LAT daily exposure over the precession
period(53.4days)rangesbetweenafewpercentand±50%(forarockingangleof50◦)dependingonthesourcedeclination.
2.2.Pylikelihoodanalysis
ResultsdescribedinthefollowingwereobtainedviaanunbinnedpylikelihoodanalysisperformedwiththeFermi-LATScienceTools
softwarepackage4versionv9r19p0.ThetoolsusedinStep1andthepylikelihoodanalysismadeuseofthesamelistofphotonswith
energyabove100 MeV and containedwithin a 20◦-in-diameterROI. The P6 V3 DIFFUSE set of instrumentresponsefunctions
(IRFs)wasemployed.TheGalacticdiffuseemissionmodelversion”gll iem v02.fit”andtheisotropic(sumofresidualinstrumental
andextragalacticdiffuse)background“isotropic iem v02.txt”(filesprovidedwithScienceTools)wereusedthroughouttheproce-
dure.Thenormalizationofbothdiffusecomponentswereleftfreeinthefitforthepylikelihoodanalysis.Foruniformityovermany
realizationsproducedunderthesameconditions,theoptimumenergywasevaluatedfromthetruefluxandspectralindexusedin
Monte-Carlosimulations.
1 TheTestStatisticisdefinedasTS =2(lnL(source)−lnL(nosource)),whereLrepresentsthelikelihoodofthedatagiventhemodelwithor
withoutasourcepresentatagivenpositioninthesky.SeeAppendixfordetails.
2 Γcanbetakenfromasourcecatalogwhenavailable(forrealdata)orsettothemeasuredvalueoveralongperiod.
3 PreliminarylightcurvesofbrightLAT-detectedblazarscomputedwithrealdataprovedthattakingaconstantphotonspectralindexinthebin
widthestimateprovidesreasonablyaccurateresultsintermsofσ .
lnF
4 http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/
2
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
3. Validationwithsimulations
Inthissection,wefirstpresenttheresultsoftheadaptive-binningmethodappliedtosimulatedvariablesources.Then,weaddress
some of the possible caveats of this method by considering simulated constant sources. For this purpose, we compare results
obtainedforadaptive-binningandfixed-binninglightcurvesregardingthefollowingissues:biasesandfluctuationsinthemeasured
fluxesandphotonindices,correlationbetweenfluxandΓandfinallythefluxcorrelationbetweenconsecutivebins.Fixed-binning
analyseswerecarriedoutwiththesametotalnumberofbinsastheadaptive-binninganalysestoenablemeaningfulcomparisons.
Thecriterionadoptedforalladaptive-binninganalysesis∆ =25%.
0
3.1.Descriptionofsimulations
Dataweresimulatedintheenergyrange0.1–200GeVwiththetoolgtobssim,partoftheFermi-LATScienceToolssoftwarepackage,
whichgeneratesphotoneventsfromastrophysicalsourcesandmodelstheirdetectionsaccordingtotheIRFs.TheP6 V3 DIFFUSE
setofIRFswasusedconsistentlywiththeanalysisdescribedabove.Thesesimulationsdonotincludeallknownsystematiceffects
oftherealdata,inparticularrelatedtothevaryinginstrumentalbackgroundduringtheorbit.However,thislackisnotbelievedto
significantlyaffecttheconclusionsdrawnthereafterduetothemoderatemagnitudeoftheseeffects(afew%intheflux, Abdoetal.
2011). The simulations were performed for an 11-month long period starting at the beginning of the scientific operation of the
Fermimission,usingtheactualLATpointinghistory.Althoughlightcurvesshowninthefollowingusefluxesabovetheoptimum
energy,thedifferentexamplesaredistinguishedaccordingtotheirtruefluxesabove100MeV(F in10−7 phcm−2 s−1)tomake
100
comparisonsclearer.Allsourcesweresimulatedatthesamehigh-Galacticlatitudeskyposition,(l,b)≃(80◦,80◦).
3.2.Lightcurvesofvariablesources
Inthissection,themethodis illustratedforthreevariablesourceswhichweresimulatedbymeansoftemplatesshowingrealistic
fluxvariationstypicalofblazars.Thesourcespectraweremodeledwithasinglepower-lawfunctionwithanaverageflux F =2
100
(correspondingtoafairlybrightblazardetectedbytheLAT)andafixedphotonindexΓ=2.4.TheoptimumenergyisE =230MeV
1
fortheseparameters.
