Clustered Calibration: An Improvement to Radio Interferometric Direction Dependent Self-Calibration

6.8 MB·English

by  Sanaz Kazemi

Mon.Not.R.Astron.Soc.000,1–18(2007) Printed11December 2013 (MNLATEXstylefilev2.2)
Clustered Calibration: An Improvement to Radio Interferometric
Direction Dependent Self-Calibration
1⋆ 2 1
S. Kazemi , S. Yatawatta , S. Zaroubi
1KapteynAstronomicalInstitute,UniversityofGroningen,P.O.Box800,9700AVGroningen,theNetherlands
2ASTRON,Postbus2,7990AADwingeloo,theNetherlands
3 11December 2013
1
0
2 ABSTRACT
Thenewgenerationofradiosynthesisarrays,suchasLOFARandSKA,havebeendesignedto
n
surpassexistingarraysintermsofsensitivity,angularresolutionandfrequencycoverage.This
a
J evolutionhasledtothedevelopmentofadvancedcalibrationtechniquesthatensurethedeliv-
eryof accurateresultsatthe lowestpossible computationalcost. However,the performance
3
ofsuchcalibrationtechniquesisstilllimitedbythecompact,brightsourcesinthesky,used

] ascalibrators.Itis importantto havea brightenoughsourcethatis welldistinguishedfrom
M the backgroundnoise levelin orderto achievesatisfactory resultsin calibration.This paper
presents“clusteredcalibration”asamodificationtotraditionalradiointerferometriccalibra-
I
. tion,inordertoaccommodatefaintsourcesthatarealmostbelowthebackgroundnoiselevel
h
intothecalibrationprocess.Themainideaistoemploytheinformationofthebrightsources’
p
measuredsignalsasanaidtocalibratefaintersourcesthatarenearbythebrightsources.Inthe
-
o casewherewedonothavebrightenoughsources,asourceclustercouldactasabrightsource
r that can be distinguished from backgroundnoise. For this purpose, we construct a number
t
s of source clusters assuming that the signals of the sources belongingto a single cluster are
a
corrupted by almost the same errors. Under this assumption, each cluster is calibrated as a
[
single source,using the combinedcoherenciesof its sourcessimultaneously.This upgrades

1 thepowerofanindividualfaintsourcebytheeffectivepowerofitscluster.Thesolutionsthus
v obtainedforeveryclusterareassignedtoeachindividualsourceinthecluster.We giveper-
3 formanceanalysisofclusteredcalibrationtoshowthesuperiorityofthisapproachcompared
3 to the traditional un-clustered calibration. We also provide analytical criteria to choose the
6
optimumnumberofclustersforagivenobservationinanefficientmanner.
0
. Keywords: methods:statistical,methods:numerical,techniques:interferometric
1
0
3
1
:
v 1 INTRODUCTION before imaging. We also consider the removal of bright sources
i fromtheobserveddatapartofcalibration,thatenableimagingthe
X Low frequency radio astronomy is undergoing a revolution as a
weakbackground sources.Forlowfrequencyobservationswitha
r new generation of radio interferometers with increased sensitiv- widefieldofview,calibrationneedstobedonealongmultipledi-
a ityandresolution,suchastheLOwFrequencyARray(LOFAR)1,
rectionsinthesky.Propercalibrationacrossthefullfieldofviewis
theMurchisonWide-fieldArray(MWA)2andtheSquareKilome-
requiredtoachievetheinterferometer’sdesiredprecisionandsen-
terArray(SKA)3 arebeingdevisedandsomearealreadybecom-
sitivitygivingushighdynamicrangeimages.
ingoperational.Thesearraysformalargeeffectiveaperturebythe
combinationofalargenumberofantennasusingaperturesynthe- Early radio astronomy used external (classical) calibration
sis (Thompsonetal. 2001). In order to achieve the full potential whichestimatestheinstrumentalunknownparametersusingara-
of such an interferometer, it isessential that the observed data is dio source with known properties. This method has a relatively
properlycalibratedbeforeanyimagingisdone.Calibrationofra- lowcomputationalcostandafastexecutiontime,butitgenerates
diointerferometersreferstotheestimationandreductionoferrors results strongly affected by the accuracy with which the source
introduced by the atmosphere and also by the receiver hardware, properties are known. In addition, existence of an isolated bright
source, which is the most important requirement of external cal-
ibration, in a very wide field of view is almost impractical, and
even when it does, external calibration would only give informa-
⋆ E-mail:[email protected]
1 http://www.lofar.org tionalong thedirectionof the calibrator.Theexternal calibration
2 http://www.mwatelescope.org wasthenimprovedbyself-calibration(Pearson&Readhead1984).
3 http://www.skatelescope.org Self-calibration has the advantage of estimating both the source
(cid:13)c 2007RAS
2 Kazemi et al.
and instrumental unknowns, without the need of having a prior putationallycomplex. Hierarchical clustering(Johnson 1967)isa
knowledgeofthesky,onlyutilizingtheinstrument’sobserveddata. well-knownclusteringapproachwithastraightforwardimplemen-
Moreover,itcanachieveasuperiorprecisionbyiteratingbetween tationsuitableforourcase.But,itscomputational costgrowsex-
theskyandtheinstrumentalparameters. ponentiallywiththesizeofitstargetdatasetwhichcanbeadis-
The accuracy of any calibration scheme, regardless of the advantagewhenthenumberofsourcesishuge.WeightedK-means
usedtechnique, isdeterminedbythelevelofSignaltoNoiseRa- clustering(Kerdprasopetal.2005;MacQueen1967)isalsooneof
tio (SNR). This limits the feasibility of any calibration scheme themostusedofclusteringschemesapplicableinclusteredcalibra-
toonlybright sources that haveahighenough SNRtobedistin- tion.Theadvantageofthisclusteringtechniqueisitslowcomputa-
guishedfromthebackgroundnoise(Bernardietal.2010;Liuetal. tionalcost,whichisproportionaltothenumberofclustersandthe
2009;Pindoretal.2010).Notethatinterferometricimagesarepro- sizeofthetargetdataset.Nonetheless,weemphasizethatthecom-
duced using the data observed during the total observation (inte- putational timetaken byanyof theaforementioned clusteringal-
gration) time. However, calibration is done at shorter time inter- gorithmsisnegligiblecomparedwiththecomputationaltimetaken
valsofthattotalduration.Thisincreasesthenoiselevelofthedata by the actual calibration. Therefore, we pursue all clustering ap-
for which calibration is executed compared to the one in the im- proachesinthispaper.However,theuseofFuzzyC-meanscluster-
ages.Inotherwords,therearesomefaintsourcesthatappearwell ing(Bezdek1981)inclusteredcalibrationrequiresmajorchanges
abovethenoiseintheimageswhiletheyarewellbelowthenoise inthecalibrationdatamodelandwillbeexploredinfuturework.
