Photometric Redshift Estimation for Quasars by Integration of KNN and SVM

6.7 MB·

by  Bo Han

ResearchinAstronomyandAstrophysicsmanuscriptno.
(LATEX: HB.tex; printedonJanuary11,2016; 1:22)
Photometric Redshift Estimation for Quasars by Integration of KNN
and SVM ∗
6
1
1 1 2 2
BoHan ,Hongpeng Ding ,YanxiaZhang ,YonghengZhao
0
2

n
a
1 InternationalSchoolofSoftware,WuhanUniversity,Wuhan,430072,P.R.China;
J

8 2 KeyLaboratoryofOpticalAstronomy,NationalAstronomicalObservatories,ChineseAcademyof

] Sciences,20ADatunRoad,ChaoyangDistrict,100012,Beijing,[email protected]
M
I Received2015June12;accepted
.
h
p
-
o
r AbstractThemassivephotometricdatacollectedfrommultiplelarge-scaleskysurveysoffer
t
s
a significantopportunitiesformeasuringdistancesofcelestialobjectsbyphotometricredshifts.
[
However,catastrophicfailureisstillanunsolvedproblemforalongtimeandexistsinthecur-

1
v rentphotometricredshiftestimationapproaches(suchask-nearest-neighbor).Inthispaper,
9
3 weproposeanoveltwo-stageapproachbyintegrationofk-nearest-neighbor(KNN)andsup-
7
portvectormachine(SVM)methodstogether.Inthefirststage,weapplyKNNalgorithmon
1
0
photometricdataandestimatetheircorrespondingz .Byanalysis,wefindtwodensere-
phot
.
1
gionswithcatastrophicfailure,oneintherangeofz ∈[0.3,1.2],theotherintherangeof
0 phot
6
z ∈[1.2,2.1].Inthesecondstage,wemapthephotometricinputpatternofpointsfalling
1 phot
: into the two ranges from original attribute space into a high dimensional feature space by
v
i
X GaussiankernelfunctioninSVM.Inthehighdimensionalfeaturespace,manyoutlierpoints
r resultingfromcatastrophicfailurebysimpleEuclideandistancecomputationinKNNcanbe
a
identifiedbyaclassificationhyperplaneofSVM andfurtherbecorrected.Experimentalre-
sultsbasedontheSDSS(theSloanDigitalSkySurvey)quasardatashowthatthetwo-stage
fusion approach can significantly mitigate catastrophic failure and improve the estimation
accuracyofphotometricredshiftsofquasars.The percentsin different|∆z| rangesand rms
(rootmeansquare)errorbytheintegratedmethodare83.47%,89.83%,90.90%and0.192,
respectively,comparedtotheresultsbyKNN(71.96%,83.78%,89.73%and0.204).
Keywords: catalogs-galaxies:distancesandredshifts-methods:statistical-quasars:gen-
eral-surveys-techniques:photometric
∗SupportedbytheNationalNaturalScienceFoundationofChina.
2 B.Hanetal.
1 INTRODUCTION
Photometric redshifts are obtained by images or photometry. Compared to spectroscopic redshifts, they
showtheadvantagesofhighefficiencyandlowcost.Especially,withtherunningofmultipleongoingmulti-
band photometric surveys, such as SDSS (the Sloan Digital Sky Survey), UKIDSS (the UKIRT Infrared
Deep Sky Survey) and WISE (the Wide-Field Infrared Survey Explorer), a huge volume of photomet-
ric data are collected, which are largerthan spectroscopicdata by two or three ordersof magnitude.The
massivephotometricdataoffersignificantopportunitiesformeasuringdistancesofcelestialobjectsbypho-
tometric redshifts. However, photometric redshifts show the disadvantages of low accuracy compared to
spectroscopic redshifts, and require more sophisticated estimation algorithms to overcome the problem.
Researchers worldwide have investigated the photometric redshift estimation techniques in recent years.