Figure1comparesthetruelightcurveswithlightcurvescalculatedusingtheadaptive-binningmethodforthethreesources.The
analysisforthelightcurveshowninFigure1topwasalsoconductedassumingareversedarrowoftime,theresultsbeingpresented
in Figure2. Statistically significantfluxescanbe determinedoververydifferenttime scales, rangingfroma few hoursto several
monthsintheseexamples.Theoccurrenceofsignificantshort-termflaringactivityduringthelow-fluxstatescanberuledout.
Anadversefeatureofthemethodistheflaretruncation(thelightcurveinFigure1cshowssuchafeatureatMET≃255000000).
Whenasignificantflareoccursafteralongperiodoflowactivity,anon-negligiblenumberofphotonsfromtheflarehastobe”con-
sumed”to counterbalancetheaccumulatedbackgroundphotonsandenablethecriterionto befulfilled.Possible waysto mitigate
thiseffectarebeingexploredandwillbeimplementedinfutureversions.Thetoolsallowlightcurvestobegeneratedbyconsidering
eitherincreasingordecreasingphotontimes,sotherobustnessofanyanalysisresults(timelags,dutycycles,riseanddecaytimes...)
canbetestedbyusingbothflavors.
Figure1cmayconveytheimpressionthattheadaptive-binningmethodwillleadtolightcurveslaggingtherealone.However,
cross-correlationanalyses have shown thatno significantlag is observedfor the lightcurvesinvestigatedhere, probablybecause
truncationaffectsonlyveryfewbinsandtheflareleadingedgesarestillwellsampled.
Figure 3 shows the distribution of relative flux uncertainties calculated from the pylikelihoodanalysis for the three variable
sources.Allthreedistributionsarecenteredatavalueclosetothetarget∆ =25%(depictedbythedashedline)withessentiallyall
0
valuesincludedintherange20%–30%withatypicalrmsof3%,demonstratingtheperformanceofthemethod.
3.3.Caveatsexploredwithsteadysources
Steady sources with three different fluxes (F =0.5, 1 and 5) and two different photon spectral indices (Γ= 2.0 and 2.4) were
100
simulatedoverthesame11-monthlongperiodasforvariablesources.Fiftyrealizationswereperformedforeachcase.Theaverage
numbersofbinsinthelightcurvesaswellastheoptimumenergiesarereportedinTable1forthedifferentcases.ForF =1and
100
Γ=2.0,alightcurvewherefluxesareestimatedafterStep1(i.e.,usingthesimplemaximum-likelihoodproceduredescribedinthe
Appendix)for a particular realization is compared to that resulting from Step 2 (i.e., where fluxes are computed with the whole
pylikelihoodanalyseswith the time intervalsestablishedin Step 1) in Figure4. Thereis a goodagreementbetweenthese fluxes,
whichholdsforallrealizationsandparametersets.ExamplesofadaptivebinninglightcurvesforthefluxareshowninFigure5for
thedifferentcases,whilelightcurvesforthephotonspectralindexaredisplayedinFigure6.Figure7comparesthefluxdistributions
obtainedviatheadaptive-andfixed-binningmethodsbysummingupallrealizations.Foragivensetofconditions,thedistributions
havesimilarmeansandrmsbutdifferentskews.Apositiveskew(typicallyequalto1σforthedistributionsshowninFigure7)is
a featureof theadaptive-binningmethod.A positivefluctuationinflux leadsto a shorter-than-averagebinassociatedwith a high
apparentflux.Anegativefluctuationhaslesschancetoberecordedsincethebinwillbeextendeduntilenoughphotonshavebeen
accumulatedtomeetthe∆ -wisecriterion.Thephotonindexdistributionshavebeenfoundtobeverysimilarforthetwomethods
0
andclosetogaussiandistributions.