level during calibration.It hasbeen agreat challenge tocalibrate This paper is organized as follows: First, in section 2 we
suchfaintsourceshavingaverylowSNR.Inthispaperwepresent presentthegeneraldatamodelusedinclusteredcalibration.Insec-
clusteredself-calibration,introducedinKazemietal.(2011a),and tion3, wepresent modified weighted K-means and divisive hier-
emphasizeitsbetterperformancecomparedtoun-clusteredcalibra- archical clustering forclustering sources inthesky. Next, insec-
tionbelowthecalibrationnoiselevel.Existingun-clusteredcalibra- tion 4, we focus on analyzing the performance of clustered cali-
tionapproaches(Intemaetal.2009;Smirnov2011)canonlyhan- bration,withanaprioriclusteredskymodel,andcomparethere-
dleahandfulofdirectionswherestrongenoughsourcesarepresent sultswithun-clusteredcalibration.Inclusteredcalibration,thereis
andweimprovethis,inparticularforsubtractionofthousandsof contentionbetweentheimprovementofSNRbyclusteringsources
sources over hundreds of directions in the sky, as in the case for andtheerrorsintroducedbytheclusteringofsourcesitself.Thus,
LOFAR(Bregman2012). we relate the clustered calibration’s performance to the effective
Theimplementationof clusteredcalibrationisperformed by SignaltoInterferenceplusNoiseRatio(SINR)obtained foreach
clusteringthesourcesinthesky,assumingthatallthesourcesina cluster. For this purpose, we use statistical estimation theory and
singleclusterhavethesamecorruptions,andcalibratingeachclus- theCramer-RaoLowerBounds(CRLB)(Kay1993).Insection5,
terasasinglesource.Attheend,theobtainedcalibrationsolutions wederivecriteriaforfindingtheoptimumnumberofclustersfora
foreverysourceclusterisassignedtoallthesourcesinthatcluster. givensky.WeusetheSINRanalysisandadoptAkaike’sInforma-
Thisprocedureimprovestheinformationusedbycalibrationbyin- tionCriterion(AIC)(Akaike1973)andtheLikelihoodRatioTest
corporatingthetotalofsignalsobservedateachclusterinsteadof (LRT)(Graves1978)toestimatetheoptimumnumberofclusters.
eachindividualsource’ssignal.Therefore,whenSNRisverylow,it Wepresentsimulationresultsinsection6toshowthesuperiority
providesaconsiderablybetterresultcomparedtoun-clusteredcal- ofclusteredcalibrationtoun-clusteredcalibrationandtheperfor-
ibration.Moreover, calibratingforasetofsourceclustersinstead mance of thepresented criteriaindetecting theoptimum number
offor alltheindividual sourcesintheskyreducesthenumber of ofclusters.Finally,wedrawourconclusionsinsection7.Through
directionsalongwhichcalibrationhastobeperformed,thuscon- this paper, calibration is executed by the Space Alternating Gen-
siderably cutting down the computational cost. However, there is eralizedExpectationmaximization(SAGE)(Fessler&Hero1994;
onedrawbackinclusteredcalibration:Thecorruptionsofeachin- Yatawattaetal. 2009; Kazemietal. 2011b) algorithm. Moreover,
dividualsourceinonegivenclusterwillalmostsurelybeslightly in our simulations, radio sources are considered to be uniformly
different from the corruptions estimated for the whole cluster by distributedintheskyandtheirfluxintensitiesfollowRaleighdis-
calibration. Wecallthisadditional error as“clustering error”and tribution,whichistheworstcasescenario.Inrealskymodels,there
in order to get the best performance in clustered calibration, we usually exist only a few (two or three) number of bright sources
shouldfindtherightbalancebetweentheimprovementinSNRas whichdominatetheemission.Inthepresenceofthesesourcesand
opposedtodegradationbyclusteringerror. thebackground noise, itisimpracticaltosolvefortheotherfaint
Recently,clusteringmethodshavegainedalotofpopularityin sourcesinthefieldofviewindividually.Therefore,obtainingabet-
dealingwithlargedatasets(Kaufman&Rousseeuw1990).There tercalibrationperformanceviatheclusteredcalibrationapproach,
arevariousclusteringapproachesthatcouldbeappliedtocalibra- comparedtotheun-clusteredone,isguaranteed.Therefore,insec-
tionbasedontheirspecificcharacteristics.Anoverviewofdifferent tion6weillustratethisusingsimulationsinwhichthebrightness
clusteringmethodsisgiveninXu&Wunsch(2008).Clusteringof distributionofsourcesisapowerlawwithaverysteepslope.On
radio sources should take into account (i) their physical distance topofthat,Kazemietal.(2012);Yatawattaetal.(submitted2012)
fromeachotherand(ii)theirindividualintensity.Thesmallerthe alsopresenttheperformanceofclusteredcalibrationonrealobser-
angular separations of sources are, the higher the likelihood that vationsusingLOFAR.
they share the same corruptions in their radiated signals. More- The following notations areused inthispaper: Bold, lower-
over, in order to get the best accuracy in the calibration results, caselettersrefertocolumnvectors,e.g.,y.Uppercaseboldletters
thereshouldbeabalancebetweeneffectiveintensitiesofdifferent refertomatrices,e.g.,C.Allparametersarecomplexnumbers,un-
clusters.Thus,intheclusteringprocedure,everysourceshouldbe lessstatedotherwise.Theinverse,transpose,Hermitiantranspose,
weightedsuitablyaccordingtoitsbrightnessintensity. andconjugationofamatrixarepresentedby(.)−1,(.)T,(.)H,and
The brightness distribution of radio source in the sky is a (.)∗,respectively.Thestatisticalexpectationoperatorisreferredto
powerlawandthespatialdistributionisPoisson.Therefore,clus- asE . .ThematrixKroneckerproductandtheproper(strict)sub-
{}
tering thesources via probabilistic clustering approaches iscom- setaredenotedby and(,respectively.I isthen nidentity
n
⊗ ×
(cid:13)c 2007RAS,MNRAS000,1–18
ClusteredRadioInterferometricCalibration 3
matrixand∅istheemptyset.TheKroneckerdeltafunctionispre- andfrequency resolutionisperformedtoestimateand correctfor
sentedbyδ .RandCarethesetsofRealandComplexnumbers, G -sandthecorrectedvisibilitiesareobtainedas
ij p
respectively.TheFrobeniusnormisshownby . . Estimatedpa-
rametersaredenotedbyahat,(.).Alllogarithm||ic||calculationsare Vpq =G−p1VpqG−qH. (4)
to the base e. The multivariate Gaussian and Complex Gaussian Theremainingerrorsareuniquetoagivendirection,butresidual
e
distributionsaredenotedby cand ,respectively. errors in G -s are also absorbed into these errors, which are de-
N CN p
notedbyJ intheusualnotation.
pi
VectorizingEq.(4),thefinalvisibilityvector ofthebaseline
p qisgivenby
−
2 CLUSTEREDSELF-CALIBRATIONDATAMODEL
K
In this section, we present the measurement equation of a po- vpq = J∗qi(θθθ)⊗Jpi(θθθ)vec(Ci{pq})+npq. (5)
larimetric clustered calibration in detail (Hamakeretal. 1996; i=1
X
Hamaker 2006).Suppose wehave aradiointerferometer consist- Stacking up all the cross correlations (measured visibilities) and
ing of N polarimetric antennas where each antenna iscomposed noise vectors in y and n, respectively, the un-clustered self-
oftwoorthogonaldual-polarizationfeedsthatobserveKcompact calibrationmeasurementequationisgivenby
sources in the sky. Every i-th source signal, i 1,2,...,K ,
causes an induced voltage of vpi = [vXpi vY∈pi]T{ at X and Y} K
dipolesofeveryp-thantenna,p 1,2,...,N .Inpractice, y= si(θθθ)+n. (6)
∈{ } i=1
e X
vpi =Jpiei, (1) InEq.(6),y,n CM,M =2N(N 1),thenoisevectoriscon-
∈ −
wJphierreepereise=nts[etXhei2eYi]2T Jieosntehsemsoautrricxe’(sHealmecatkreicrefitealdl.v1e9c9t6o)rcaonrd- ΠΠsΠid,enre∼dtNoh(0av,ΠΠeΠaMz×eMro)m,aenadntGheaunsosnialinnedaisrtfruibnucttiioonnwsii(thθθθ)ciosvdaerifiannecde
× as
respondingtothedirection-dependentgaincorruptionsintheradi-
a(awttghheaeditcisbohienglaan(mtaterlro.sopThnoah,spepiesnh,eetschryioicssrtresauemnpcdttiifoioronennsq,ouawserpienlhcloeybrrieigrceeinsxdpapiotsletanodisnreftei,rodoenmitsnc,.t)mheetaocnir.n)edspdttrrheuotemapiepler.rntoitpeas-l si(θθθ)≡ JJ∗2∗3ii((θθθθθθ))⊗⊗JJ11ii((θθθθθ...θ))vveecc((CCii{{ppqq}})) . (7)
The signal vp obtained at every antenna p is a linear su-  J∗Ni(θθθ)⊗J(N−1)i(θθθ)vec(Ci{pq}) 
perposition of the K sources corrupted signals, v where i  
pi ∈ CalibrationisessentiallytheMaximumLikelihood(ML)esti-
1,2,...,K ,plustheantenna’sthermalnoise.Themultitudeof
{ } mationoftheunknownparametersθθθ(P complexvaluesor2P real
ignoredfaintersourcesalsocontributestothisaddietivenoise. values),oroftheJonesmatricesJ(θθθ),fromEq.(6)andremovalof
Thevoltagescollectedattheinstrumentantennasgetcorrected
theKsources.Notethatcalibrationmethodscouldalsobeapplied
forgeometricdelays,basedonthelocationoftheirantennas, and
totheuncorrectedvisibilitiesof(4)toestimateG andG errors
p q
someinstrumentaleffects,liketheantennaclockphasesandelec-
aswell.
