Using hydrodynamical simulations of stellar atmospheres for periodogram standardization : application to exoplanet detection

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by  Sophia Sulis

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USING HYDRODYNAMICALSIMULATIONSOF STELLAR ATMOSPHERESFOR
PERIODOGRAMSTANDARDIZATION:APPLICATIONTO EXOPLANET DETECTION
S.Sulis,D. Mary,L. Bigot
LaboratoireLagrange, UMRCNRS 7293,Universite´ Coˆted’Azur,
ObservatoiredelaCoˆte d’Azur, CS34229,06304Nice, France
6
1
0
2 ABSTRACT has to be estimated from the data. This can be done through non-
n Our aim is to devise a detection method for exoplanet signatures parametric methods (e.g. local SNR [9], modified periodogram
smoothers [10, 11], robust M-estimators [12, 13]) or parametric
a (multiplesinusoids)thatisbothpowerfulandrobusttopartiallyunk-
methods (e.g. iterative Yule-Walker [14], ratio of autoregressive
J nownstatisticsunderthenullhypothesis.Intheconsideredapplica-
7 tion,thenoiseismostlycreatedbythestellaratmosphere,withsta- (AR)spectralestimates[15],balancedmodeltruncation[16]).Even
if some of these estimators are asymptotically unbiased, the una-
2 tisticsdependingonthecomplicatedinterplayofseveralparameters.
voidable injection of estimation noise in the denominator of the
Recentprogressesinhydrodynamic(HD)simulationsshowhowever
standardized periodogram makes the statistical characterization of
] thatrealisticstellarnoiserealizationscanbenumericallyproduced
T off-line by astrophysicists. We propose a detection method that is theteststatisticsdifficult.OnecanresorttoMonteCarloorboots-
trap simulations [17] to evaluate the thresholds empirically (see
S calibratedbyHDsimulationsandanalyzeitsperformances.Acom-
[18],Chap.7forexamplesofgapsbetweentheoreticalandempirical
. parisonofthetheoreticalresultswithsimulationsonsyntheticand
h thresholds),butthisproceduremaynotbetractableformassivetime
realdatashowsthattheproposedmethodispowerfulandrobust.
t series(largeN).
a
m IndexTerms— Detection,periodogram,colorednoise,standar- InAstronomy,preprocessingstagesaimedat“whitening”thenoise
dization,statistics. (e.g.withlocaltrendfilteringorlineremovalwiththeCLEANme-
[
1. INTRODUCTION thod[19])orbasedonvariousARMA(AR-MovingAverage)noise

models[20]aremostoftenappliedtothedatabeforeconductingthe
1 This study is motivated by the long-standing challenge of de-
v tecting rocky low mass exoplanets. In this aim, future instruments detectiontest.Suchproceduresposethesamequestionofrobustness
5 withextremely low detector noise are being developed, giving ac- abouttheactualFArateatwhichthetestisconducted.Inpresence
7 cesstomassivetimeseriesofobservations(highsamplingratesof ofunknown colorednoise, thereliabilityof aclaimeddetectionof
3 typicallyfive samples per minute, for several months toyears, see alowmasstelluricplanetisthusdifficulttoassess(seeforinstance
7 e.g.[1,2,3]).Wefocusontheso-calledradialvelocitydata,where therecentandcontroversialcaseofαCentauriB[21,22]).
0 Inthepresentwork,wedonotattempttobuilddedicatedpara-
theplanet signature(ifpresent) appears asoneorafewsinusoidal
. metricnoisemodels.Wechooseinsteadtoexploitrecentprogresses
1 components of weak amplitudes w.r.t. the noise level [4]. For the
inHDsimulations[23].Theseresultsdemonstratethatreliabletime
0 considerednewobservingfacilities,themainnoisesourcewillnot
series of the stellar noise can be simulated by numerical codes,
6 comefromtheinstrumentbutfromthestochasticactivityofthestar
which account for the complex interplay of various astrophysical
1 itself(convection,magneticdynamo,stellarspotsandoscillations).