The distributions of relative statistical uncertainties obtained with the adaptive-binning method (resulting from Step 2) are
presentedinFigure8.Thesedistributionsaremoregaussian-likethanthoseobtainedforvariablesources,withameanvalueclose
to 25% as required and a typical rms of about 1.7%. Simulations performedfor a very low flux of F =0.1 (Γ=2.0), where the
100
3
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
10−6
a
−1 )
s
−2
m
c
h 10−7
p
] (E1
>
E
x[
u
Fl
10−8
×106
240 245 250 255 260 265 270
MET (s)
10−6
−1 ) b
s
−2
m
c
h
p
] (E110−7
>
E
x[
u
Fl
10−8
×106
240 245 250 255 260 265 270
MET (s)
10−6
c
−1 )
s
−2
m
h c10−7
p
] (E1
>
E
x[
u
Fl
10−8
×106
240 245 250 255 260 265 270
MET (s)
Fig.1.Simulatedlightcurves(blue)withtheadaptive-binninglightcurvessuperimposed(Step1:blackopensquares,Step2:red
solidcircles)forthreevariablesources.METstandsforMissionElapsedTime.
methodyieldsasinglebinoverthe11-monthperiod,showthatthemeanofthedistributionremainswithin10%ofthetargetvalue
eveninthatextremecase.
4
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
10−6
−1 )
s
−2
m
c
h 10−7
p
] (E1
>
E
x[
u
Fl
10−8
×106
240 245 250 255 260 265 270
MET (s)
Fig.2.SameasFig.1abuttheanalysiswasconductedassumingareversedarrowoftime.
20 22 20
18 20 18
16 18 16
14 16 14
Number of bins11028 Number of bins1110248 Number of bins11028
6 6 6
4 4 4
2 2 2
010 15 20 25 30 35 40 010 15 20 25 30 35 40 010 15 20 25 30 35 40
Relative flux uncertainty (%) Relative flux uncertainty (%) Relative flux uncertainty (%)
Fig.3.Distributionofrelativefluxuncertaintiesgivenbythepylikelihoodanalysisforthreevariablesources.Thedashedlinedepicts
thetarget∆ =25%.
0
Γ=2.0 Γ=2.4
F <N > E (MeV) <N > E (MeV)
100 bins 1 bins 1
0.5 14 396 8 299
1 37 332 23 253
5 268 246 192 205
Table1.Averagenumberoftimebinsandoptimumenergiesforthedifferentsimulationconditions.
Thepossiblebiasesinducedbythismethodareexploredinthefollowing.
3.3.1. Biasandstatisticalfluctuationsinfluxandphoton-indexmeasurements
Thepylikelihoodanalysisisexpectedtoprovidethecorrectmeanfluxandphotonspectralindexoverallpossibletimeintervals.We
neverthelesscheckfortheabsenceofbiasesinthemeasurementoftheseparameterswhenthetimeintervalsaredeterminedwiththe
presentmethod.Furthermore,possibleadditionalfluctuationsinthemeasuredparameterscouldarisefromtheintrinsiccorrelation
betweenbinwidthandphotoncontentsofthebin.Botheffectsareaddressedhere.
Onedefines∆F (∆Γ)asthedifference,expressedinsigma,betweenthemeasuredflux(index)andtheaveragevaluemeasured
over the whole 11-month period. Summing over all realizations, the distributions of the means (left) and rms (right) of the ∆F
(top)and∆Γ(bottom)distributionsaredisplayedinFigure9foraparticularsetofparameters.Theresultsoftheadaptive-(solid)
and fixed- (dashed) binning methods are compared in this figure. The means < ∆F > and < ∆Γ > are listed in Tables 2 and 3
respectively,whilethermsvaluesaregiveninTables4and5.
A small bias is observed in the mean flux values. Since similar features are seen for both adaptive- and fixed-binning light
curves,onecansafelyconcludethatthisbiasarisesfromsmallsystematicseffectspertainingtoourpylikelihoodanalysisandisnot
createdbytheadaptive-binningapproach.
5
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
×10−9
50
1)
−
s
−2 m 40
c
h
p 30
1) (
E
>
E 20
x(
u
Fl
10
×106
0
240 245 250 255 260 265
MET
Fig.4.Comparisonoffluxfromthetool,i.e.,afterStep1(opendiamonds)andfluxresultingfromthepylikelihoodanalyses,i.e.,
afterStep2(filledcircles)forasimulatedconstantsourcewithF =1andΓ=2.4.
