Artificial neural network based calibrations for the prediction of galactic [NII] $\lambda$ 6584 and H$\alpha$ line luminosities

8.4 MB·

by  Hossein Teimoorinia

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed30January2014 (MNLaTEXstylefilev2.2)
Artificial neural network based calibrations for the prediction of
galactic [NII] λ 6584 and Hα line luminosities.
4
1 Hossein Teimoorinia1 & Sara L. Ellison1
0 1DepartmentofPhysics&Astronomy,UniversityofVictoria,FinnertyRoad,Victoria,BritishColumbia,V8P1A1,Canada.
2

n
a 30January2014
J

9
2 ABSTRACT

] Theartificialneuralnetwork(ANN)isawell-establishedmathematicaltechniquefordata
A
prediction,basedontheidentificationofcorrelationsandpatternrecognitionininputtraining
G
sets.WepresenttheapplicationofANNstopredicttheemissionlineluminositiesofHαand
. [NII]λ6584ingalaxies.Theseimportantspectraldiagnosticsareusedformetallicities,active
h
galactic nuclei (AGN) classification and star formation rates, yet are shifted into the infra-
p
red for galaxies above z ∼ 0.5, or may not be covered in spectra with limited wavelength
-
o coverage.TheANNistrainedwithalargesampleofemissionlinegalaxiesselectedfromthe
r SloanDigitalSkySurveyusingvariouscombinationsofemissionlinesandstellarmass.The
t
s ANNistestedforgalaxiesdominatedbybothstarformationandAGN;inbothcasestheHα
a
and[NII]λ6584lineluminositiescanbepredictedwithascatterσ <0.1dex.Wealsoshow
[
thattheperformanceoftheANNdoesnotdependsignificantlyonthecoveringfraction,mass

1 ormetallicityofthedata.Polynomialfunctionsarederivedthatalloweasyapplicationofthe
v ANNpredictionstodetermineHαand[NII]λ6584lineluminosities.AnANNcalibrationfor
4 theBalmerdecrement(Hα/Hβ)basedonlineequivalentwidthsandcoloursisalsopresented.
8 The effectiveness of the ANN calibration is demonstrated with an independentdataset (the
3
GalaxyMassandAssemblySurvey).Wedemonstratetheapplicationofourlineluminosities
7
to thedeterminationofgas-phasemetallicitiesandAGN classification.TheANN technique
.
1 yields a significant improvementin the measurementof metallicities that require [NII] and
0 Hα when compared with the function based conversions of Kewley & Ellison. The AGN
4 classificationissuccessfulfor86percentofSDSSgalaxies.
1
: Key words: Astronomical databases: catalogues- Methods: data analysis- Methods
v
:statistical-galaxies:ISM
i
X
r
a
1 INTRODUCTION 2006).However,theweakstrengthofthe[OIII]λ4363linemeans
thatthis‘direct’techniqueisonlyeasilyappliedtorelativelymetal-
poorgalaxies.Alargenumber ofindirectmetallicitycalibrations,
Thephysical propertiesof starsandgalaxiesareencoded intheir
which use stronger emission lines, have therefore been proposed
spectral properties. Transcribing observational properties such as
(see Kewley&Ellison 2008 for a review), based either on pre-
colourandspectrallinecharacteristics,intoquantitiessuchasstel-
dictionsfromtheoreticalmodels,orcalibrationfromobservational
larmass,velocitydispersion,dust,temperature,densityandchem-
data.
icalabundancesisacornerstoneofastronomicaltechniques.How-
ever,accesstokeydiagnosticfeaturesisoftenlimitedbyeitherat- Whereasthestronglinemetallicitycalibrationsaremotivated
mospherictransmission,instrumentalcapability(e.g.limitedwave- bythedetectionthresholdofaweakspectralfeature,othercalibra-
lengthcoverage) ordetectionthresholds.Tocircumventtheseim- tionsaredrivenbywavelengthcoverage.Forexample,oneofthe
posedlimitations,itiscommonpracticetodetermineindirectmea- most common techniques for deriving star formation rate (SFR)
sures of the underlying physical quantities via calibrations with inlocal galaxiesisviatheluminosityof Hα(e.g.Kennicuttetal.