tronicgains. Then, they arecorrelated inthearray’s correlator to
TheJonesmatrixJ ,foreveryi-thdirectionandateveryp-th
pi
generatevisibilities(Hamakeretal.1996).Thevisibilitymatrixof
antenna,isgivenas
thebaselinep−q,E{vp⊗vqH},isgivenby
J E Z F . (8)
pi pi pi pi
K ≡
Vpq =Gp Jpi(θθθ)Ci{pq}JHqi(θθθ) GHq +Npq. (2) In Eq. (8), Epi, Zpi, and Fpi are the antenna’s voltage pattern,
i=1 ! ionospheric phase fluctuation, and Faraday Rotation Jones matri-
X
ces,respectively. Inpractice,theE,Z,andFJonesmatricesob-
In Eq. (2), θθθ CP, P = 4KN, is the unknown instrumental
∈ tainedfornearbydirectionsandforagivenantennaarealmostthe
andskyparametervector,Npq istheadditive2 2noisematrix
× same.Thus,foreveryantennap,ifthei-thandj-thsourceshavea
of thebaseline p q,andC istheFouriertransformof the
i-th source cohere−ncy matrixi{Cpqi}= E{ei ⊗eHi } (Born&Wolf smallangularseparationfromeachother,wemayconsider
i1s9I9i9,;thHeanmCakie=retI2ailI.21.9C9o6n).siIdferthinegit-hthisssoouurrcceertaodhiaatvioeneqinutaetnosriitayl Jpi∼=Jpj. (9)
coordinate, (RightAscensionα,Declinationδ),equal to(αi,δi), Thisistheunderlyingassumptionforclusteredcalibration.
and the geometric components of baseline p q tobe (u,v,w), Clusteredcalibrationfirstassignssourceclusters,Li fori
then − 1,2,...,Q whereQ K,onwhichtheskyvariationisconsid∈-
{ } ≪
eredtobeuniform.Then,itassumesthereexistsauniqueJ which
Ci{pq} =e−2π(ul+vm+w(√1−l2−m2−1))Ci, (3) issharedbyallthesourcesofthei-thclusterLi,i 1,2,p.i..,Q ,
∈{ }
atreceiverp,p 1,2,...,N .Basedonthat,thevisibielityatev-
where2 = 1and, ∈{ }
− erybaselinep q,givenbyEq.(4),isreformulatedas
−
l=sin(αi−α0)cos(δi), Q
m=cos(δ0)sin(δi)−cos(αi−α0)cos(δi)sin(δ0), Vpq = Jpi(θθθ){ Cl{pq}}JHqi(θθθ)+Npq. (10)
arethesourcedirectioncomponentscorrespondingtotheobserva- Xi=1 lX∈Li
e e e e e e
tionphasereferenceof(α0,δ0)(Thompsonetal.2001).Theerrors In Eq. (10), Npq is the clustered calibration’s effective noise at
commontoalldirections,suchasthereceiverdelayandamplitude baselinep qwhichwillbeexplicitlydiscussedatsection4.Note
−
errors,aregivenbyGp andGq.Initialcalibrationatafinertime that clusteredecalibration estimates the new unknown parameter
(cid:13)c 2007RAS,MNRAS000,1–18
4 Kazemi et al.
θθθ CPe where P = 4QN. Denoting the effective signal of ev- For defining thecentroids, weassociate aweight toevery source
∈
eryi-thclusteratbaselinep qby xi,as
−
e e I
Ci{pq} ≡ Cl{pq}, (11) wi =w(xi)≡ I∗i, fori∈{1,2,...,K}, (16)
e lX∈Li whereIiisthesource’sintensityandI∗ =min I1,I2,...,IK .
theclusteredcalibrationvisibilityvectoratthisbaseline(vectorized { }
Applyingtheweightstotheclusteringprocedure,thecentroidsof
formofEq.(10))is
theclustersleanmostlytowardsthebrightestsources.Thatcauses
a tendency in faint sources to gather with brighter sources close
Q
vpq = J∗qi(θθθ)⊗Jpi(θθθ)vec(Ci{pq})+npq. (12) tboeitnhgemaddiendtouponweitchlusostmere.bTrhiguhst,erthseoiurrwceesa’ksigsinganlas.lsMaorreeopvroerm,voeterdy
i=1
X
Finally,stackingupteheveisibilietieseofalltheeinstrumeent’sbaselines strongsourceswillbeisolatedsuchthattheirsignalsarecalibrated
individually,withoutbeingaffectedbytheotherfaintsources.
invector y,clusteredcalibration’sgeneral measurement equation
We cluster radio sources using weighted K-means
isresultedas
(Kerdprasopetal. 2005) and divisive hierarchical clustering
Q (Johnson 1967) algorithms. Since the source clustering for cal-
y= si(θθθ)+n. (13) ibration is performed offline, its computational complexity is
Xi=1 negligiblecomparedwiththecalibrationprocedureitself.
InEq.(13)s isdefinedsimilarteoseinEeq.(7)whereJandCare
i i
replacedbyJandC,respectively.
3.1 WeightedK-meansclustering
Becausee of the similarity between the clustered and the un-
clustered caelibratioen’s measurement equations presented by Eq. Step1.SelecttheQbrightestsources,x1∗,x2∗,...,xQ∗,andini-
(13) and Eq. (6), respectively, they could utilize the same cali- tializethecentroidsofQclustersbytheirlocationsas
bration techniques. Thus, the only difference between these two
typesofcalibrationisthatclusteredcalibrationsolvesforclusters cq ≡[αq∗,δq∗], forq∈{1,2,...,Q}, q∗ ∈{1∗,2∗,...,Q∗}.
(17)
ofsourcesinsteadoffortheindividualones.Thatupgradesthesig-
Step2.Assigneachsourcetotheclusterwiththeclosestcentroid,
nalswhichshouldbecalibratedforfromEq.(3)toEq.(11).
definingthemembershipfunction
mLq(xi)=(10,, iOfthde(rxwi,isceq)=min{d(xi,cj)|j =1,...,Q}
3 CLUSTERINGALGORITHMS
Step3.Updatethecentroidsby
Clusteringisgroupingasetofdatasothatthemembersofthesame
g1r9o9u0p).(Tchluisstesirm) ihlaarvietysiosmdeefisniemdilbaaristeiedso(nKtahuefmapapnl&icaRtioounssoefeuthwe cq = PKi=Ki=11mmLLqq(x(xi)i)wwixii, forq∈{1,2,...,Q}. (18)
clusteringmethod.