Basically, these techniques are categorized into two types: template-fitting models and data mining ap-
proaches. Template-fitting model is the traditional approach for estimating photometric redshifts in as-
tronomy.It extractsfeaturesfromcelestialobservationalinformation,suchas multibandvalues, andthen
matches them with the designed templates constructed by theoretical models or real observations. With
featurematching,researcherscanestimatephotometricredshifts.Forexample,Bolzonellaetal.(2000)es-
timatedphotometricredshiftsthroughastandardSEDfittingprocedure,whereSEDs(spectralenergydis-
tributions)wereobtainedfrombroad-bandphotometry.Wuetal.(2004)estimatedthephotometricredshifts
ofalargesampleofquasarswiththeχ2minimizationtechniquebyusingderivedtheoreticalcolor-redshift
relationtemplates.Rowan-Robinsonetal.(2008)proposedanapproachusingfixedgalaxyandquasartem-
plates applied to data at 0.36-4.5 µm, and on a set of four infrared emission templates fitted to infrared
excess data at 3.6-170µm. Ilbert et al. (2009) applied a template-fitting method (Le Phare) to calculate
photometricredshifts in the 2-deg2 COSMOS field. Experimentalresults from the abovetemplate-fitting
methodsshowedthattheirestimationaccuracyreliedonthetemplatesconstructedbyeithersimulationor
realobservationaldata.
Dataminingapproachesapplystatistics andmachinelearningalgorithmsona setoftrainingsamples
and automatically learn complicated functionalcorrelationsbetween multibandphotometricobservations
andtheircorrespondinghighconfidenceredshiftparameters.Thesealgorithmsaredata-drivenapproaches,
ratherthantemplate-drivenapproaches.Theexperimentalresultsshowedthattheyachievedmuchaccurate
photometricestimations in manyapplications.For example,Ball et al. (2008)applied a nearestneighbor
algorithmtoestimatephotometricredshiftsforgalaxiesandquasarsusingSDSSandGALEX(theGalaxy
EvolutionExplorer)datasets.Abdallaetal.(2008)estimatedphotometricredshiftsbyusinganeuralnet-
workmethod.Freemanetal.(2009)proposedanon-linearspectralconnectivityanalysisfortransforming
photometriccolorsto a simpler,morenaturalcoordinatesystem whereintheyappliedregressiontomake
redshiftestimations.Gerdesetal.(2010)developedaboosteddecisiontreemethod,calledArborZ,toesti-
matephotometricredshiftsforgalaxies.Wayetal.(2012)proposedanapproachbasedonSelf-Organizing-
Mapping(SOM)toestimatephotometricredshifts.Bovyetal.(2012)presentedtheextremedeconvolution
techniqueforsimultaneousclassificationandredshiftestimationofquasarsanddemonstratedthattheaddi-
tionofinformationfromUVandNIRbandswasofgreatimportancetophotometricquasar-starseparation
and essentially the redshift degeneracies for quasars were resolved. Carrasco et al. (2013) presented an
PhotometricRedshiftEstimation 3
algorithmusingpredictiontreesandrandomforesttechniquesforestimatingphotometricredshifts,incor-
poratingmeasurementerrorsinto the calculationwhilealso efficientlydealingwith missingvaluesin the
photometricdata.Bresciaetal.(2013)appliedtheMultiLayerPerceptronwithQuasiNewtonAlgorithm
(MLPQNA)toevaluatephotometricredshiftsofquasarswiththedatasetfromfourdifferentsurveys(SDSS,
GALEX,UKIDSS,andWISE).