processes in the star’s interior. We assume that a training data set
: Sinusoiddetectionisanotherlong-standingproblem.Invarious
v (intheformoftimeseries)isavailableandusethisinformationto
fields,includingindeedStatisticsandAstronomy,aparticularlyrich
i standardize the periodogram. The data set is considered unbiased,
X amountofmethodsexists,acentralpieceofwhichisthe(Schuster’s)
but possibly limitedinsize, because HD simulations areheavy. In
periodogram[5]:
ar P(ν):= N1 XN X(tj)e−i2πνtj 2. (1) fcaaclct,ulsaimtiounlamtinognt1h0s0ovdeary1s2a0tChiPgUh.sCamonpsleinqguernattley,rreeqauliirsetiscanruomunbder3s
(cid:12)j=1 (cid:12) ofsimulatedlightcurvesavailableinpracticewillnotbemorethan,
InEq.(1),X(t)willbeanev(cid:12)enlysampledtimes(cid:12)eriesobtainedatN say,ahundred.Thisraisesthequestionoftheimpactofestimation
(cid:12) (cid:12)
epochstj,withsamplingrate∆t=tj+1−tj,∀j ∈{1,...,N−1}. noiseintheproposedstandardizationapproach.
ThetypeIerrors(orfalsealarms,FA)ofanytestbasedontheordi- Addressing this question first requires, for the two hypotheses of
natesofP dependonthestatisticsoftheseordinatesunderthenull ourmodel(Sec.2),theinvestigationofthestatisticsoftheclassical,
hypothesis. In practice, the statistics of the noise are often not (or averaged and standardized periodograms. This is the purpose of
only partially) known, and so are those of the P ordinates. In this Sec.3,wheretheuseofasymptoticresultsismotivatedbythelarge
case,itisdifficulttoassesshowreliableisanyclaimeddetection. duration and high sampling rate (large N) considered here. The
InStatisticalsignal processing, some classical tests(e.g.[6,7, second step istoselect several tests(Sec.4) for whichthe benefits
8])partiallycopewiththisissuebyguaranteeingthecontrolofthe gained from the proposed standardization can be highlighted and
FA rate whatever the unknown noise variance, provided that the quantified. Weopt for a sample of classical tests covering the dif-
noise is Gaussian, independent and identically distributed (i.i.d.). ferentcasesofsingleandmultiplesinusoidsdetection.Atthisstage
Whenthenoiseisnotwhitebutcoloredinsomeunknownmanner, weareinpositiontoderivethetestsstatisticsandthecorresponding
aclassicalapproach consistsincalibratingP(ν)bysomeestimate FA(Sec.5)anddetection(Sec.6)rates.Thelaststepisanumerical
P(ν), leading to a frequency-wise standardized periodogram of evaluation of the theoretical results, the method performances and
the form P(ν). Usually, the noise Power Spectral Density (PSD) itsactualrobustness(Sec.7).
b
P(ν)
b
2. STATISTICALMODEL Notethatinthissetting,anysourceofbias(causedforinstanceby
imperfect HD simulations) is left out of scope of this study. The
Weconsidertwohypotheses:
focusisonthestochasticestimationnoisecausedbythefiniteness
H0:X(tj)=ǫ(tj) ofT,whichisencapsulatedinLandimpactsthedistributionofP.

 H1:X(tj)=XNs αisin(2πfitj+φi)+ǫ(tj) (2) 3.3. Standardizedperiodogram
where X(t = j∆t) is thi=e1evenly sampled data time series and Thestandardizedperiodogramconsideredhereisdefinedas:
j
ǫ(tj)azero-meansecond-orderstationaryGaussiannoisewithPSD P(ν ):= P(νk). (7)
Sǫ(ν) and autocorrelation function rǫ, for which inf(Sǫ(ν)) > 0 k P(νk)
and |r (k)| < ∞. The N amplitudes α , frequencies f and
phaseskφ rǫepresenttheplanetssignaturesandaireunknown. i As the numerator and deenominator are independent variables with
i
P knownasymptoticdistributions,assessingthedistributionoftheirra-
3. PERIODOGRAMS’STATISTICS:ASYMPTOTICS tioisstraightforward.Theratiooftwoindependentrandomvariables
3.1. Classical(Schuster’s)periodogram (r.v.)V1∼χ2d1 andV2∼χ2d2 followsaFisher-Snedecorlawnoted
Without loss of generality and tosimplify thepresentation, N F(d1,d2)with(d1,d2)degreesoffreedom: VV21//dd21∼F(d1,d2)[26].