We need to define clustering schemes in which two radio RepeatstepPs2and3untiltherearenoreassignmentsofsourcesto
clusters.
sources merge to a single cluster based on the similarity in their
direction dependent gain errors ( Eq. (8) ). Radiations of sources
that are close enough to each other in the sky are assumed to be 3.2 Divisivehierarchicalclustering
affectedbyalmostthesamecorruptions(Eq.(9)).Basedonthat,
weaimtodesignsourceclusterswithsmallangulardiameters.On Step 1. Initialize the cluster counter Q′ to 1, assign all the K
theotherhand,everycluster’sintensityisthesumoftheintensities sourcestoasingleclusterL1and∅toasetofnullclustersA.
ofitsmembers(Eq.(11)).Inordertokeepabalancebetweendif- Step2.ChooseclusterLq∗,forq∗ 1,2,...,Q′ A,withthe
∈{ }−
ferentclusters’intensities,weintendtoapplyweightedclustering largestangulardiameter
techniquesinwhichthesourcesareweightedproportionaltotheir D(Lq∗)=max D(Lq)q 1,2,...,Q′ A . (19)
intensities. { | ∈{ }− }
Suppose that the K sources, x ,x ,...,x have Step 3. Apply the presented weighted K-means clustering tech-
1 2 K
(α1,δ1),(α2,δ2),...,(αK,δK) equatorial coordinates, re- niquetosplitLq∗ intotwoclusters,L′q∗ andL′q′∗.
spectively. The aim is to provide Q clusters so that the objective
fduianmcteiotenrfof=clusPteQqr=L1qD,f(oLrqq)∈is{m1,i2ni,m..iz.e,dQ.}D,d(Lefiqn)eidsatshe angular SLRteqe∗ppe≡a4t.LsItf′qe∗pD,s(L2LQ,′q3′∗,≡)an+LdD′q4′∗(u,Lnant′q′i∗dl)QA<′==D∅Q(,L.oqt∗h)e,rwthiesnesseettAQ′==AQ∪′{+q∗1},.
D(Lq) max d(xi,xj)xi,xj Lq , (14)
≡ { | ∈ }
andd(.,.)istheangularseparationbetweenanytwopointsonthe 3.3 Comparisonofclusteringmethods
celestialsphere.Havingtworadiosourcesaandbwithequatorial
Hierarchical clustering method tends to design clusters with al-
coordinates(αa,δa)and(αb,δb),respectively,theangularsepara-
mostthesameangulardiameters,whereas,theK-meansclustering
tiond(a,b),inradians,isobtainedby
methodtendstokeepthesamelevelofintensityatallitsclusters.
tan−1pcos2δbsins2in∆δaαs+inδ[bco+sδcaossiδnaδcbo−sδsbicnoδsa∆coαsδbcos∆α]2, Iinndperdacictiactein,gsitnhceeshaimerearscohluictiaolncslutostseoriunrgcemseinthsomdamllackleusstleersss,ietrproerrs-
(15) formsbetterthanWeightedK-meansclusteringinaclusteredcali-
where∆α=αb αa. brationprocedure.But,whenthenumberofsources(andclusters)
−
(cid:13)c 2007RAS,MNRAS000,1–18
ClusteredRadioInterferometricCalibration 5
42 42
41 41
40 40
g.]39 g.]39
e e
d d
δ [38 δ [38
37 37
36 36
35 35
278 276 274 272 270 278 276 274 272 270
α [deg.] α [deg.]
Figure1.Asimulated8by8degreesskyoffiftypointsourceswithinten- Figure 2. Fifty point sources are clustered into ten source clusters by
sitiesbelow3Jy.Thesourcepositions andtheirbrightnessarefollowing WeightedK-meansclustering technique. Thereisnotagoodbalancebe-
uniformandRayleighdistributions,respectively.Themarkersizesarepro- tweendifferentclustersangulardiameters.
portionaltosourcesintensities.
42
isverylarge(Q>100),itsprohibitivecomputationalcostsmakes
thefastK-meansclusteringmethodpreferable.
41
3.3.1 Example1:WeightedK-meansandhierarchicalclustering 40
Wesimulatean8by8degreesskywithfiftypointsourceswithin-
tensitiesbelow3Jansky(Jy).Thesourcepositionsandtheirbright- g.]39
nessfollow uniform andRayleigh distributions, respectively. The e
d
resultisshownbyFig.1inwhichthesymbolsizesareproportional δ [38
tointensitiesofsources.WeightedK-meansanddivisivehierarchi-
calclusteringmethodsareappliedtoclusterthefiftysourcesinto 37
tensourceclusters.TheresultsarepresentedinFig.2andFig.3,
respectively. Fig. 2 shows that the Weighted K-means clustering
36
coulddesignsourceclusterswithconsiderablylargeangulardiam-
eters. Assigning the same calibration solutions to the sources of
35
theselargeclusterscouldcausesignificanterrors.However,asFig. 278 276 274 272 270
3 shows, this isnot thecase for the hierarchical clustering and it α [deg.]
constructsclusterswithalmostthesameangulardiameters.
Sincethenumberofsourcesinthissimulationisnotthatlarge Figure3.Fiftypointsourcesareclusteredintotensourceclustersviahier-
(K = 50),thedifferencebetweenexecutiontimeofthetwoclus- archicalclusteringmethod.Differentclustershavealmostthesameangular
diameters.
teringmethodsisnotsignificant.Henceinsuchacase,theuseofhi-
erarchicalclusteringmethod,ratherthantheWeightedK-means,is
advised.However,thisisnotthecasewhenwehavealargenumber
SNR (Kazemietal. 2012). This superiority is in the sense of
ofsources,andsubsequentlyalargenumberofsourceclusters,in
achievinganunprecedentedprecisioninsolutionswithaconsider-
thesky.Todemonstratethis,weusethetwoclusteringtechniques
ablylowcomputationalcomplexity,giventheoptimumclustering
forclusteringthousandofsources(K =1000)intoQsourceclus-
scheme.Inthenextsection,wepresentdifferentcriteriaforfinding
ters,Q 3,4,...,100 .Themethods’computationaltimesver-
theoptimumnumberofclustersatwhichtheclusteredcalibration
∈{ }
susthenumber of clustersareplottedinFig.4.AsFig.4shows,
performsthebest.
forlargeQs,thecomputationalcostofWeightedK-meansismuch
cheaperthantheoneofthehierarchicalclustering.Thatcanmake
the Weighted K-means clustering method more suitable than the 4.1 Cramer-RaoLowerBounds
hierarchicalclusteringforsuchacase.
The most fundamental assumption in clustered calibration is that
thesourcesatthesamecluster haveexactlythesamecorruptions
in their radiated signals. This assumption is of course incorrect,
4 PERFORMANCEANALYSIS
nonetheless, it provides us with a stronger signal, the sum of the
In this section, we explain the reasons for clustered calibration’s signalsinthewholecluster.Wepresentananalyticcomparisonof
betterperformance,comparedtoun-clusteredcalibration,atalow clustered and un-clustered calibration where we use the Cramer-
(cid:13)c 2007RAS,MNRAS000,1–18
6 Kazemi et al.
Inother words, thevarianceof anyunbiased estimatorof theun-

knownparametervectorθθθisboundedfrombelowbythediagonal
Weighted K−means
800 elementsof[ (θθθ)]−1.