Thoughtemplate-fittingapproachesanddataminingapproachescanroughlyestimatephotometricred-
shifts,theybothsufferthecatastrophicfailureprobleminestimatingphotometricredshiftsofquasarswhen
thespectroscopicredshiftislessthan3(Richardsetal.2001;Weinsteinetal.2004;Wuetal.2004).Zhang
etal.(2013)practicallydemonstratedthatwithcross-matchedmultibanddatafrommultiplesurveys,such
asSDSS,UKIDSSandWISE,k-nearestneighbor(KNN)algorithmcanlargelysolvethecatastrophicfail-
ureproblemandimprovephotometricredshiftestimationaccuracy.Themethodbecomesmoreimportant
as the developmentof multiple large photometricsky surveysand the comingof the age of astronomical
bigdata.However,duringthedatapreparationprocess,weneedtocross-matchmultibandinformationof
quasarsfrommultiplephotometricsurveys.Thenumberofmatchedquasarrecordsisfarlessthantheorig-
inal quasar numberin a single survey.For example,there are 105,783quasar samples available in SDSS
DR7.However,thenumberofcross-matchedsamplesfromSDSS,WISEandUKIDSSisonly24,089.The
cross-matchedsampleisaroundonefourthofSDSSquasardata.Thisshortcominggreatlylimitstheappli-
cationscopeofthisestimationapproachtoonlyasmallportionofcross-matchedquasarsobservedbyall
surveys.
Inthispaper,weproposeanoveltwo-stagephotometricredshiftestimationapproach,i.e.,theintegra-
tionofKNN(k-nearestneighbor)andSVM(supportvectormachine)approaches,tomitigatecatastrophic
failureforquasarsbyusingrelativefewbandattributesonlyfromasinglesurvey.Thepaperisorganized
as follows. Section 2 describes the data used. Section 3 presents a brief overview of KNN, SVM and
KNN+SVM.Section4givestheexperimentalresultsbyKNN+SVM.Theconclusionsanddiscussionsare
summarizedinSection5.
2 DATA
Our experimentsare based on a dataset generated from the Sloan Digital Sky Survey(SDSS; York et al.
2000),whichlabelshighlyreliablespectroscopicredshiftsandhasbeenwidelyusedinphotometricredshift
estimation. The dataset was constructed by Zhang et al. (2013) for estimating photometric redshifts of
quasars. Theyused the samples of the Quasar CatalogueV (Schneideret al. 2010)in SDSS DR7, which
included 105,783 spectrally confirmed quasars. In each quasar record, five band features u,g,r,i,z are
provided.SimilartoZhangetal.(2013),inourexperiments,weusethesefiveattributesu−g,g−r,r−
i,i−z,r(shortfor4C+r)astheinputandthecorrespondingspectroscopicredshiftasaregressionoutput.
3 METHODOLOGY
Firstly, we study the characteristics of catastrophic failure for quasars and observe that the outlier points
by KNN are clustered into two groups: one group’s spectroscopic redshift z is between 0.2 and 1.1,
spec
while its photometric redshift z is between 1.2 and 2.1, and the other group’s z is between 1.4
phot spec
4 B.Hanetal.
6
5
4
ot
h 3
p
Z
2 Group1
1
Group2
0 µ
0 1 2 3 4 5 6
Zspec
Fig.1 PhotometricredshiftestimationbyKNNestimation.ThepointsinGroup1andGroup2
areoutlierpoints.µistheparameterrepresentingtheerrortolerantscope.Theslantdottedlines
givethecorrectedestimatedrangeofphotometricredshifts.Thehorizontaldottedlinesdescribes
thezonesofGroup1andGroup2.
and 2.3, while its z is between 0.3 and 1.2 (shown in Figure 1). Some points with z falling into
phot phot
Group1actuallyhavez closetotherangeofGroup2,buttheyarewronglyestimatedbyKNNandare
spec
mixedintoGroup1,andviceversa.Thetwogroupslookalmost180-degreerotationallysymmetricalong
the45-degreediagonallineinthez vs.z diagram.ThetwooutlierclustersshowthatKNNmethod
phot spec
cannoteffectivelydistinguishoutlierpointsfromgoodestimationpointsusingEuclideandistanceinthetwo
regions.Next, we proposea two-stage integrationapproachby fusionof k-nearestneighbors(KNN) and
supportvectormachine(SVM)methods.Inthefirststage,we applyKNN algorithmonphotometricdata
andestimate their correspondingz . In thesecondstage, we mapphotometricmultibandinputpattern
phot
ofpointsfallingintothetworangeswithz ∈[0.3,1.2]andz ∈[1.2,2.1]fromanoriginalattribute
phot phot
spaceintoahighdimensionalfeaturespacebyGaussiankernelfunctioninSVM.Inthehighdimensional
feature space, many outlier points can be identified by a classification hyperplane in SVM and further
be corrected.Since mostpointsofcatastrophicfailurehave beenidentifiedand corrected,ourintegration
approachcanimprovethephotometricredshiftestimationaccuracy.