isevenandtheconsideredfrequenciesinEq.(1)belongtothesub-
Consequently,from(3)and(6),theasymptoticdistributionofP un-
set of N2 − 1 Fourier frequencies νk = {Tk} where k ∈ Ω := derH0is:
b{u1t,.in.c.o,nN2sis−ten1t}e.sAtismymateptooftitchaellPy,StDhe[2p4e]r.ioHdoowgreavmer,Punisdearntuhnebaibaosevde P(ν |H )∼ Sǫ(νk)χ22/2 ∼F(2,2L), ∀k ∈ Ω. e (8)
assumptionsonǫ,theperiodogramordinatesatdifferentfrequencies k 0 S (ν )χ2 /2L
ǫ k 2L
νkandνk′ areasymptoticallyindependent[18]. e
UnderH ,theasymptoticdistributionofP is([24],theorem5.2.6): Similarly, under H , we deduce the asymptotic distribution of P
0 1
with(4)and(6):
S (ν )
P(ν |H )∼ ǫ k χ2, ∀k ∈ Ω. (3) χ2 /2 e
k 0 2 2 P(ν |H )∼ 2,λk ∼F (2,2L), ∀k ∈ Ω, (9)
k 1 χ2 /2L λk
Theχ2isaχ2atk=0,N.WerestricttoΩforsimplicity. 2L
Under2H1, th1edistributio2n of P(νk) can be found in the complex withλk geiven by (5). These results call for two remarks. First, an
casein[18],Corollary6.2.UsingthiscorollaryandEuler’sformula F distributionsimilartothatof(8)wasobtainedin[27]whenusing
for the sines in model (2), we obtain that, for large N, P(νk) is ratiosoftheformP(νk)/P(νl),k 6= l(forsymmetrytestingpur-
approximatelydistributedas: poses). Second, (8) shows that the distribution of the standardized
periodogram isindependent ofthe(partiallyunknown) noisePSD,
P(ν |H )∼ Sǫ(νk)χ2 , ∀k ∈ Ω, (4) whichisindeedanecessaryconditionforarobustdetectiontest.
k 1 2 2,λk
4. STATISTICALTESTS
withλ =λ(ν )anon-centralityparameterexpressingthespectral
k k
leakageofallsignalfrequenciesatlocationν .Generally,thislea- The first three tests below are considered for comparison pur-
k
kageiscausedbythefactthatthe{f }donotbelongtothenatural poses.Thethreeothertestsarethecounterpartofclassicaltestsap-
i
2 pliedtoP insteadofP (manymoresuchtestscouldbedevised).
FouriergriddefinedbyΩ.DenotingbyK (ν)= sin(Nπν) the
N Nsin(πν) Inthefollowing,theorderstatisticsoftheperiodogramswillbeno-
Feje´rkernel(orspectralwindow),theasymptotice(cid:16)xpression(cid:17)ofthe tedwitheparentheses.Forinstance,theorderstatisticsofP are:
parametersλk,k∈Ωformodel(2)writes: mkinP(νk)=P(1) <P(2) <...<P(N2−1) =mkaxP(νk).