morereadilymeasuredquantities.Anexcellentexampleisthede- 1994). At z > 0.5 this line shifts out of the optical window,
termination of the gas phase metallicity in galaxies. Ideally, the motivating calibration against bluer lines such as [OII] λ3727
metallicityisdeterminedthroughthemeasurementoftheelectron (e.g. Gallagheretal. 1989; Kennicutt 1992; Kewleyetal. 2002;
temperature. This requires detection of emission lines with very Kennicutt&Evans 2012). At a similar redshift, the [NII]λ6584
differentexcitationpotentials,suchas[OIII]λ4363and[OIII]λλ line is also lost from the optical window. [NII] λ 6584 is used
4959,5007,whichinturnpermitsasolutionoftheelectrontemper- in a number of diagnostics, including some metallicity calibra-
ature, density and oxygen abundance (e.g. Osterbrock&Ferland tions (e.g. Denicolo,Terlevich&Terlevich 2002; Pettini&Pagel
2 Teimoorinia&Ellison
2004) and the very widely used active galactic nucleus (AGN) 2 REGRESSIONMETHODS
classification technique of Baldwin,Phillips&Terlevich (1981),
Themodellingofdataisamathematicalproblemthatisfundamen-
or the ‘BPT’ technique. Alternative AGN classification schemes
talinthescientificmethod.Linearmodelsarenotalwayssufficient
that can be applied to z > 0.5 galaxies use a combi-
todescribephysicalphenomena,sothatnon-linearrepresentations
nation of bluer emission lines, the 4000 Å break (D ),
n4000 may be required to find more complicated relations between dif-
colour and mass (e.g., Stasinskaetal. 2006; Juneauetal. 2011;
ferent parameters in a given data set. In particular, when there is
Trouille,Barger&Tremonti 2011; Yanetal.2011). For example,
insufficient information on the nature of the data under study, or
in the absence of near-infrared spectroscopy, Juneauetal. (2011)
thefunctiondescribingtheconnection betweentheparametersof
showthatcombining[OIII]λ5007/Hβandstellarmasssuccessfully
thedataisnotknownapriori,non-linearmethodsareparticularly
distinguishesbetweenstarformationandAGNemission.
powerful,
Althoughtheapplicationofcalibratedmetallicities,SFRsand
Ingeneral,regressionmethodsdealwithafunctioninthefol-
AGN classification is straight-forward and has proven a tremen-
lowingform:
dously useful tool in the study of galactic properties, each diag-
nostic is designed for a single purpose prediction. In this paper, Tar= f(A,X). (1)
i i
weinvestigatetheuseofartificialneuralnetworks(ANNs)forpre-
For a defined function with known input vectors X the aim
dictingtherawlineluminositiesofHαand[NII]λ6584,quantities i
istofindparametersA inordertoretrievethetargetvector Tar.
whichcanthenbeappliedintheuser’schoiceofcalibration.More- i
Whentheanalyticformof f isnotexplicitlyknown,thenlinearor
over, the line luminosities themselves contain information about
non-linearrepresentationsofthefunctionmustbederivedinorder
physical conditions within the galaxy such as ionization parame-
to identify the optimal form. In this regard, non-linear functions
teranddustextinction(Grovesetal.2012;Brinchmannetal.2008;
may be preferable, since they can solve more complicated prob-
Kewleyetal.2013a),whichcanbedeterminedfromtheANNpre-
lems.Selectingaspecialformforthefunction(andoptimizingthe
dictedluminosities.
valuesofA)isreferredtoasaregressionmethod.Amongstregres-
The artificial neural network is a non-linear method that i
sionmethods,artificialneuralnetworksandbasisfunctionregres-
‘learns’fromtrainingsetsofdatatopredictthepropertiesofother
sion(BFR)aretwoofthemostcommonlyused.
samples. Therehas been an increasing interest inusing ANNsin
astronomy; recent applications include star-galaxy discrimination
and galaxy classification (Lahavetal. 1996; Andreonetal. 2000; 2.1 ArtificialNeuralNetworks
Cortiglionietal.2001;Balletal.2004;Teimoorinia2012).Essen-
tiallyanyproblemthatisbasedonpatternrecognitionorcorrela- ANNsarewidelyusedinalargevarietyofapplicationsinvarious
tionscanbetackledwithANNs.Thepotentialdownsideisthatthe disciplines.Aneuralnetworkis,inessence,anattempttosimulate
ANN machinery is complex and requires significant overhead to thebrain.Itisamathematicalmodelthattriestosimulatethefunc-
constructandtrain.Thisisincontrastwithmanyofthecalibration tionalities of biological neural networks. Its basic building block
approachesdescribedabovethatprovidesimpleanalyticformulae isasimplemathematicalmodel.Aneuralnetworkisacomputing
topredictphysicalquantitiesbasedonlinearcorrelations. systemmadeupofanumberhighlyinterconnectedprocessingele-