PhotometricRedshiftEstimation 5
TheKNNalgorithmgenerallyappliesEuclideandistanceofattributes(showninEquation1)tocompute
distancebetweenpointmandpointn,
d = (f −f )2+(f −f )2+...+(f −f )2 (1)
m,n q m,1 n,1 m,2 n,2 m,k n,k
wheref (f ) denotesthe jth attribute among4C +r inputpatternforthe mth (nth)points,k repre-
m,j n,j
sents the total number of attributes. The points in Group 1 and Group 2 show that those outlier quasars
cannotbe correctlyidentified in an Euclidean space. In other words, we cannothave a simple plane as a
usefulseparatingcriterionbetweenpointsinGroup1andGroup2.Basedonthepresentdata,theprovided
informationis notenoughto give goodestimation of the outlier points. Now there is a question whether
thoseoutlierpointscanbelinearlyseparableinahigh-dimensionalnon-Euclideanfeaturespace?Thereby,
we explorethe kernelfunctionin SVM and map features into a high dimensionspace and test if we can
correctlyclassifyoutlierpointsinGroup1andGroup2.Fromtheanalysisabove,weproposeatwo-stage
integrationapproachbyfusionofestimationwithKNNandclassificationwithSVM.
3.1 EstimationwithK-Nearest-Neighbor
K-Nearest-Neighbor(KNN)algorithmisalazypredictorwhichrequiresatrainingsetforlearning.Itfirst
finds the nearest neighbors by comparing distances between a test sample and training samples that are
similartoitinafeaturespace.Next,itassignstheaveragevalueofthenearestneighborstothetestsample
asitspredictionvalue.Ingeneral,thedistanceiscomputedasEuclideandistancedescribedinEquation1.
Intheeraofbigdata,wehavebeencollectingmoredatathaneverbeforeandKNNachievesmuchaccurate
predictions(Zhangetal.2013).Thereby,wealso useKNN inourresearch.OnedisadvantageofKNN is
thehighcomputationcost.WeapplyKD-treetoefficientlyimplementKNNalgorithm.
3.2 ClassificationwithSupportVectorMachine
SupportVectorMachine(SVM)isaneffectiveclassificationalgorithmbasedonstructuralriskminimization
principleproposedbyVapnik(1995).Givenatrainingdatasetwithnrecords,eachrecordhasthepattern
of(x ,y )fori=1,2,...,n,weaimtobuildalinearclassifierwiththefollowingEquation2,
i i
f(x)=w·x+b (2)
Here, w and b are weight vector and bias respectively. Figure 2 illustrates that several lines can separate
twocategoriesofpoints.InSVM,forminimizingtheclassificationerrorriskforothertestdatasets,weaim
to finda line (shownasthe dashline inFigure2) with themaximizedmarginto separatethe two classes
of points. This principlemakesSVM have a better classification accuracythan other competingmachine
learningmodelsinmanyclassificationtasks.
Sometimes,aclassificationtaskishardandnotlinearlysolvable.TheleftgraphinFigure3showsone
suchcase.Insuchcase,byVapnik-Chervonenkindimensiontheory,SVMappliesakernelfunctiontopro-
motetheoriginalflatspaceintheordinaryinnerproductconcepts.Bythetheoryofreproducingkernels,we
canmaptheoriginalEuclideanfeaturespacetothehigh-dimensionalnon-EuclideanfeaturespaceinSVM
classification algorithm.Thereby,some ofnon-linearityproblemsin the originallow-dimensionalfeature
spaceℜd becomelinearlysolvableinthehigh-dimensionalspaceℜD.TherightgraphinFigure3showsa
6 B.Hanetal.
2
Margin=
||w||
w•x+b=1
w•x+b=0
w•x+b=-1
Fig.2 Maximizingclassification marginis aimed in SVM. The pointson the dotted lines are
calledassupportvectors.ThedistancebetweenthetwodottedlinesisnamedasMargin.When
theMarginismaximized,theclassificationaccuracyachievesbest.