N Ns
λ(νk)= 2Sǫ(νk)Xi=1α2ihKN(fi−νk)+KN(fi+νk) 4.1.PCerlhaaspsiscathletemstsost classical reference test (including in Astro-
+2Nsins(iNn(ππ((ffii−−ννkk))))Nsins(iNn(ππ((ffii++ννkk))))cos(2π(N+1)fi+2φi)i. nomy)istheFisherHt1est: P(N−1)
(5) T ≷ γ, with T := 2 , (10)
Fi Fi
3.2. Averagedperiodogram H0 P(k)
WeassumethatatrainingdatasetT ofindependentrealizations whereγ ∈ R+ isathresholdthatdetermiknXe∈sΩtheFArate.Thistest
ofthestellarnoiseisavailable.ThissetisobtainedbyHDsimula-
is robust to an unknown noise variance [28], but the noise must
tionsandcomposedwithLtimesseriesX sampledonthesamegrid
ℓ bewhiteGaussian. Whenthisisthecase, thedenominator of (10)
astheobservations:T ={X (t ),j =1,...,N},ℓ=1,...,L.
ℓ j is an asymptotically unbiased estimate of the PSD (to a constant)
Astraightforward,consistentandunbiasedestimateofthenoisePSD
[11]. Under H ,for amodel involvingasinglesinusoid f onthe
canbeobtainedbytheaveragedperiodogram[25]: 1 1
Fourier grid, the Generalized Likelihood Ratio (GLR) test corres-
P(ν|H ):= 1 L 1 N X (t )e−i2πνtj 2, pboensdesentoasPa(Ns/t2a−nd1)ar≷diHHza10tγion[2o9f].thSeo,GuLnRdebryHth0e, tehsetimFiastheedrvtaersitancacne
0 L N ℓ j σˆ2:∝ P(ν ).
Xℓ=1 (cid:12)Xj=1 (cid:12) k∈Ω k
(cid:12) (cid:12) Inthecaseofmultiplesinusoids(N >1),theperformancesof
whoseasymptoticdistributioncan(cid:12)beeasilyobtainedus(cid:12)ing(3)as: P s
theFishertestareknowntodecrease,owingtotheperturbationsof
sinusoidsinthenoisevarianceestimation.Manyalternativesexist,
S (ν )
P(νk|H0)∼ ǫ2Lk χ22L, ∀k ∈ Ω. (6) e.g.[30,11].Thesetestsattempttobetterestimatethenoiselevelby
excludinganumberN ofordinatespresumablycontaminatedbythe r.v.areindependentandfollowaF distributionwithknowndensity
c
sinusoids(see[11,18,24]).Twosuchtests,offeringthesamerobust- ϕ [26].UsingtheincompleteBetafunctionB(d ,d )andnoting
F 1 2
nessagainstunknownvariance,aretherobustFishertest[18]andthe thatB(1,L)= 1(1−t)L−1dt= 1 weobtain:
Chiutest[7]definedbyT ≷H1γandT ≷H1γ,where: 0 L
Fi,rob H0 Ch H0 R 1 1 γ −L−1 γ −L−1
T :=b (N −1)r P(N2−1) , T := P(N2−Nc) , ϕF(γ,2,2L)= B(1,L) · L · 1+ L = 1+ L .
Fi,rob r 2 N2−1−Nc Ch N2−1−Nc It can be checked that the norm(cid:16)of ϕ (cid:17)(γ,2,2L)(cid:16)is 1. Tu(cid:17)rning to
P P F
(k) (k) theCDFΦ (γ,2,2L)ofT(n),weobtainbyusingF(γ,d ,d )=
F 1 2
Xk=1 Xk=1 (11) I d1γ (d21,d22)[26]orbydirectintegrationofϕF(γ,2,2L):
wthiatththre=lastN2tw−N21o−−a1Npcpraonadchbersp=os1e+therq−u1e(s1ti−onro)flothge(1ch−oicre)o.fNNotce. d1γ+d2 ΦF(γ,2,2eL)=1− γ+LL L. (16)
NotfixingNcinadvancebutestimatingthisparameterfromthedata ThePFAcanbecomputedthankstothe(cid:16)asympt(cid:17)oticindependenceof
mayleadtoamorepowerfultest,butatthecostofaweakercontrol
the{T(n)}:
of the FA rate (as N is random). To avoid this complication, the
c
testswillbecomparedinthenumericalstudyforN settoN .