In this paper, we aim to combine the power of the artificial mentsthatarecalledneurons.AnANNcontainsthreelayers:input,
neuralnetworkwiththeuser-friendlinessofananalyticsolutionin hiddenandoutput,whereeachlayerhasadifferentnumberofneu-
order to provide ANN-calibrated equations that predict the lumi- rons.Theinputlayerandoutputlayersareconnectedtotheinput
nositiesofHαand[NII]λ6584.Thepaperispresentedasfollows. andtargetdata,respectively.Thehiddenlayerconnectsthesetwo
In Section 2 we describe the basics of regression techniques, in- layersinamorecomplicatedway.Inthisway,allpossiblecombi-
cluding their mathematical foundations and the specific cases of nationsoftheinputandthetargetdatacanbeconsidered.Choos-
theANNandbasisfunctionregression.Section3presentsasimple ing thenumber of neurons depends on the nature of theproblem
case study of how the ANN can be applied to spectral data. The under study. Usingthismodel, each input datumisweightedand
Balmer decrement (Hα/Hβ) of star-forminggalaxies in theSloan thenaddedupwithabias,B,tomakeadesiredtarget,Tar,byan
Digital Sky Survey (SDSS) is predicted and compared to the re- activationfunction, f.Anon-linearactivationfunctionintroduces
centlypublishedlinearcalibrationofGrovesetal.(2012).Readers non-linearityintothenetwork, whichusuallyisselectedasasig-
whoaresolelyinterestedinthepredictionofHαand[NII]λ6584lu- moid function (e.g., see Vanzella et al 2004). In vector form we
minositiesmaywishtoskipdirectlytoSections4and5wherewe havethefollowingrelationship:
present theANN calibrationsfor star-forminggalaxiesandemis- Tar= f(W*X +B) (2)
tr
sionlinegalaxiesofunknownclassificationrespectively.InSection
6wedescribethewidelyapplicablepolynomialandmatrix-based Anetworkcancontainseverallayerswithdifferent f,Wand
forms of our ANN outputs. The robustness of the calibrations in B.Theoutput ofeachlayercanbeconsideredastheinput ofthe
tested in Section 7, and the application to gas-phase metallicities next layer. In this way, each element of the input vector Xtr (the
andAGNclassificationaredemonstratedinSection8. inputtrainingdataset)isconnected toeachneuroninputthrough
ThroughoutthepaperweassumeacosmologywithΩ ,Ω , theweightmatricesW.Anetworkcanbetrainedusingsupervised
tot M
Ω =1.0,0.3,0.7andH =70kms−1Mpc−1.Linefluxes(corrected orunsupervisedmethods(Haganetal.1994;Kohonen1997).Ina
Λ 0
for Galactic extinction and underlying stellar continuum), stellar supervisedmode,forexample,Vanzellaetal.(2004)usethemag-
masses,equivalentwidthsandothergalacticparametersaretaken nitudesofgalaxiesusingtheHDF-SandSDSSdataasinputdata,
fromtheMPA/JHUcatalogs(unlessotherwisenoted),witherrors Xtr,totrainanetworkusingtheredshiftinformationofthegalaxies
scaledaccordingly1. astargetvalues,Tar.
tion corrections and error scalings can be found at
1 The MPA/JHU catalogs, details on Galactic extinc- http://www.mpa-garching.mpg.de/SDSS/DR7/raw data.html
ANN emissionlinecalibrations 3
Briefly,aftercollectingthedata(i.e.,TarandX )anetwork andaLevenberg-Marquardtalgorithmtofitthedata.IfXisanN-
tr
iscreatedandthenconfigured. Configurationisthestepinwhich dimensionalcolumnvector,thenwecanwriteaquadraticfunction
thenetwork isarrangedtobecompatible withtheproblemunder inageneralformas:
study (e.g., a fitting or a classification problem). Initialization of
N N
theANNparameters(weightsandbiases)andtrainingthenetwork f(C,X)= C XX + C′X +C′′. (6)
(findingsuitableweightsandbiases)arethenextsteps.Thelatter ij i j i i
iX,j=1 Xi=1
canbedoneusingoneofseveralalgorithms.Forexample,inagra-
dientdescentalgorithm,andinaniterativeway,theaimmaybeto Equivalently, in matrix form we have: f(C,X) = XTCX +
improve(orupgrade) weightsandbiasesinthedirectionwherea XTC′ +C′′. N is the number of physical parameters used as in-
performancefunction(e.g.,ameansquareerrorfunction)decreases put data and Cij is a symmetric matrix. For example, for a cer-
mostrapidly(i.e.,inthedirectionofnegativegradient).IftheWk tainrunwhichutilizesthreephysicalparameterssuchasX1=mass,
arethecurrentweightsandbiasesandGisthecurrentgradientthen X2=OIIIandX3 =Hβ,N=3andthereisatotalof10independent
thealgorithmmaybewrittenas: coefficientstobedetermined.Bythisequation,weconsiderdiffer-
entconnectionsbetweenthephysicalparametersinamorecompli-
W =W −αG , (3)
k+1 k k k catedform.Wewillgiveanexampleofusingthisequationinthe
where α isa learning rate that is an adjustable parameter. α nextsection.