Divisive Hierarchical I
s) Using(23)and(26),theFisherinformationmatrixofthevis-
me (600 ibilityvectoryisobtainedas
al ti I(θθθ)=2σ−2Re(JsHJs), (27)
on400 whereJsistheJacobianmatrixofswithrespecttoθθθ
ati
ut 2 ∂
mp200 Js(θθθ)= ∂θθθ{J∗qi⊗Jpi}[I4⊗vec(Ci{pq})]. (28)
o i=1
C X
Thus,variationsofanyunbiased estimatorofparameter vectorθθθ,
0
letssayθθθ,isboundedfrombelowbytheCRLBas

0 20 40 60 80 100
Number of clusters b Var(θθθ)>[2σ−2Re(JsHJs)]−1. (29)
Figure4.WeightedK-meansanddivisivehierarchicalclusteringmethods Letstrytoboundtheerrorvariationsoftheclusteredcalibra-
b
computational costs.Forsmallnumberofsourceclusters,thereisnodif- tion parameters assuming that the two sources construct a single
ferencebetweenexecutiontimesofthetwoclusteringmethods.But,when cluster,calledclusternumber1.WereformEq.(20)as
thenumberofsourceclustersislarge,thecomputationalcostofWeighted
K-meansbecomesmuchcheaperthantheoneofthehierarchicalclustering. Vpq =Jp1(C1{pq}+C2{pq})JHq1+Γ1{pq}+Γ2{pq}+Npq, (30)
where Γ , referred to as the “clustering error” matrices, are
ie{pq} e
Rao Lower Bound (CRLB) (Kay 1993) as a tool to measure the givenby
performanceofthecalibration. Γi{pq} =JpiCi{pq}JHqi−Jp1Ci{pq}JHq1, (31)
andJp1(θθθ)istheclusteredcalibrationseolutionatreeceiverp.
4.1.1 EstimationsofCRLBfortwosourcesatasinglecluster
Eq.(30)impliesthatwhatisconsideredasthenoisematrixN in
pq
Forsimplicity,firstconsiderobservingtwopointsourcesatasingle thecelusteeredcalibrationdatamodel,Eq.(10),isinfact
baseline,letssaybaselinep q.BasedonEq.(4),thevisibilities e
aregivenby − Npq ≡Γ1{pq}+Γ2{pq}+Npq. (32)
Vectorizing Eq. (30), the clustered calibration visibility vector is
Vpq =Jp1C1{pq}JHq1+Jp2C2{pq}JHq2+Npq, (20) obtainedby e
binilitthyevuenc-etcolruissteredcalibrationstrategy.VectorizingVpq,thevisi- y=J∗q1⊗Jp1vec(C1{pq}+C2{pq})+npq, (33)
y=J∗q1⊗Jp1vec(C1{pq})+J∗q2⊗Jp2vec(C2{peq})+npq. wherWeneppqo=intvoeeuct(Nthapteqd).ependingontheobservationeaswellasthe
Assumingnpq (0,σ2I4),wehave positionesofthetwoesourcesonthesky,theclusteringerrorΓi{pq}
∼CN willhavedifferentproperties.However,inordertostudytheper-
y (s(θθθ),σ2I4), (21) formance of theclusteredcalibrationinastatisticalsense, and to
∼CN
simplifyouranalysis,wemakethefollowingassumptions.
where
s(θθθ)≡ J∗qi(θθθ)⊗Jpi(θθθ)vec(Ci{pq}). (22) and(i)ovCerondsifidfeerrensttastkisytirceaalliezxaptieocntsatwiohneroevtehredsioffuerrceenstaorbesrearnvdaotimonlys
i=1,2
X distributedonthesky.Inthatcase,almostsurelyE J E J
UsingEq.(21),thelog-likelihoodfunctionofthevisibilityvector { } → { }
andconsequently
yisgivenby
e
E Γ 0. (34)
(θθθy)= 4ln π σ−2(y s(θθθ))H(y s(θθθ)). (23) { i{pq}}→
L | − {σ2}− − − Inotherwords,weassumetheclusteringerrortohavezeromean
CRLBisatightlowerboundontheerrorvarianceofanyun- overmanyobservationsofdifferentpartsofthesky.
biasedparameterestimators(Kay1993).Basedonitsdefinition,if (ii) Weassumethatthecloserthesourcesaretogetherinthesky,
thelog-likelihoodfunctionoftherandomvectory, (θθθy),satis- thesmallertheerrorsintroducedbyclusteringwouldbe.Therefore,
L |
fiesthe“regularity”condition givenasetofsources,theclusteringerrorwillreduceasthenum-
berofclustersincrease.Infactthiserrorintroducedbyclustering
∂
Ey ∂θθθL(θθθ|y) =0, forallθθθ, (24) vanishes when the number of clusters is equal to the number of
(cid:20) (cid:21) sources(eachclustercontainsonlyonesource).Therefore,givena
setofsources,thevarianceofΓ willdecreaseasthenumber
thenforanyunbiasedestimatorofθθθ,θθθ, i{pq}
ofclustersincrease.
var(θθθi)>[I−1(θθθ)]ii, fobri∈{1,...,M}, (25) Using Eq. (34) and bearing in mind that E Npq = 0, we
whereI(θθθ)istbheFisherinformationmatrixdefinedas can consider npq ∼ CN(0,σ2I4) where E{np{qnHpq}} = σ2I4.
Therefore,
∂2 (θθθy)
I(θθθ)=−Ey ∂Lθθθ∂θθθ|T . (26) e y e (s,σ2I4), e e e
(cid:20) (cid:21) ∼CN
e (cid:13)ec 2007RAS,MNRAS000,1–18
ClusteredRadioInterferometricCalibration 7
The CRLBs obtained for the un-clustered and clustered cal-
2000
ibrations in Eq. (29) and Eq. (35), respectively, are both almost
Clustered calibration equaltotheinverseoftheSignaltoInterferenceplusNoiseRatio
Un−clustered calibration (SINR), SINR−1. In un-clustered calibration, the effective signal
1500 for the faintest source is C where the noise is N . There-
2{pq} pq
fore,SINRforthissourceis
CRLB1000 SINR2 = ||CN2{ppqq}2||2. (36)
|| ||
But, in clustered calibration, the effective signal and noise are
500 C{pq} ≡ C1{pq} +C2{pq} and Npq, respectively. Thus, SINR
fortheclusteris
e e
C 2
00 1000 2000 3000 4000 5000 SINRc = || N{pq}|2| . (37)
Clustered calibration effective noise power ||e pq||
Clusteredcalibrationhasanimprovedperformancewhen
Figure5.Clusteredandun-clusteredcalibrationsCRLB.Whentheeffec- e
tivenoisepoweroftheclusteredcalibration,||Ne||2,issmallenough,then
SINR SINR . (38)
c 2
its CRLB is lower than of the un-clustered calibration’s and it reveals a ≫
superiorperformance. Considerthetwopossibleextremesinaclusteredcalibration
procedure:
s≡J∗q1⊗Jp1vec(C1{pq}+C2{pq}), sm(ai)llCnulumstbeerrinogfmclaunsytersso.uIrncetshiisncaaslea,rgtehefiaenldguolfarvideiwamteotearvoefrya
andsimilartoeEq.e(29),weehave clusterisprobablytoolargefortheassumptionofuniformcorrup-
tionstoapply.Subsequently,dedicatingasinglesolutiontoallthe
Var(θθθ)>[2σ−2Re(JesHJes)]−1. (35) sourcesofeveryclusterbyclusteredcalibrationintroducescluster-
ingerror matricesΓΓΓ witha largevariance (see Eq. (31)). Having
We use numerical sibemulationes to compare the un-clustered and high interference power, the clustered calibration effective noise
clustered calibrations performances via their CRLBs which are
NofEq.(32)becomesverylarge.Therefore,clusteredcalibration
givenbyEq.(29)andEq.(35),respectively.