c s PFA(eT,γ):=Pr(maxT(n)>γ|H0)=1− Pr(T(n)≤γ|H0)
n
4.2. Testsbasedonthestandardizedperiodogram n
Y
UnderH0 andintheasymptoticregime,P in(7)isi.i.d.since =1−e Φ (γ,2,2L) eN2−1 =1− 1− L Le N2−1,
theratiocancelsoutthefrequencydependenceonthePSD. F γ+L
IfNs =1,thefollowingsimpletestisthuslikeelytobepowerful: (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) (17)
whichisaremarkablysimpleexpression.
H1
T H≷0γ, with T :=P(N2−1). (12) Wtipeletusrinnunsoowidstodtehteectteiostns).TTFhi,eroabs,yTmCphtoatnicddTiNstcrib(duetisoignnseodfftohremfiursl-t
Similarly, the disecussion above sugegestseto consider Fisher’s ap- twotestsareknownundertheWGNassumpteion[7].Usingthesame
proachandtoapplytest(10)to(7): reasoningasaboveforT ,itiseasytoseethatthenoisePSDaf-
Fi
H1 P(N−1) fects the distribution of the order statistics involved in these tests,
TFiH≷0γ, with TFi:= e 2P(k). (13) twhiethnuumncboenrtrKolloafbolerdiimnaptaecstPonlathrgeePrFthAa.nTγurfnoilnlogwtosaTNbicn,ormemiaalrdkistthrai-t
e e kX∈Ω bution:K ∼B(N −1,1−Φ (γ,2,2L)),fromwehich,with(16),
Finally,inthecaseof several sinusoids, thegooed resultsof Chiu’s 2 F
weobtain: e
testsuggesttolookfordeviationsintheregionoftheNcthlargestP Nc−1
ordinate,P(N2−Nc),insteadofthelargestone,P(N2−1): e PFA(TNc,γ):=Pr(TNc>γ|H0)=1− i=0 Pr(K=i)
e TNcHH≷10γ, with TNc :=P(N2e−Nc). (14) =1e−Nc−1 N/2e−1 L Li 1X−( L )L N2−1−i.
i ! γ+L γ+L
5. STATISTeICSUNDERH0AeNDFALeSEALARMRATE Xi=0 (cid:16) (cid:17) (cid:16) (cid:17) (18)
Theaccuratecontrolofthefalsealarmprobability(PFA)incase 6. STATISTICSUNDERH1ANDDETECTIONRATE
ofpartiallyunknowncolorednoiseisacriticalpoint.Wenowshow
Under H , the same approach shows that one can not control
thatwhilethiscontrolis(notsurprisingly)problematicwithclassical 1
testslike(10,11),theproposedtests(12,14)allowsuchacontrol. thedistributionofther.v.TFi(n)andTFi(n)andconsequentlythe
associateddetectionprobabilities(P ).Incontrast,from(9)and
WeinvestigatefirstthetestsT ,T andT (designedforsinglesi- DET
Fi Fi
nusoiddetection).UnderH0,theirteststatisticsinvolvethelargest usingthesamereasoningasforthePFAe,thePDETofT is:
valueofasetof N −1r.v.,T (ne),T (ne)andT(n),whosedefi-
nitionsanddistrib2utionsaregiFveinby,uFsiing(4)and(8): PDET(T,γ):=Pr(mnaxP(n)>γ|H1)=1− ΦFλek(γ,2,2L).