´(cid:71) ´(cid:39)
f(x)=0
(cid:1131)
Fig.3 Linearindistinguishablepointsinalowdimensionalspace(ℜd)canbeseparatedinahigh
dimensionalspace(ℜD)bytheapplicationofkernelfunction(∅)inSVM.f(x) = 0represents
thehyperplanethatseparatesthetwoclasses.
dimensionmappingbykernelfunctiontosolvetheproblem.Therefore,Equation2canbetransformedto
thefollowingformbyfeaturemappingfunction∅,
f(x)=w·∅(x)+b (3)
Inthisway,wecanhavethefollowingobjectivefunctionandconstraintsforaSVMclassifierasbelow,
andminimize
n
||w ||2 +C ǫ
X i
i=1
PhotometricRedshiftEstimation 7
subjectto
y ·(<w,∅(x )>+b)≥1−ǫ (4)
i i i
here,C isaregularizationparameterandǫ isaslackvariable.
i
Byrepresentedtheorem,wehave,
n
f(x)= α y ∅(x )T∅(x)+b (5)
X i i i
i=1
α is a parameter with the constraint that α ≥ 0. For solving Equation 5, SVM introducesa kernel
i i
functiondefinedas,
K(x ,x)=∅(x )T∅(x) (6)
i i
In this paper, we practically apply Gaussian kernel (shown in Equation 7) to achieve the non-linear
classification.
−||x1−x2||2
K(xi,x)=e 2σ2 (7)
wherex ,x representsvectorsofmultibandattributesorinputpatternsobservedfromasinglesurvey,σ
1 2
isafreeparameter.
In this way, we aim to apply a SVM classifier to distinguish the mixture pointsin Group1 and other
pointsaroundtheminordiagonalwithz ∈ [1.2,2.1]inthez vs.z diagram.Similarly,wecan
phot phot spec
distinguishpointsinGroup2andotherpointswithz ∈[0.3,1.2].
phot
3.3 IntegrationofKNNandSVMforPhotometricRedshiftEstimation
ThephotometricredshiftestimationalgorithmbyintegrationofKNNandSVMispresentedinthefollow-
ing.Toobtaintherobustaccuracymeasureforourintegrationapproach,werepeattheexperimentsfornum
rounds.In eachround,the data sets willexperiencethe initializationstep, KNN step, SVM trainingstep,
SVMteststep,correctionstepandevaluationstep.Ininitializationstep,werandomlydividetheSDSSdata
setintoseparatetrainingset,validationsetandtestset.InKNNstep,weapplyKNNalgorithm(k = 17)
toestimatezphot−validationandzphot−testbasedontrainingsetandtheunionoftrainingsetandvalidation
setrespectively.InSVM trainingstep, we aim tobuild two SVM classifiers:SVM1 andSVM2to distin-
guishgoodestimationandoutlierpointswithzphot−validation ∈ [1.2,2.1]andzphot−validation ∈[0.3,1.2],
respectively.ThegoodestimationoroutlierpointsisdefinedinthefollowingEquation8,
|z −z | ≤ µ good
 spec phot
(8)

|z −z | > µ bad
spec phot

here,µistheparameterwhichmeanstheerrortolerantscopederivedfromthevalidationset.
Visually,thegoodestimationpointswillfallintoareacloseto45-degreediagonallineinthediagram,
whiletheoutlierpointswillfallintoGroup1andGroup2inFigure1.