100
×10−6 ×10−6
0.5
−1)s 0.4 −1)s 0.5
−2 m −2 m 0.4
h c 0.3 h c 0.3
(pFE>E1 00..21 (pFE>E1 00..21
0 240 245 250 MET2 55 260 265 ×106 0 240 245 250 MET2 55 260 265 ×106
×10−9 ×10−9
−1)s 50 −1)s 50
−2 m 40 −2 m 40
h c 30 h c 30
p p
(FE>E1 1200 (FE>E1 1200
0 240 245 250 MET2 55 260 265 ×106 0 240 245 250 MET2 55 260 265 ×106
×10−9 ×10−9
20 14
−2−1) sm 111468 −2−1) sm 1102
(ph cFE>E1 114026802 240 245 250 MET2 55 260 265 ×106 (ph cFE>E1 40268 240 245 250 MET 255 260 265×106
Fig.5. Examples of light curves for the different sets of parameters : Γ=2.0 (left) and Γ=2.4 (right), F =5, 1, 0.5 from top to
100
bottom.Thesolidlinedepictsthetruefluxvalue.
F Adaptive(Γ=2.0) Fixed(Γ=2.0) Adaptive(Γ=2.4) Fixed(Γ=2.4)
100
5 -0.20±0.01 -0.15±0.01 -0.19±0.01 -0.15±0.01
1 -0.13±0.02 -0.09±0.02 -0.14±0.03 -0.13±0.03
0.5 -0.06±0.03 -0.07±0.03 -0.12±0.05 -0.13±0.05
Table2.Comparisonof<∆F >foradaptiveandfixedbinnings.
Theaveragermsvaluesarefoundtobeclosetotheexpectedvalueof1forallcases(withslightlylowervaluesfortheadaptive-
binningapproachrelativetothefixed-binningone).Onecanthusexcludethattheadaptive-binningapproachgivesrisetosignificant
fluctuationsinthemeasuredparameters.
3.3.2. Flux-indexcorrelation
Anotherissueworthinvestigatingisthecorrelationbetweenmeasuredphotonindexandfluxinagiventimeinterval.Allphotonsdo
notcontributeequallytothefulfillmentofthecriterion,withhigher-energyphotonscontributingmore.Thedetectionofaseriesof
photonswithhigher-than-averageenergieswillthusleadtothecriterionbeingsatisfiedfasterandconsequentlytotheintervalunder
considerationbeingclosedearlier.Inordertoevaluatewhethertheadaptive-binningmethodcreatesadditionalcorrelationbetween
thephotonspectralindexandflux (asmeasuredin Step 2),thecorrespondingcorrelationfactoris computedforeachrealization.
6
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
3 3.5
2.8
2.6 3
2.4
Γ 2.2 Γ 2.5
2
1.8
1.6 2
1.4
1.2 ×106 1.5 ×106
240 245 250 MET2 55 260 265 240 245 250 MET2 55 260 265
3 3.6
2.8 3.4
2.6 3.2
2.4 3
Γ 2.22 Γ 22..68
1.8 2.4
1.6 2.2
1.4 2
1.2 ×106 1.8 ×106
240 245 250 MET2 55 260 265 240 245 250 MET2 55 260 265
2.8 3.2
2.6 3
2.4
2.8
2.2
Γ 2 Γ 2.6
1.8 2.4
1.6 2.2
1.4 ×106 2 ×106
240 245 250 MET2 55 260 265 240 245 250 MET 255 260 265
Fig.6.Temporalevolutionofthemeasuredphotonspectralindex:Γ=2.0(left)andΓ=2.4(right),F =5,1,0.5fromtoptobottom.
100
Thesolidlinedepictsthetruephotonindex.
800
500
700
600 400
mber of bins450000 mber of bins300
Nu300 Nu200
200
100
100
00 0.5 1 1.5 2 2.5 3 00 0.5 1 1.5 2 2.5 3
Normalized Flux Normalized Flux
100
60
80
50
Number of bins4600 Number of bins3400
20
20
10
00 0.5 1 1.5 2 2.5 3 00 0.5 1 1.5 2 2.5 3
Normalized Flux Normalized Flux
50
25
40
20
Number of bins2300 Number of bins1105
10 5
00 0.5 1 1.5 2 2.5 3 00 0.5 1 1.5 2 2.5 3
Normalized Flux Normalized Flux
Fig.7.Normalizedfluxdistributionsforadaptive(solid)andfixed(dashed)binningsandsteadysourceswithΓ=2.0(left)andΓ=2.4
(right),F =5,1,0.5fromtoptobottom
100
Since E is chosen as the optimum energy, for which the correlation between photon index and integral flux is minimum by
min
definition, the observed correlation factor is expected to be small. The correlation-factor distributions obtained for adaptive and
fixedbinningsareshowninFigure10.Forallcases,thedistributionsareverysimilarforthetwoapproachesandareallcentered
atsmallvalues,withlargerfluctuationsforlowerfluxes.Therefore,themeasuredphotonspectralindicesandfluxesdonotappear
morecorrelatedwhenusinganadaptivebinningrelativetoafixedbinning.