controlsthesizeoftheweightandbiaschangesduringthelearn-
ingprocess.WeusetheLevenberg-Marquardtoptimizationmethod
(Marquardtetal.1963)toupdateweightandbiasvalues.Finally,
3 ACASESTUDY:ANANNCALIBRATIONOFTHE
after the validation of a network, it will be ready to use (or test)
BALMERDECREMENT
for new data. In the validation step, the trained network (i.e.,the
networkwithdefinedweightsandbiases)isgenerallytestedonan InordertodemonstratetheapplicationoftheANNtospectraldata,
independent dataset,Xva, settoverifythenetwork’s performance andtotestitsperformanceagainsttheanalyticcalibrations,weap-
isrobust. plytheANNtechniquetothepredictionoftheBalmerdecrement
Out = f(W*X +B). (4) (Hα/Hβ). Since the ANN predicts the line luminosities that we
va va
would expect tomeasure inthe observed spectrum, thepredicted
If the output of a network, Outva, isclose to the actual val- HαluminositycanbecombinedwiththeobservedHβluminosity
ues then this means that the network is well trained. Generally, todeterminetheamountofdustextinction(seebelow).Thedeter-
this also shows that there has been a good correlation between minationofextinctionisacriticalcomponent ofspectralanalysis
Xtr andTar.Insomecases,wewishtoparameterizethemapping since, once determined, a correction can be applied to other line
from D-dimensional inputs to the desired output. In other words, fluxes.
wehavemulti-dimensionalinputsandaknownoutput.Theques- The performance of the ANN can be compared directly to
tionofwhichinputdata-set(fromnavailableinputvectors)make the analytic prediction of the Balmer decrement presented by
thiscorrelationthestrongestisanimportantissue.Specifically,in Grovesetal. (2012). A major component of the work presented
thiswork,wewillconsiderwhichcombinationsofstrongemission hereisthetranslationoftheANNsolutionsintopolynomialequa-
lines,andstellarmass,canbebestcombinedtogiveareliablepre- tions that can be simply applied by any user. We therefore also
dictionoftheHαand[NII]lineluminosities. present and test a second order polynomial basis function that is
Inequation2,theinputdatagenerallycontaindifferentpieces based on the ANN, i.e., Eq. 6, to find the associated coefficients
ofphysicalinformation: thatcanbeusedtopredictHα/Hβforagivensetofinput.
Under the assumption of either an optically thin or thick
X =X (X ,X ,...,X ). (5)
tr tr m1 m2 mn
medium, and the temperature of the region, it is possible to pre-
Inotherwords,Xtrcanbeamatrixcontainingn(column)vec- dict the intrinsic ratio of the Balmer emission line series (e.g.
tors. For example, it can comprise the information of colour (as Dopita&Sutherland 2003; Osterbrock&Ferland 2006). The ob-
vectorXm1)aswellasinformationofequivalent width(EW)ofa servedratioof fluxescanthereforebecombined withpredictions
certainemissionline(Xm2) of adatasetof m galaxies. Again,to from extinction laws (e.g. Cardelli,Clayton&Mathis 1989; Pei
giveaconcreteexample,Grovesetal.(2012)showthattheBalmer 1992;Calzettietal.1994)toderivetheattenuationthroughapar-
decrement(Hα/Hβ)canbemeasuredfromtheemissionlineEWs ticular region. As described in detail by Grovesetal. (2012), the
andcolours. Therefore, inaneural network approach, X1 andX2 accurate measurement of Hα and Hβ line fluxes requires suffi-
canbeconsideredasEWsandcolours,respectively. ciently high S/N and resolution to be able to account for under-
lyingabsorptioninthestellarcontinuum.Equivalentwidths,how-
ever,arerelativelyeasytomeasure,butdonotaccountforthere-
2.2 BasisFunctionRegression
quired absorption corrections. For this reason, some studies (e.g.
Inregressionproblems, modelshavethesamebasicformof Y = Kobulnicky&Phillips2003)haveproposedtheuseofEWsinstead
f(C,X)inwhichYandXaretargetandindependent inputdata, offluxes,withafixedcorrectionforunderlyingstellarabsorption.
i i
respectively. In principle, f can have any form. In any selected AmoresophisticatedapproachisproposedbyGrovesetal.(2012).
form,coefficientsC shouldbefoundaccordingtoaminimization Based on a calibration of Hα and Hβ EWs, continuum-corrected
i
algorithm. In a basis function regression we can select a polyno- fluxes and colours, Grovesetal. (2012) provide a calibration be-
mial function in which f(C,X) can, for example, be a second or tween EWs and the Balmer decrement that statistically accounts
thirdorderpolynomialfunction.Inordertoprovideauser-friendly forthestellarabsorption.
methodfortheapplicationoftheANNpredictions,wewillpresent Based on a sample of emission line (star-forming and AGN
thepolynomial functionsthatbestencapsulate thenetworks’ pre- dominated) galaxies in the SDSS, the analysis of Groves et al.
dictions. To thisend, we usea second order polynomial function (2012) determines the Balmer decrement by first quantifying an
4 Teimoorinia&Ellison
absorptioncorrectionfactortotheHαandHβEWs,andthenuses
therest-framefibreg−r colour asaproxy forthestellarcontin- 0.075 10<EW(Hα)<30 [A]
uum.Grovesetal.(2012)hencedeterminethefollowingprediction (G12−formula)
oftheBalmerdecrement(whereEWsareinÅ): 0.07
)
x
log(Hα/Hβ)=log[EEWW((HHαβ))++AA1]+[A3(g−r)+A4]. (7) σ (de0.065
2
Generally, the absorption EW is not the same for all the 0.06
Balmer lines, i.e., A , A . Grovesetal. (2012) use a constant
1 2
Balmer absorption correction for both EW(Hα) and EW(Hβ). In 0.05 0.1 0.15 0.2 0.25 0.3 0.35
z
thiscaseacorrectionfactor,whichdependsalsoontheEWrange
ofthesample,isderived.Thecorrectionfactorisdeterminedfrom
minimizing the offset between the SDSS galaxies’ EW Balmer
decrementsandthemeasuredHα/Hβ(withanerror-basedweight-
ingscheme,asdescribedinGrovesetal.(2012).