SINRwillbeverylowanditdoesnotproducehighqualityresults.
e(ii) Clustering sources in a small field of view to a very large
numberofclusters.Inthiscase,thevarianceofΓΓΓmatricesareal-
4.1.2 Example2:CRLBfortwosourcesandonecluster
mostzerowhilethesignalpowersofsourceclustersarealmostas
Wesimulateatwelvehour observation of twopoint sources with lowasoftheindividualsources.Therefore,theSINRofclustered
intensitiesI =11.25andI =2.01Jyatskycoordinates(l,m) calibrationisalmostequaltotheun-clusteredcalibrationSINRand
1 2
equalto( 0.014, 0.005)and( 0.011, 0.010)radians,respec- the calibration performance isexpected tobe almost the same as
− − − −
tively.Weusetheuv-coverageofWesterborkSynthesisRadioTele- well.
scope(WSRT)with14receiversinthissimulation. Thus,thebestefficiencyofclusteredcalibrationisobtainedatthe
We Consider the J Jones matrices in Eq. (20) to be diagonal. smallestnumberofclustersforwhichEq.(38)issatisfied.Weuse
Their amplitude and phase elements follow (0.75,0.95) and theSINRofEq.(38)asanefficientcriterionfordetectingtheopti-
U
(0.003,0.004)distributions,respectively.Thebackgroundnoise mumnumberofclusters.
U
isN (0,10I). Jones matricesof the clustered calibration,
∼ CN
Jp1 for p = 1,2,areobtainedasJp1+ (0.02,0.40)ejU(0.5,2).
U 4.1.4 Generalizationtomanysourcesandmanyclusters
For20realizationsofJmatrices,wecalculatedCRLBoftheun-
celustered and clustered calibrations using Eq. (29) and Eq. (35), Forthevisibilityvectoryofun-clusteredcalibration’sgeneraldata
respectively.TheresuletsarepresentedinFig.5.Asshowninthis model,presentedbyEq.(6),wehave
figure, for small enough errors matricesΓΓΓ of Eq. (31), the clus-
tered calibration’s CRLB stands below the un-clustered calibra- K
tion’sCRLB.Ontheotherhand,withincreasingpoweroferrorma- y∼CN( si(θθθ),ΠΠΠ). (39)
trices,orthepowerofeffectivenoiseN,theun-clusteredcalibra- Xi=1
tion’sCRLBbecomeslowerthantheclusteredcalibration’sCRLB. Therefore,theCRLBofun-clusteredcalibrationis
e
K K −1
var(θθθ)> 2Re ( Jsi(θθθ))HΠΠΠ−1( Jsi(θθθ)) ,
4.1.3 AnalysisofCRLB (cid:20) (cid:26) i=1 i=1 (cid:27)(cid:21)
X X
Generally, if source 1 is considerably brighter than source 2, whereJsi istheJacobianmatrixofsiwithrespecttoθθθ.
C C , and if the weak source power ismuch Computing the exact CRLB is more complicated when we
|| 1{pq}|| ≫ || 2{pq}||
lower than the noise level, C N , then clustered haveclusteredcalibration.Intheclusteredcalibrationmeasurement
|| 2{pq}|| ≪ || pq||
calibration’s performance is better than un-clustered calibration. equation,givenbyEq.(13),wehave
Note that the worst performance of both calibrations is at the
K
faintestsourceandwearemoreconcernedtocomparetheCRLBs
n ΓΓΓi+n, (40)
forthissource. ≡
i=1
X
e
(cid:13)c 2007RAS,MNRAS000,1–18
8 Kazemi et al.
wherenistheun-clusteredcalibration’snoisevector, x 10−10
5
ΓΓΓi=[vec(ΓΓΓi{12})T...vec(ΓΓΓi{(N−1)N})T]T, (41) Clustered calibration
Un−clustered calibration
andΓΓΓ isgivenbyEq.(31).Duetotheexistenceofthenuisance
i{pq} 4
parametersΓΓΓi intheclusteredcalibrationdatamodel, calculation
ofitsconventionalCRLBisimpractical.Thisleadsustotheuseof
B
Cramer-Raolikeboundsdevisedinthepresenceofthenuisancepa- L3
rameters(Gini&Reggiannini2000).WeapplytheModifiedCRLB R
C
(MCRLB)(Ginietal.1998)totheperformanceof clusteredcali- M
bration. 2
TheMCRLBforestimatingtheerrorsofθθ˜θinthepresenceof
thenuisanceparametersΓΓΓ(clusteringerror)isdefinedas
b 1
∂ ∂ −1
var(θθ˜θ)> E E ln P(yΓΓΓ;θθ˜θ) ,
(cid:20) y,ΓΓΓ(cid:26)− y|ΓΓΓ(cid:26)∂θθ˜θ∂θθ˜θT { | }(cid:27)(cid:27)(cid:21) 0 10 Num20ber of clu3s0ters 40 50
(42)
b
whereP(yΓΓΓ;θθ˜θ)istheProbabilityDensityFunction(PDF)ofthe Figure 6. Clustered calibration’s MCRLB. Forvery small Q,where the
|
visibilityvector y assumingthat theΓΓΓmatricesofEq. (41) area effectofinterferenceislarge,MCRLBishigh.Byincreasingthenumber
prioriknown.Sincen (0,ΠΠΠ),fromEq.(13)wehave of clusters, MCRLB decreases and reaches its minimum where the best
∼CN performanceoftheclusteredcalibrationisexpected.Afterthat,duetothe
Q K dominanteffectofthebackgroundnoise,MCRLBstartstoincreaseuntilit
yΓΓΓ ([ ˜si+ ΓΓΓi],ΠΠΠ), (43) reachestheun-clusteredcalibrationCRLB.
| ∼CN
i=1 i=1
X X
andthereforeinEq.(42), E ∂ ∂ ln P(yΓΓΓ;θθ˜θ) ,which
− y|ΓΓΓ ∂θθθ˜∂θθθ˜T { | }
(cid:20) (cid:21) −6
iscalledthemodifiedFisherinformation,isequalto x 10
10
Q K Q K Clustered calibration
2Re [ J (θθ˜θ)+ J (θθ˜θ)]HΠΠΠ−1[ J (θθ˜θ)+ J (θθ˜θ)] . 9
{ ˜si ΓΓΓi ˜si ΓΓΓi } Un−clustered calibration
i=1 i=1 i=1 i=1
X X X X 8
Note that E in Eq. (42) could be estimated by Monte-Carlo
y,ΓΓΓ
method.
R 7
Asaruleofthumb,reducingtheheavycomputationalcostof N
MCRLB,onecaninterprettheSINRtestofEq.(38)asfollows:If SI 6
inaveragetheeffectiveSINRofclusteredcalibration,SINR ,gets
c
higherthantheeffectiveSINRofun-clusteredcalibrationobtained 5
fortheweakestobservedsignal,SINR ,
w
4
E SINRc E SINRw , (44)
{ }≫ { }
3
thenclusteredcalibrationcanachieveabetterresults.InEq.(44), 0 10 20 30 40 50
Number of clusters
theexpectationistakenwithrespecttothenoiseN,errormatrices
ΓΓΓ,andallthebaselines. Figure 7.Clustered calibration’s SINR. SINRis low for smallQ,when
the interference is large. By increasing the number of clusters the SINR
increasesandgetsitshighestlevelforwhichthebestperformanceofthe
4.1.5 Example3:MCRLBandSINRestimations calibrationisexpected.Afterthat,itdecreasesduetothedominanteffectof
thebackgroundnoise,andconvergestotheun-clusteredcalibrationSINR.
We simulate WSRT including N = 14 receivers which observe
fifty sources with intensities below 15 Jy. The source positions
andtheirbrightnessfollowuniformandRayleighdistributions,re-
spectively. The background noise is N (0,15IM), where
∼ CN
M = 2N(N 1) = 364. We cluster sources using divisive estimatedresultsofMCRLBandE SINRc arepresentedbyFig.