TFi(n) := kX∈PΩ(νPn()νk) as,n∼i.eni.d. kX∈χΩ22Sχeǫ22(Snǫ)(/k2)/2, With(1e7)tγh(eTre,PlaFtiAo)ns=hiLpeγ(1P−FA()1f−orPTeFAc)aηn1 b−ekYd∈L1eΩ−riv1ed,as: ((1290))
TeFi(n):= XPe(νPn()νk) as,n∼i.i.d. XF(F2,(22L,2)L), (15) whereη= N2 e−1.With(1h9(cid:16))and(20),wede(cid:17)duce: i
k∈Ω e k∈Ω
T(n) := P(n) as,∼i.i.d. F(2,2L), PDET(T,PFA)=1− YΦFλk(L[(1−(1−PFA)η1)−L1 −1],2,2L),
e k∈Ω
e e (21)
where“as,ni.ni.d.”meansasymptoticallynonindependent andnon
whichcanbeusedtocomputetheROCcurves.
identicallydistributed.Inthefirstcase,eachr.v.T (n)clearlyde-
Fi
Thefunctionγ 7→ P (γ)ofthetestT canbededuced simi-
pends on the unknown PSD. In the second case, each r.v. T (n) DET Nc
Fi larlyto(18),withthedifferencethatKisnolongerbinomialowing
isaratioofaF variableoverasumofF variables.Toourknow-
ledge,thereisnoanalyticcharacterizationoftheresultingdiestribu- to the λk. Denote by Ω(i) one particularecombination of i indices
tion. Hence, in both cases, assessing the distribution of the maxi- taken in Ω and Ω(i) := Ω\Ω(i) the set of remaining indices. Let
mumof these variablesisproblematic. Inthethirdcase, the T(n) {Ω(1i),...,Ω(ii)}(resp.{Ω(1i),...,Ω(Ni)−1−i})denotetheindicesin
2
e
{twΩo(is)u}c(hrecsopm.obfinalalti{oΩns(,i)a}n).dWleitthΩtihe(sreesnpo.taΩtiio)nbsewteheobsteatinof:all the /H]z 102 NSioginseal 0.18 Te (L=5)
PDET(TeNc,γ)=1−NXic=−01ΩX(i)kY=i1(cid:16)1−ΦFλΩ(γ(k,i2),2L)(cid:17)N2kY−′=11−iΦFλΩ(γ(k,i′2),2L(2)2,) 22νgS()[m/s 110001 PDET000...246 TTTeeeFF(iiL((LL===10510)0)0)
which is typographically heavy but can be used to compute ROC lo 10-10 2 4 6 8 00 0.5 1 TeNc (L=5)
curves(inEq.(22)thenon-centralityparametersaregivenby(5)). ν[mHz] PFA TeNc (L=100)
7. NUMERICALSIMULATIONS TFi Te TFi
ForthepurposeofmakingempiricalROCcurves,wefirstconsi-
deranoisecorrespondingtoanautoregressiveprocess:AR(6).The 400 30 TFi,rob
shape of its PSD is representative of a stellar PSD, with local va- # 200 #1200 TCh
riations (stellar oscillations) and higher energy at low frequencies 0 0
0 2 4 6 8 0 2 4 6 8
(convection, magnetic activity). Under H1, we consider the most ν[mHz] ν[mHz]
generalcaseNS > 1.Firstofall,weevaluatethereliabilityofthe Fig.2.(a)PSDofARnoise(blue)andsines(red).(b)ROCcurves
PFA(γ),PDET(γ)andPDET(PFA)expressionsfortheT andTNc incaseofsignalfrequencies fallingintothePSD“valley”region:
tests(Fig.1).Wecomparehereourtheoreticalresults(5),(17),(18), N =N =5,α =0.08m/sforallsines(theapparentamplitude
c s i
(19), (21) and (22) with 104 Monte Carlo (MC) simulaetions. Tehe differenceiscausedbythedifferentleakage affectingthesesines),
theoretical expressions are in agreement with the MC simulations f = [5,5.5,5.75,6,6.5]mHz,with104 simulations, N = 1024,
i
andthetestperformanceslogicallyincreasewithL(astheestima- ∆t=60s,L= [5,100].(c,d)Histogramsofthefrequencydistri-
tionnoisedecreases).
butionunderH ofT =maxT (n)andT =maxT(n).