Here, we select a subset of the full data that have 10 <
EW(Hα) < 30 Å, for which A = A = 3.5 Å (see also Fig.4 of
1 2
Grovesetal.2012).ThevaluesofA andA aredeterminedfrom
3 4
thebest-fitbetweencontinuumfluxesmeasuredattheHαandHβ
wavelengths and fibre rest-frame g−r colour, and are found by
Grovesetal.(2012)tobe0.39and−0.26,respectively.
It is useful to begin our analysis with a test of the Balmer
decrementpredictionfromEq.7basedonEWsextractedfromthe
Figure 1. The top panel: the standard deviation between the predicted
MPA/JHUcatalogs,observedPetrosianmagnitudesobtainedfrom
log(Hα/Hβ)andobservedratioversusredshift.Thehorizontaldashedline
the SDSS and our own corrections to rest-frame quantities. Us-
showsscatterobtainedbyEquation7fromGrovesetal.(2012).Thelower
ing Eq. 7, in Fig. 1 we plot the standard deviation between the panels: the estimated log(Hα/Hβ) obtained from Equation 7 versus the
predicted ratio of log(Hα/Hβ) and the observed ratio as a func- valuepresentedintheMPA/JHUcatalogsfortwocases,z<0.1(left)and
tion of redshift. The top panel of Fig. 1 shows that the scatter z<0.35(right).Thecoloursandassociatedcolourbarsindicatethenumber
agrees with that quoted by Grovesetal. (2012) for z < 0.1 and ofgalaxiesineachcolouredpoint.
this EW range: σ = 0.058 dex. However, for higher redshifts,
thescatterinthecomparisonbetweentheobservedandpredicted
Balmerdecrementsincreasessteadily.Thiseffectishighlightedin andaveragetheresultsforthe10besttrainednetworksintermsof
thelower twopanels of Fig. 1which compares the observed and performance.
predictedratiooflog(Hα/Hβ)intworedshiftranges,z<0.1(left) WebeginwiththesamesampleusedbyGrovesetal.(2012)
andz< 0.35(right).Thesepanelsincludeallemissionlinegalax- describedabove:allemissionlinegalaxiesintheSDSSDR7inthe
ieswithaS/N>3(basedonthecriterionappliedbyGroveset al. MPA/JHU catalogswithHαand HβS/N>3,whichyieldsasam-
2012) in each of the Hα and Hβ lines. The colours and associ- pleof∼95,000 galaxies. Ofthe∼95,000 galaxiesinthesample
ated colour bar indicate the number of galaxies in each coloured werandomly selecthalf ofthemfortrainingthenetwork andthe
point.WespeculatethatthedeclineinperformanceofEq.7forour restareusedasavalidationset.Thevalidationstepcomparesthe
datasetmightbecausedbyadifferenceincalculationofrest-frame performance of the ANN on target data (i.e. data not used in the
colours.Regardlessofthesourceofthisredshiftdependence,inor- training step) to ensure that accuracy is preserved in its applica-
dertofairlycomparetheperformanceoftheANNmethodandthat tion.Ademonstrationofthetrainingandvalidationsetsisshown
ofGrovesetal.(2012),wesimplyrestrictourselvestotheredshift inSection7.1.
rangez<0.1. TheANNissetupwithsixdifferentruns,whichrepresentsix
differentsetsoftrainingdata.Althoughthedefaultcomparisonwith
Grovesetal.(2012)usesEW(Hα),EW(Hβ),gandr,itisinstruc-
tivetoinvestigateotherparametercombinations.Thedatausedin
3.1 TheANNandbasisfunctionapproaches
theinputrunsarelistedinTable1,whererunR3usesthesameset
WewillnowcomparetheperformanceoftheANNandalsotheba- ofinput dataasGrovesetal.(2012). Fig.2compares theperfor-
sisfunction(polynomial)methodtothesameapplication,namely manceoftheANNandbasisfunctionmethods,witheachrowrep-
the prediction of the Balmer decrement based on the observed resentingadifferentcombinationofinputparameters,summarised
equivalent widthsofHαandHβandrest-frameg−rcolour.The by the runs in Table 1. The ANN predicted values are compared
artificialneural network iscalibratedby setsof training datathat tothosemeasuredintheMPA/JHUcatalogsintherightpanelsof
aredefinedbythevectorX ,seeequation2.Forexample,torepli- Figure2.Theleftpanelsshowtheresultsobtainedfromthebasis
tr
cate the input data of the Grovesetal. (2012) analysis we define functionapproach,denotedbyf(x).