Specifically, we use those outlier points with zphot−validation ∈ [1.2,2.1] and zphot−validation ∈
[0.3,1.2] to construct datasets: Group1_trainingdata and Group2_trainingdata, respectively. In the two
datasets,inputsarepatterns4C+randz directlyfromKNN,andtheoutputisz .
phot spec
InSVMteststep,weapplyclassifiersSVM1andSVM2toidentifyoutlierpoints.
8 B.Hanetal.
In correction step, we use KNN algorithm based on Group1_data to compute z for those outlier
phot
pointsdistinguishedbySVM1intestdata.SinceGroup1_dataandthoseoutlierpointshavethesimilarpat-
ternwhiletheoutputofGroup1_trainingdataisz ,theKNNalgorithmcanimprovethez estimation.
spec phot
Similarly,we canuse Group2 totraindataandthencorrectoutlierpointsdistinguishedbySVM2 intest
data.
Inevaluationstep, we applythe percentsin different|∆z|rangesandrootmean square(rms)errorof
∆ztotestourphotometricredshiftestimationapproach.Thedefinitionof∆zislistedinEquation9.
z −z
spec phot
∆z = (9)
1+z
spec
Thedetailedstepsofthetwo-stagemethodareasfollowing.Tobemuchclearer,theflowchartofthe
wholeprocessisshowninFigure4.
LoopId=1;
DowhileLoopId≤num;
InitializationStep:
Randomlyselect1/3samplefromtheSDSSquasarsampleasthetrainingset,another1/3sampleas
thevalidationset,andtheremaining1/3sampleasthetestset.
KNNStep:
1. Based on training set, we apply KNN (k = 17) algorithm to estimate zphot−validation for each
sampleinvalidationset;
2.Basedontheunionoftrainingsetandvalidationset,weapplyKNN(k =17)algorithmtoestimate
zphot−testforeachsampleintestset.
SVMTrainingStep:
1.For those sampleswith zphot−validation ∈ [1.2,2.1]in validationset, we train a classifier SVM1
with Gaussian kernel, which distinguishes good estimation and outlier points by Equation 8. With those
outlierpoints,webuildadatasetGroup1_trainingdata,whichiscomposedof4C+randz astheinput
phot
andz astheoutput;
spec
2.Similarly,forthosesampleswithzphot−validation ∈[0.3,1.2]invalidationset,wetrainaclassifier
SVM2 with Gaussian kernel, which distinguishes good estimation and outlier points. With those outlier
points,webuildadatasetGroup2_trainingdata,whichiscomposedof4C +r andz astheinputand
phot
z astheoutput.
spec
SVMTestStep:
1.Forthosesampleswithzphot−test ∈ [1.2,2.1]intestset,weapplytheclassifierSVM1todistin-
guishgoodestimationandoutlierpoints;
2.Similarly,forthosesampleswithzphot−test ∈ [0.3,1.2]intestset,weapplytheclassifierSVM2
todistinguishgoodestimationandoutlierpoints.
CorrectionStep:
1.Forthoseoutlierpointswithzphot−test ∈[1.2,2.1]intestset,weapplytheKNNalgorithmbased
onthedatasetGroup1_trainingdata;
2.Forthoseoutlierpointswithzphot−test ∈[0.3,1.2]intestset,weapplytheKNNalgorithmbased
onthedatasetGroup2_trainingdata.
PhotometricRedshiftEstimation 9
EvaluationStep:
By comparing zphot−test and zspec for all samples in test set, we compute the popular accuracy
measuresforredshiftestimationandrmserrorof∆z.
LoopId=LoopId+1;
Enddo.
Output the mean and standard error for the percents in different |∆z| ranges and rms error of ∆z to
evaluatetheaccuracyofourproposedintegratedapproachKNN+SVM.
4 EXPERIMENTALRESULTS
Inthe experiments,we adopttheinputpattern4C +r as attributes,whichare widelyacceptedbyrecent
researchesonphotometricredshiftestimation.Inourdesignedalgorithm,wepracticallysetnum=10and
repeattheexperimentsfor10times.