− { }
hierarchical clustering, into Q number of clusters where Q 6andFig.7,respectively.AswecanseeinFig.6,forverysmall
∈
3,4,...,50 .Clusteredcalibration’sJonesmatrices,J,aregener- Q,wheretheeffectofinterferenceislarge,MCRLBishigh.Byin-
a{tedas (0.9},1.1)ejU(0,0.2).Sinceforsmallernumberofclusters, creasingthenumberofclusters,MCRLBdecreasesandreachesits
U
weexpectlargerinterference(errors)inclusteredcaliberation’sso- minimumwherethebestperformanceoftheclusteredcalibrationis
lutions,foreveryQ,weconsider 5i=01Γi ∼CN(0,1Q50IM).The expected.Afterthat,duetothedominanteffectofthebackground
choiceofthecomplexGaussiandistributionfortheerrormatrices noise,MCRLBstartstoincreaseuntilitreachestheCRLBofun-
P
ΓΓΓisduetothecentrallimittheoremandtheassumptionsmadein clustered calibration. Thesame result isderived fromE SINRc
{ }
section4.1.1. plotofFig.7.AsFig.7shows,E SINRc islowforverysmallQ,
{ }
Weproceed tocalculate theclustered calibration’sMCRLB, whentheinterference(i.e.,theerrorduetoclustering)islarge.By
given by Eq. (42), and E SINRc , utilizing the Monte-Carlo increasingthenumberofclusters,E SINRc increasesandgetsits
{ } { }
method.JacobianmatricesforMCRLBarecalculatednumerically peak for which the clustered calibration performs the best. After
andincomputationofE SINRc ,signalpowerofeveryclusteris that, it decreases and converges to the E SINR of un-clustered
{ } { }
obtainedonlyusingthecluster’sbrightestandfaintestsources.The calibration.
(cid:13)c 2007RAS,MNRAS000,1–18
ClusteredRadioInterferometricCalibration 9
4.2 Computationalcost and“solver noise”which isreferredtoaserrorsproduced bythe
calibrationalgorithmitself.WeassumethatthetrueJonesmatrices
Inthemeasurementequationofun-clusteredcalibration,presented
along different directions (clusters) at the same antenna are sta-
in Eq. (6), we have M = 2N(N 1) constraints given by the
− tisticallyuncorrelated.Notethatthisassumptionisonlymadefor
visibility vector y, and need to solve for P = 4KN unknown
theLRTtoproduce reasonable resultsand thisassumption isnot
parametersθθθ.IfP >M,thenEq.(6)willbeanunder-determined
neededforclusteredcalibrationtowork.Therefore,ifsuchcorre-
non-linearsystem.Thisclarifiestheneedofhavingasmallenough
lations exist, they are caused by the aforementioned errors. Con-
N(numberofantennas)andalargeenoughK(numberofsources)
sequently,themoreaccuratetheclusteredcalibrationsolutionsare,
forestimatingθθθ.However,clusteredcalibration,Eq.(13),hasthe
thesmallertheirstatisticalsimilaritieswouldbe.Basedonthisgen-
advantageofdecreasingthenumberofdirections,K,relativetothe
eralstatement,thebestnumberofclustersinaclusteredcalibration
numberofsourceclusters,Q K.Thisconsiderablycutsdown
≪ procedureistheonewhichprovidesuswiththeminimumcorrela-
thenumberofunknownparametersP thatneedstobecalibrated,
tionsinthecalibratedsolutions.Notethatforafixedmeasurement,
thusreducingthecomputationalcostofcalibration.
thecorrelationduetothesystemnoiseisfixed.Therefore,differ-
encesinthestatisticalsimilaritiesofsolutionsobtainedbydifferent
clusteringschemesareonlydueto“clusteringerrors”and“solver
5 SELECTIONOFNUMBEROFCLUSTERS noise”.
Toinvestigatethestatisticalinteractionbetweenthegainsolu-
Consideraclusteredcalibrationprocedurewithapredefinedclus-
tionsweapplytheLikelihood-RatioTest(LRT).
tering scheme. There is no guarantee that the calibration results
forQnumber ofclusters,whereQ 1,2,...,K israndomly ConsidertheclusteredcalibrationsolutionJpi(θθθ)fordirectionsi,
chosen, isthemostaccurate.Thus,w∈e{seektheopti}mumnumber i 1,2,...,Q ,atantennasp,wherep 1,2,...,N ,
∈{ } ∈{ }
ofclustersatwhichtheclusteredcalibrationperformsthebest.In e e
J J
ttheriisosnec(AtioICn,)w(Aekdaeiskceri1b9e7t3h)e,uassewoefl:l(ai)sA(iki)aiLkiek’esliIhnofoordm-RaatitoionTCersi-t Jpi =" Je1211,,pp Je1222,,pp #i. (49)
(LRT) (Graves 1978), in finding this optimum Q for a given ob- e
Then,theparametervectorθθθpei(corresepondingtothei-thdirection
servation. Some other alternative criteria could also be found in
andp-thantenna)isobtainedby
Wax&Kailath(1985). Notethat for different clustering schemes
e
theoptimumQisnotnecessarilythesame. θθθpi=[Re(J11,p)Im(J11,p) ... Re(J22,p)Im(J22,p)]Ti . (50)
Letusdefineforeachantennapandeachpairofdirectionskand
e e e e e
5.1 Akaike’sInformationCriterion(AIC) l,wherekandlarebelongto 1,2,...,Q ,avectorzpklas
{ }
Weutilize Akaike’s Information Criterion(AIC) tofind the opti- z =[θθθT θθθT]T. (51)
pkl pk pl
mumQforclusteredcalibration.
Infact,weareconcatenatingthesolutionsofthesameantennafor
Considerhavingn ∼ CN(0,σ2IM)inthegeneraldatamodelof twodifferentdirections(clusterse)togeether.
clusteredcalibration,Eq.(13).Then,thelog-likelihoodofthevisi-
WedefinethenullH modelas
bilityvectoryisgeivenby e 0
H : z (m,Σ ). (52)
(θθθy) = M logπ M logσ2 0 pkl ∼N 0
L | − −
where
Q Q
1
e − σ2(y− si(θθθ))He(y− si(θθθ)). (45) m=[m¯(θθθpk)T m¯(θθθpl)T]T, (53)
i=1 i=1
X X
Themaximumlikelihoodesetimeationofthenoieseevarianceσ2is and e e
e
σ2= M1 (y− Q si(θθθ))H(y− Q si(θθθ)). e (46) Σ0 =" s2(θθ0θpk) s2(0θθθpl) #. (54)
i=1 i=1 e
SubstitutingceEq.(46)inEqX.(4e5),ewearriveaXttheemeaximumlikeli- InEq.(53)andEq.(54),m¯ ands2 aredeenotingsamplemeanand
samplevariance,respectively.Notethathavingalargenumberof
hoodestimationofθθθ,
samplesinhand,theassumptionofhavingaGaussiandistribution
for solutions is justified according to the Central Limit theorem.
(θθθy) = Melogπ M
L | − − ThestructureofthevariancematrixΣ tellsusthatthestatistical
0
Q Q
be − Mlog{M1 (y− si(θθθ))H(y− si(θθθ))}.(47) bcoetrwreelaetniotnhebseotwluetieonntshθθθecoamnpdoθθθnen,tissozfertoh.eTrahnisdiosmthveeicdteoarlzcqaksle,oinr
i=1 i=1 pk pl
X X whichtherearenoestimationerrors.