X =log(EW(Hα)),X =log(EW(Hβ)),X =g,andX =r,asfourin- ThetoppanelofFigure2(runR1)showsthatanANNwith
1 2 3 4
dividualcomponentsofX .Thesefourcomponents(N=4)canbe inputdataX =log(EW(Hα))andX =log(EW(Hβ))alonedoesnot
tr 1 2
alsousedinEquation6tofindthe15independent coefficientsof contain sufficient information to accurately reproduce the target
this equation. At the start of the training procedure, the network value, Hα/Hβ. The standard deviation for this run is relatively
must be initializedand the choice of theinitializationparameters large, almost 0.08 dex. Adding information about the brightness
may affect the outcome of the trained network. We therefore re- of the galaxy, in the case of run R2, the r-band absolute magni-
peat the training procedure 20 times with different initializations tude,doesnotsignificantlyimprovethesituation.RunR3hasthe
ANN emissionlinecalibrations 5
Table1.Balmerdecrementtrainingruns;allmagnitudesareextinctioncor-
rected,absolute,rest-frameandcalculatedwithintheSDSSfibre.EWsare
inÅ.
Runs Inputvectors(X)
R1 log(EW(Hα)),log(EW(Hβ))
R2 log(EW(Hα)),log(EW(Hβ)),r
R3 log(EW(Hα)),log(EW(Hβ)),g,r
R4 log(EW(Hα)),log(EW(Hβ)),i,z
R5 log(EW(Hα)),log(EW(Hβ)),log(Mfib/M⊙)
R6 log(EW(Hα)),log(EW(Hβ)),u,g,r,i,z
0.08
ANN
0.07 f(x)
x)0.06 σ
e
d G12
(0.05
σ
0.04
0.03
1 2 3 4 5 6
Run (index)
Figure3.ThestandarddeviationofthesixdifferentrunslistedinTable1
andshowninFig.2forANN(bluesolidline)andthebasisfunction(red
dashedline). Forcomparison, thestandard deviation oftheGroves etal.
(2012)calibrationisshownasahorizontaldottedline.Runswhosestandard
deviationsarebelowthisline(R3–R6)showasmallerscatterbetweenthe
ANNpredictionsandmeasuredvaluesthanGrovesetal.(2012).
same set of input data as the calibration of Groves et al. (2012):
EW(Hα),EW(Hβ),gandr(absolutemagnitudes)andnowthethird
panel fromtopofFigure2showsagoodcorrelationbetweenthe
ANNpredictionandmeasuredBalmerdecrement.Asdescribedby
Grovesetal.(2012),thisisexpectedasthecolourprovidesinfor-
mation on the stellar continuum. The standard deviation for run
R3 using the ANN is 0.038 dex, compared to 0.058 dex for the
Grovesetal. (2012) predictions from equation 7 (Figure 1). Al-
though this improvement is incremental, it does demonstrate the
powerandapplicabilityoftheANNapproach.TheANNalsohas
theadvantageofavoidinganexplicitfitbetweencontinuummea-
sured at the Hα and Hβ wavelengths and colour; the coefficients
A andA areproducednaturallyaspartoftheANNoutput.Run
3 4
R4substitutesiandz-bandmagnitudesforthegandrusedinR3;
thecorrelationisstillgood(σ=0.046dex),butnotasgoodasin
R3.Usingthefibrestellarmass(alsofromtheMPA/JHUcatalog)
insteadof colour informationalsoperformsmorepoorly thanR3
withaσ = 0.052dex.Therunwiththeleastscatterbetweenthe
predictedandobservedvaluesisR6,whichusesall5SDSSpho-
tometricbands.RunR6performsmarginallybetterthanR3witha
standarddeviationof0.037dex.
Figure2.Eachpanelshowsthevalueoflog(Hα/Hβ)estimated fromthe Figure3summarisestheperformanceofeachofthesixruns;
basisfunction(leftpanel)andANN(rightpanel)versusthemeasuredvalue
foreachrunthestandarddeviationbetweenthepredictedandmea-
intheMPA/JHUcatalog.Thefigurecomprisesthesixdifferentrunslisted
suredBalmerdecrement isplotted.ThevalueoftheGrovesetal.
inTable1.Run3usesthesameinputdataasthecalibrationofGroveset
(2012) calibration is shown for reference as a horizontal dotted
al.(2012),usingX1=log(EW(Hα))andX2=log(EW(Hβ)),g,andr,i.e.the
line.Thebluesolidandreddashedlinesshowtheresultobtained
panellabelled‘R3-ANN’isdirectlycomparablewithFig.1.Thecolours
and associated colour bars indicate the number of galaxies in each data fromANNandf(x),respectively.Figure3showsthatrunsR3–R6
point. allyieldlowerscatterthanthecalibrationofGrovesetal.(2012).