For classification, we apply the widely used tool LIBSVM (Chang, 2011).By using Gaussian kernel
function, we train classifiers SVM1 and SVM2 for the sample with zphot−validation ∈ [1.2,2.1] and for
the sample with zphot−validation ∈ [0.3,1.2] in the validation set, separately. To optimize the estimation
accuracy,weadjusttwoparameterscontrollingtheGaussiankernelinSVM,acostcoefficientC thatmea-
sures the data unbalanceand a factor γ depicting the shape of the high dimensionalfeature space. Other
parametersaresettothedefaultvaluesinLIBSVM.Inordertoobtainthebestmodelparameters,thegrid
search is adopted.The grid search in SVM1 and SVM2 is indicated in Figure 5. For SVM1, the optimal
modelparameterCis2,γis8,meanwhile,theclassificationaccuracyis94.12%.ForSVM2,thebestmodel
parameterC is128,γ is0.5,theclassificationaccuracyamountsto90.04%.
Withtheoptimizedparametersandtheunionofthetrainingsetandthevalidationsetasanewtraining
set, we compare the estimation accuracy between original KNN (k = 17) algorithm and our integration
approach KNN+SVM. The parameter µ is a factor to determine whether a point has good estimation or
not. We change the value of µ to check its influence on the estimation accuracy.The results are listed in
Table1.ForKNN,theproportionsof|∆z|< 0.1,0.2,0.3andrmserrorofpredictedphotometricredshifts
are71.96%,83.78%,89.73%and0.204,separately;forKNN+SVM,theseoptimalmeasuresare83.47%,
89.83%,90.90%and0.192,respectively,whenµ=0.3,whichareboldinTable1.Obviously,thesecriteria
for photometric redshift estimation are all significantly improved with the new method. It suggests that
the integration approach can effectively correct those outlier points with z ∈ [1.2,2.1] and z ∈
phot phot
[0.3,1.2].Thereby,itcansignificantlymitigatecatastrophicfailureandimprovetheestimationaccuracyof
photometricredshifts.
Theexperimentalresultsalsoshowthatwithoutcross-matchingmultibandobservationsfrommultiple
surveys, we can effectively apply Gaussian kernel function in SVM to identify outlier points in Group 1
andGroup2fromcatastrophicfailurebymappingattributesfromasingledatasourceintoahighdimen-
sionalfeaturespace.Theidentificationhelpsuscorrectthoseoutlierpointsandtherebyimproveestimation
accuracy.
In order to compare the performanceof photometricredshift estimation by KNN algorithmwith that
by KNN and SVM approach, the photometric redshift estimation with these two methods is shown in
10 B.Hanetal.
Id=1
No Output
Id<=num ?
results
Yes
Divide the quasar dataset into 3 separate
subsets randomly: training, validation and test
Apply KNN(training) to compute Zphot of the
validation subset based on 4C+r
Search for Group1 and Group2
subsets from the validation result
Apply 4C+r+Zphot as input, whether
|Zphot-Zspec|<=margin as an output class, build SVM1/SVM2 classifier for
Group1/Group2; those outliers in Group1/Group2 are combined with
Zspec to form the correction dataset Outlier1/Outlier2
Apply KNN(training+validation) to compute
Id=Id+1
Zphot of the test subset based on 4C+r
Search the test results for
Group1 and Group2 subsets
For those estimations in Group1/Group2, apply
SVM1/SVM2 classifier to compute if they are outliers
For those outliers from SVM1/SVM2, apply KNN on
Outler1/Outler2 to compute Zphot with 4C+r
Evaluate estimation accuracy
Fig.4 TheflowchartofphotometricredshiftestimationbytheintegrationofKNNandSVM.
Figure6andFigure7,respectively.AsindicatedbyFigures6-7,wecanseeclearlythattheoutlierpoints
inbothGroup1andGroup2havebeensignificantlydecreasedbyadoptingthenewmethodKNN+SVM.
Itintuitivelyprovesthatourproposedapproachiseffective.