UsingEq.(47),theAICisgivenbye e e e
Toinvestigatetheevalidityeofthenullmodelcomparedwiththe
caseinwhichthereexistssomecorrelationbetweenthesolutions,
AIC(Q)= 2 (θθθy)+2(2P). (48)
− L | wedefinethealternativeH1modelas
TheoptimumQisselectedastheobenethatmineimizesAIC(Q). H1: zpkl (m,Σ1), (55)
∼N
wherethevariancematrixΣ isgivenby
1
5.2 Likelihood-RatioTest(LRT)
s2(θθθ ) Cov(θθθ ,θθθ )
pk pk pl
Errorsinclusteredcalibrationoriginatefromthesystem(skyand Σ = . (56)
1
instrumental)noise,“clusteringerrors”introducedinsection4.1.1,  Cov(θθθ ,θθθ )T s2(θθθ ) 
pek pl e ple
 
(cid:13)c 2007RAS,MNRAS000,1–18 e e e
10 Kazemi et al.
1994;Yatawattaetal.2009;Kazemietal.2011b)withnineitera-
Cov(θθθ ,θθθ )inEq.(56)isthe8 8samplecovariancematrix. tions.PlotsoftheaveragedFrobeniusdistancebetweenthesimu-
pk pl
×
Usingtheabovemodels,theLikelihood-Ratioisdefinedas lated(true)Jonesmatricesandtheobtainedsolutionsisshownin
e e Fig.11.AswecanseeinFig.11,forbothclusteringschemes,in-
Likelihoodfornullmodel
Λ= 2ln , (57) creasingthenumberofclustersdecreasesthisdistanceandthemin-
− (cid:18)Likelihoodforalternativemodel(cid:19) imumisreachedatapproximatelythirtythreeclusters(Q = 33).
Beyondthisnumberofclusters,itincreasesuntilthefiftyindividual
whichhasaχ2 distributionwith64degreesoffreedom.AsΛbe-
sourcesbecomeindividualclusters.Thisshowsthatthebestperfor-
comessmaller,thenull model,inwhichthestatisticalcorrelation
manceofboththedivisivehierarchicalandtheweightedK-means
betweenthesolutionsiszero,becomesmoreacceptableratherthan
clusteredcalibrationsisapproximatelyatthirtythreeclustersand
thealternativemodel.Therefore,thesmallertheΛis,thelessthe
issuperiortothatoftheun-clusteredcalibration.
clusteredcalibration’serrorsare,andvice-versa.
The Frobenius distance curves in Fig. 11, the MCRLB curve in
Fig.6,andtheSINRcurveinFig.7illustratethatclusteredcalibra-
tionwithanextremelylownumberofclustersdoesnotnecessarily
performbetterthantheun-clusteredcalibration.Thereasonisthat
6 SIMULATIONSTUDIES
whenthereareonlyasmallnumberofclusters,theinterference,or
We use simulations to compare the performance of un-clustered theso-called“clusteringerrors”introducedinsection4.1.1,isrela-
andclusteredcalibration.Workingwithsimulationshastheadvan- tivelylarge.Therefore,theeffectofthisinterferencedominatesthe
tageofhavingthetruesolutionsavailable,whichisnotthecasein clustering of signals. On theother hand, wereach the theoretical
real observations. That makes the comparison much more objec- performancelimitapproximatelyaftertwentyfivenumberofclus-
tive.Nevertheless,thebetterperformanceoftheclusteredcalibra- ters and therefore increasing the number beyond this point gives
tionincomparisonwiththeun-clusteredonesincalibratingforreal highlyvariableresults,mainlybecausewearelimitedbythenum-
observations of LOFAR is also shown by Kazemietal. (2011a); berofconstraintsasopposedtothenumberofparametersthatwe
Yatawattaetal.(submitted2012). needtosolvefor.But,thisisnotthecasefortheplotsinFig.6and
WesimulateanEast-Westradiosynthesisarrayincluding14 Fig.7.ThereasonisthattheMCRLBresultsofFig.6aswellasthe
antennas (similartoWSRT)and an8by 8degrees skywithfifty SINRresultsofFig.7areobtained byMonte-Carlomethodwith
sourceswithlowintensities,below3Jy.Thesourcepositionsand iterationsoverfiftydifferentskyandnoiserealizations.However,
theirbrightnessfollowuniformandRayleighdistributions,respec- Fig.11islimitedtothepresentedspecificsimulationwithonlyone
tively. The single channel simulated observation at 355 MHz is skyandonenoiserealization.
showninFig.8. Theresidualimagesoftheun-clusteredcalibrationaswellas
Weproceedtoaddgainerrors,multiplyingsourcecoherencies thedivisivehierarchicalandweightedK-meansclusteredcalibra-
bytheJonesmatrices,asitisshowninEq.(2),tooursimulation. tionsforQ = 33areshownbyFig.12andFig.13,respectively.
TheamplitudeandphaseoftheJonesmatrices’elementsaregener- AsitisshownbyFig.12,intheresultofun-clusteredcalibration,
atedusinglinearcombinationofsineandcosinefunctions.Weaim thesources arealmost not subtracted at all and there isasignifi-
atsimulatingaskywithalmostuniformvariationsonsmallangular cantresidualerrorremaining.TheresidualshaveasymmetricGaus-
scales.Inotherwords,weprovideverysimilarJonesmatricesfor siandistributionwithvarianceσ2 = 82.29 whichismuchlarger
sourceswithsmallangularseparations.Toaccomplishthisgoal,for thanthesimulated(true)noisevarianceσ2 =10.85.Ontheother
everyantenna,wefirstchooseasingledirectionasareferenceand hand,thesourceshavebeenperfectlysubtractedinthecaseofclus-
simulate its Jones matrix as it is explained before. Then, for the teredcalibration,Fig.13,andtheresidualsconverge tothesimu-
remaining forty nine sources, at that antenna, the Jones matrices lated background noise distribution. The residuals of the divisive
(amplitudeandphaseterms)arethatinitialJonesmatrixmultiplied hierarchical and Weighted K-means clustered calibrations follow
by the inverse of their corresponding angular distances from that symmetriczeromeanGaussiandistributionswithσ2 =20.17and
reference direction. The result of adding such gain errors to our σ2 = 18.76,respectively.Thesevariancesareclosertothesimu-
simulationisshowninFig.9. latedoneσ2=10.85andthisindicatesthepromisingperformance
ofclusteredcalibration.Aswecansee,hierarchicalclusteredcal-
ibrationprovidesaslightlybetterresultcomparedtotheK-means
one. Thisisdue tothe fact that hierarchical clustering constructs
6.1 PerformancecomparisonoftheClusteredand
clustersofsmallerangulardiametersandthusitassignsthesame
un-clusteredcalibrationsatSNR=2
calibration solutions to the sources with smaller angular separa-
Weaddnoisen (0,σ2I)withσ2 =28,asitisshowninEq. tions.
∼CN
(6),tooursimulation.TheresulthasaSNR=2andispresented WealsocalculatetheRootMeanSquaredErrorofPrediction
inFig.10.WehavechosentopresentthecaseofSNR=2since (RMSEP)toassesstheperformanceofclusteredandun-clustered
forthisparticularsimulatedobservationboththedivisivehierarchi- calibrations’ non-linear regressions. The results of log(RMSEP),
calandtheweightedK-meansclusteredcalibrationsachievetheir presentedbyFig.14,alsojustifythatthebestefficienciesofthehi-
bestperformancesatthesamenumberofclusters,aswillbeshown erarchicalandK-meansclusteredcalibrationsareobtainedatthirty
laterinthissection. three number of clusters. But, note that there is a difference be-
Weapplyun-clusteredandclusteredcalibrationonthesimu- tween the behavior of log(RMSEP) plot of Fig. 14 and the plots
lationtocomparetheirefficiencies.Thefiftysourcesaregrouped ofMCRLB,SINR,andFrobeniusdistancebetweenthesimulated
into Q 3,4,...,49 number of clusters, using the proposed JonesmatricesandsolutionsinFig.6,Fig.7,andFig.11,respec-
∈ { }
divisivehierarchicalandweightedK-meansclusteringalgorithms. tively.InFig.14,log(RMSEP)ofclusteredcalibrationislessthan
Self-calibration is implemented via Space Alternating General- thatofun-clusteredcalibration,evenforextremelylownumberof
izedExpectationMaximization(SAGE)algorithm(Fessler&Hero clusters.Thismeansthatevenwiththoselownumber ofclusters,
(cid:13)c 2007RAS,MNRAS000,1–18