Figure3alsoshowsthat thescatter for each ANNrunisslightly
6 Teimoorinia&Ellison
smallerthantheresultobtainedbythebasisfunction,althoughthe predictionoftheHαand[NII]λ6584luminosities2.Themethodol-
differenceisalwayslessthan0.01dex.Theaccuracyofpredicted ogyisthesameasintheprevioussection.Firstaseriesoftraining
BalmerdecrementsobtainedfrombothANNandf(x)approaches runsareestablished,eachonewithadifferentsetofparametersto
can clearly match (or exceed) that of the formula presented by beusedinthenetwork.Thetrainingsetsincludehalfoftheavail-
Grovesetal.(2012). ablenumberofgalaxies.Theotherhalfisusedintheverification
The coefficients derived for runs R3 and R5 (the latter of phase,whereweconfirmthattheANNpredictsthelineluminosi-
which is useful in the absence of good optical photometry) are tieswiththeexpected scatter.Finally,inSection6and Appendix
givenbelow.AllEWsareinÅ.Duetothelargenumberofcoeffi- A,weprovideanalyticcalibrationsandmatrixcoefficientsthatrep-
cients,amatrixrepresentationofEq.6ismostconvenient.These resenttheANNoutputtobeusedbythegeneralcommunity.
coefficientsthereforepermitastraightforwardanalyticmethodfor Webeginbyconsideringonlystarforminggalaxiesinthissec-
calculatingtheBalmerdecrement.ForrunR3,withXas: tion,butinthenextsectionwegoontoapplyourtechniquetothe
general sampleof emissionlinegalaxies, regardlessoftheirclas-
logEW(Hα)
sification. The [NII]/Hα ratios differ markedly between the star-
X= logEW(Hβ)  forming and AGN populations due to the more significant con-
andwhegrre f(C,X)=XTCX+XTC′+C′′,CandC′ aregiven tbcroaibcmkupgtiaroroneudnodwf.iftTohhrbAeiGdddiNffenefroelrinmnteslsitnhieneflbthuaexsisprarfetoisoresmncianensyotaforf-aftohhreamrwdinegilol-nrkeingzaoioiwnnngs
AGNBPTclassificationschemes. Whenonlystar-forminggalax-
bythefollowingmatrices:
iesareconsidered, there isarelativelytight relationship between
0.32005 −0.10277 0.14572 −0.14146 the [OIII]/Hβ ratio and that of [NII]/Hα, which is largely driven
C= −0.10277 −0.03811 −0.08066 0.07727  by a trend in metallicity (e.g. Dopita et al. 2006; Stasinska et al.
 −00..1144517426 −00..0078702676 −00..0033828832 −00..0043385863  2p0re0d6i)Ic.ntiTothhniisssotsfiegc[hNttioIsIne]qwauneednsHceelα,ecaltusmawlliengowasliialtlixesiseesein,flrseotnamdr-sftohitresmeSlifDngtSogSeaxtlhacaxetlileaesrn.et
−0.38036 classifiedasstar-formingaccordingtothecalibrationofStasinska
C′ = 0.11262  et al. (2006) and have available total and fibre stellar masses in
andC′′=−000..5.33120163020335.  ttchiaceliebMxrtaPitnAecd/tJiHaosnU,pbcauartttaonlofogtt.hiLentieAnreNnaflNluxepxerstoinchceatsvisoe.nbB,eesefinnocrceeorttrhheeicstieimsdpefoosssrietGniotainalalolcyf-
anyS/Nrequirements,thesebasiccriteriayield120,444galaxies.
Twelveseparatetrainingrunsareestablished,containingbetween
The scatter found using this relation to predict the Balmer
1 and 8 parameters in different combinations, see Table 2. Each
decrementisσ=0.042dex.
ofthe11runsbetweenR2andR12requiresatleastoneemission
line so the ANN is tested separately for 3 different S/N criteria:
ForrunR5withXas:
S/N> 3, S/N> 5 and S/N> 8. R1 uses only fibre stellar mass so
logEW(Hα) does not need a S/N requirement. For a given S/N requirement,
X= logEW(Hβ)  onlythelinesrequiredasinputinthatrunarerequiredtopassthe
Thecoeloffigc(iMenfitbs/aMre⊙:)  cthrieteSr/iNonc.rFitoerrieoxna,mapnlde,thineRsa2moenliyst[rOueIIIi]nλr5u0n0R75i.sHreoqwueirveedr,toinprausns
R8, both [OIII]λ 5007 and Hβ are required to pass the criterion.