7
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
5000 3000
4000 2500
23 Number of bins000000 12 Number of bins500000
1000
1000
500
010 15 20 25 30 35 40 010 15 20 25 30 35 40
Relative flux uncertainty (%) Relative flux uncertainty (%)
600
350
500
300
Number of bins340000 Number of bins122505000
200
100
100 50
010 15 20 25 30 35 40 010 15 20 25 30 35 40
Relative flux uncertainty (%) Relative flux uncertainty (%)
250
120
200
100
Number of bins110500 Number of bins6800
40
50
20
010 15 20 25 30 35 40 010 15 20 25 30 35 40
Relative flux uncertainty (%) Relative flux uncertainty (%)
Fig.8.Distributionofσ for∆ =25%(depictedbyadashedline)forthedifferentparametersets:Γ=2.0(left)andΓ=2.4(right),
lnF 0
F =5,1,0.5fromtoptobottom.
100
F Adaptive(Γ=2.0) Fixed(Γ=2.0) Adaptive(Γ=2.4) Fixed(Γ=2.4)
100
5 0.02±0.01 0.02±0.01 0.02±0.01 0.01±0.01
1 0.02±0.03 0.05±0.03 0.04±0.03 -0.01±0.03
0.5 0.06±0.03 0.05±0.03 0.03±0.06 0.03±0.05
Table3.Comparisonof<∆Γ>foradaptiveandfixedbinnings.
F Adaptive(Γ=2.0) Fixed(Γ=2.0) Adaptive(Γ=2.4) Fixed(Γ=2.4)
100
5 0.99±0.01 1.06±0.01 0.97±0.01 1.03±0.01
1 0.96±0.02 1.01±0.01 0.93±0.02 0.98±0.02
0.5 0.90±0.03 1.02±0.03 0.81±0.03 0.87±0.03
Table4.Comparisonoftheaveragermsofthe∆F distributionsforadaptiveandfixedbinnings.
F Adaptive(Γ=2.0) Fixed(Γ=2.0) Adaptive(Γ=2.4) Fixed(Γ=2.4)
100
5 1.02±0.01 1.03±0.01 1.05±0.01 1.03±0.01
1 1.00±0.02 1.04±0.02 1.05±0.02 1.05±0.02
0.5 1.00±0.03 1.03±0.02 1.05±0.04 1.01±0.05
Table5.Comparisonoftheaveragermsofthe∆Γdistributionsforadaptiveandfixedbinnings.
3.3.3. Interbincorrelation
In the adaptive-binning method, the starting time of a given bin depends on the starting time and flux-dependent width of the
previousbinandthusbyrecurrenceonthewholefluxhistorysincethestartofthelightcurve.Inthissection,weexploretheimpact
oftheinterbincorrelationonthemeasuredsourceparametersbycomparingtheseintwoconsecutivebins.Acomparisonwillagain
be made with the results of a similar analysisperformedwith a fixed binning,for whichfluxesmeasuredin consecutivebinsare
independentbynature.Theflux(index)inonebinisplottedversustheflux(index)inthenextbininFigure11left(right)forthe
wholelightcurveofaparticularrealization.
8
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
22
20
12
18
10 16
s s
e e
c c
r r14
u u
o 8 o
s s12
of of
er 6 er 10
b b
m m
8
u u
N N
4 6
4
2
2
0 0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
<(F−F )/ σ > RMS (F−F )/ σ
11 months F 11 months F
18
14
16
12
14
s s
e e
c c
r10 r12
u u
o o
s s
of 8 of 10
r r
be be 8
m 6 m
u u
N N 6

4
4
2
2
0 0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
<(Γ−Γ )/ σ > RMS (Γ−Γ )/ σ
11 months Γ 11 months Γ
Fig.9.Distributionsofmeanvalues(left)andrms(right)forflux(top)andphotonindex(bottom)obtainedinadaptive-(solid)and
fixed-binning(dashed)lightcurves.Inthisexample,Γ=2.4andF =5.