0 Fi Fi
100 100 Finally, we apply the tests n(12) and (14) to realnsolar data [31]
(Fig.3). These data have been collected for 1e8 years weith an even
10-1 10-1 sampling rate. As this data set is the largest one currently avai-
A T lable we use it to compute ROC curves on shorter time series of
PF10-2 DE10-2 N = 1000 samples extracted from the data set. In order to be as
P
10-3 10-3 closeaspossibletotheconsideredsetting(perfectHDsimulations
ofthestellarnoise),weusepartofthedatatostandardizetheperio-
10-4 101 γ 102 10-4 101 γ 102 d(bolgurea)mwiinth(1th).eTihnetroledfutcpeadnesligonfaFlsig(.r3eds)u.pTerhiempriogshetspthaneeslodlaisrpPlaSyDs
theROCscurves.TheproposedstandardizedtestsT,T andT
1 e Fi Nc
0.9 Empirical T (L=5) (black,yellowandred)aremorepowerfullthantheothers.
00..78 TEmhepoirreictiaclalTTee ((LL==150)0) 102 Noise 1 e e Te (eL=5)
PDETFig000000.......12345601. ValidatPio0Fn.5Aof the the1oretical reTETETsmmhhhueeelpptooosiirrrrreeeiifcctttoiiiaacccrllaaaTlllTTTTTeeeeeaNNNNnccccd(((((LLLLLT=====Nc151510)0)0b000y)))MC 22νlogS()[m/s/H]z110001 Signal PDET0000....2468 TTTTTTeeeeeFFNNFiiicc(((((LLLLL=====252520)0)0)))
simulations.Parameters:N = 1024,∆t = 60s,Ns = Nc = 3, 0 TFi,rob
ILn=a s[5e,co1n0d0],exαpie=rim0e.1ntm, w/sefoprlaacllesitnhees,dfifife=ren[5t,es5i.g7n5a,l6ef.r5e0q]umenHcize.s 0 5 1ν0[mH1z5] 20 0 P0F.5A 1 TCh
Fig.3.TestonGOLFdata.Parameters:N =103,4452simulations
ina“valley”ofthenoisePSD(Fig.2.a).Wecalculatetheempirical
for L = 5and 1272 simulations for L = 20, ∆t = 20 s, N =
ROC curves (Fig.2.b) for the tests under study (see (10) to (14)). s
N =10,α =0.17m/s,{f }equallyspacedin[10;20]mHz.
Thefigureshowsthat theperformances of testsusingP arebetter c i i
thanthosebasedonP inthisconfiguration(violet,greenandcyan
8. CONCLUSION
curves on the diagonal). To gain more insight on the reelative tests
Wehaveinvestigatedadetectionmethodbasedonperiodogram
performances, wecompare thefrequency distributionunder H of
0 standardizationthrough HD-simulationtocounteract theimpactof
TFi and T (Fig.2.c, d). For TFi,the FA repartitionisnot uniform colored noise. We have analyzed its statistical performances and
and increases in the PSD regions of larger energy (ν < 1 mH ). highlightedtheshortcomingsoftestsignoringthefrequencydepen-
z
Whensignealsfrequencieshappentofallintothesezones,testsbased dence of the noise PSD. In contrast, the proposed standardization
onP arefavored,butwhentheyfalloutsidesuchregionstheirpower leadstoarobust detection method inthesensethat thePFA isre-
liableandindependentofthenoisePSD.Alltheoreticalresultshave
vanish. In contrast, the T test allows a good detectability over all
been obtained in the asymptotic regime, but they appears to be a
thefrequencyrange,withperformancesclosetotheasymptoticone goodapproximationforrelativelylowvaluesofN.
(L→∞,noestimationneoise)forL≈102.Inbrief,thetestsT and
T allowtocontrolpreciselytheP ,theyhavegoodpowerand Acknowledgement Wearegrateful toThales AleniaSpace, PACAregion
thNeicrperformancesincreasewithL. FA e andCNRSprojectDETECTION/Imag’Inforsupportingthiswork.
e
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