100
ThedistributionofcorrelationfactorsobtainedwiththedifferentrealizationsarepresentedinFigure12.Nosignificantdiffer-
ences are observed between distributions obtainedwith adaptive and fixed binningsfor all cases. Both distributionsare centered
close to 0 (as expected for the fixed-binning case), demonstrating that the interbin dependence intrinsic to the adaptive binning
methodhaslittleimpactonthecorrelationbetweenparametersmeasuredinconsecutivebins.
4. High-levelanalysiswithvariablesources
4.1.Dutycycle
Onewaytodescribevariationsinalightcurveisviaadutycycle,correspondingtothefractionoftimeasourceisina“bright”state.
Moregenerallythiscanbepresentedasafluxdistributionoritsintegral,thecumulativedistributionfunction(CDF).InFigure13we
haveplottedflux,normalizedtoameanof1,versus1-CDF,forthelightcurvesdisplayedinFigure1.Thecurvesgivethefraction
oftime(1-CDF)thatasourceisaboveaparticularfluxlevel.Fourcurvesareshownforeachlightcurve,the‘true’lightcurve,the
adaptive-binninglightcurve,thefixed-binninglightcurve(withthesamenumbersofbinsastheadaptive-binninglightcurve)and
thefixed-binninglightcurvewithhigher(16times)timeresolution.Thetwofixed-binninglightcurvesarecomplementaryinthat
thehighresolutiononeworkswellatthehigh-fluxendbutfailsatthefaint-endwherethe noisegovernsthe distributionbecause
of the low signal-to-noise ratio of the individual points. With adaptive binning the bin widths can be optimized such that in the
presenceofnoise,thefluxdistribution,ordutycycle,canbedescribedoveramuchwiderrangethanispossiblewhenafixedbin
widthisused.Thelargertherangeinfluxesthemoreadvantageoustheadaptivebinningisforthistypeofanalysis.
9
B.Lottetal.:Adaptive-binningmethodforFermi-LATlightcurves
30
30
es25 es25
c c
sour20 sour20
er of 15 er of 15
b b
um10 um10
N N
5 5
0−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
14 12
s s
ce12 ce10
our10 our
s s 8
of 8 of
er er 6
b 6 b
m m
u u 4
N 4 N

2 2
104−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 09−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
s12 s 8
e e
c c 7
ur10 ur
o o 6
s s
of 8 of 5
er 6 er 4
b b
m m 3
u 4 u
N N 2

2
1
0−1 −0.8 −0.6 −0.4 −0.2 C0 0.2 0.4 0.6 0.8 1 0−1 −0.8 −0.6 −0.4 −0.2 C0 0.2 0.4 0.6 0.8 1
Γ/F Γ/F
Fig.10.Distributionsofcorrelationfactorbetweenspectralindexandfluxforadaptive(solid)andfixed(dashed)binnings.Γ=2.0
(left),Γ=2.4(right),F =5,1,0.5fromtoptobottom.
100
2 2
1.8 1.8
1.6 1.6
1.4 1.4
F>1.2 Γ>1.2
< <
/11 /11
+ +
Fi0.8 Γi0.8

0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
F/<F> Γ/<Γ>
i i
Fig.11. Fluxes (left) and photon spectral indices (right) measured in consecutive bins plotted one against another for a given
realization.
4.2.PowerDensitySpectrum(PDS)
ThePDSisoneofthemaintoolsusedtocharacterizelightcurvevariability.ForevenlysampleddatathePDSiscomputeddirectly
via the Fouriertransformof the lightcurve.For lightcurveswith unevensamplingthe Lomb-Scarglealgorithmcan be used and
the PDS distortions caused by the sampling are usually modeled by comparison with simulated light curves (Doneetal. 1992;
Uttleyetal. 2002). This method cannot easily be applied to adaptively binned light curves since the bin widths depend on flux,
makingtime samplingdifferentforeach simulation.Itis still possibleto make a qualitativeestimate ofthe PDS foran adaptive-
binninglightcurve.Todothisweoversamplethelightcurvewithalinearinterpolationbetweenthecenterofeachoftheadaptive
timebins.ThePDSisthencomputedwithastandardFouriertransform.Thebinningandinterpolationmodifythepowerdensity
aroundandabovethefrequenciescorrespondingtothewidthsofthetimebins.Tocheckatwhatfrequenciesthisbecomesimportant
foraparticularlightcurvewecomputethedistributionofNyquistfrequencies(equalto0.5/dtwheredtisthebinwidth)associated
10