0.09724 −0.05493 0.04130 Table3liststhenumber of galaxiesineachsample,according to
C= −0.05493 −0.03587 −0.01420  therunnumberandS/Nthreshold.Recallthathalfofthesesamples
 0.04130 −0.01420 0.02065  willbTehuesdediffienrethnettAraNinNintgrariunninsgRp1ro–ceRs1s.2arechosentobothpro-
−0.74362 videinsightintothemostrelevantparametersrequiredforanaccu-
′
C = 0.29786  ratepredictionofthe[NII]andHαluminosities,butalsowithcon-
C′′ =2.2−806.9349,3y1i9eldingascatterofσ=0.053dex. sviedsetirgaatitoenthoefitmhepmacotsotfliSk/eNlyopnrathceticAaNlcNh,oaicneds.wFhoertheexrapmeprfleo,rmwaenicne-
isimprovedwhenbothmembersofagivendoublet(suchas[OII]
Theaboveformatsareparticularlywellsuitedtomatrixori-
λλ 3727, 3729) are used. Preference is given to relatively strong
ented software (such as IDL and MATLAB), although the coef-
emission lines with blue rest wavelengths, under the assumption
′ ′′
ficients C , C and C also can be directly used by considering
ij i thatthesewillbethemostreadilydetectableinopticalspectra.We
thesummation formof Eq. 6. Forpresentation purposes, wewill
do, however, include [SII] λλ 6716, 6731 in the last of the runs.
present coefficients in the matrix format for the remainder of the
AlthoughthisdoubletisredwardsofHαand[NII]λ6584,runR12
paper.
couldpotentiallybeusefulinthecaseofoneofthelinesbeingcon-
taminated,forexamplebytelluricfeatures.Massisalsoincluded
4 THEPREDICTIONOF[NII]ANDHαLUMINOSITIES
INSTAR-FORMINGGALAXIES 2 The ANN was also tested for the prediction of line fluxes, as well as
luminosities.However,therangeinredshiftintheSDSSwassufficientto
AfterthedemonstrationintheprevioussectionofourANNtech- increasethescatterinthepredictedfluxes,soluminositiesareadoptedfor
nique, wenow movetothemainobjectiveof thecurrent work:a thiswork.
ANN emissionlinecalibrations 7
Table2.TheANNrunparametercombinations.
Runs Inputvectorslog(X)
R1 Mfib/M⊙
R2 OIIIλ5007
R3 OIIλ3727
R4 Hβ
R5 Mfib/M⊙+OIIIλ5007
R6 Mfib/M⊙+OIIλ3727
R7 Mfib/M⊙+Hβ
R8 Mfib/M⊙+OIIIλ5007+Hβ
R9 Mfib/M⊙+OIIλ3727+Hβ
R10 Mfib/M⊙+OIIIλ5007+OIIλ3727+Hβ
R11 R10+OIIλ3729,OIIIλ4959
R12 R11+SIIλλ6717,6731
Table3.ThenumberofgalaxiesineachrunwithdifferentS/Nthresholds
forstar-forminggalaxies.
Runs Ngal(S/N>3) Ngal(S/N>5) Ngal(S/N>8)
R1 120444 120444 120444
R2 93365 69062 48249
R3 79309 50262 26660
R4 118383 110186 87838
R5 93365 69062 48249
R6 79309 50262 26660
R7 118383 110186 87838
R8 92970 68149 46169
R9 79268 50246 26650
R10 76329 48989 26309
R11 47368 29143 16481
R12 46910 28841 16270
asaninputparameterinsomeoftheruns,duetoitscorrelationwith
manyothergalaxyproperties(suchasSFRandmetallicity)thatin
turncorrelatewithemissionlinestrengths(orratios).
Figures4and6showthecomparisonofpredictedversusmea-
suredHαand[NII]lineluminosities,respectively.OnlytheS/N>5
runsareshownforbrevity.InFigures5(Hα)and7([NII])asum-
maryofthestandarddeviationsisshownforeachoftherunsforall
3S/Nthresholds. Letusfirstconsider theANN’sperformancein
predictingthefluxoftheHαline(Figures4and5).Thefirstitem
ofnoteisthatthetrainingandthevalidationsetsareidenticallybe-
haved(Figure5)showingthatthetrainingofthenetworkisrobust
andrepeatable.Ascanbeseen,fromFigures4and5,thephysical
parameterthatisthesinglemostimportantingredientintheaccu-
ratepredictionoftheHαluminosityis,unsurprisingly,Hβ.Allof
therunsthat includetheHβlinehavestandard deviations <0.07
dex.Interestingly,thescattervarieslittlebetweenthedifferentS/N
thresholdswithlessthan0.01dexdifferenceinthescattersbetween
theS/N>3andS/N>8versionsofagivenrun.RunR8isperhaps
oneofthemostusefulcombinationsfromapracticalperspective, Figure4.TheANNpredictedHαluminositiescomparedtotheobserved
valuesfortheS/N>5caseforstar-forminggalaxies.Theinputdataassoci-
includingtwostrongemissionlines([OIII]λ5007andHβ)thatare
atedwitheachrunarelistedinTable2.Alltheluminositiesareinunitsof
relativelycloseinwavelength,plusfibrestellarmass.Forthisrun,
erg/s.
theANNisabletopredicttheHαluminositywithastandardde-
viationofonly0.04dex,evenwhentheS/Nthresholdisaslowas
3.
WeturnnexttotheANNpredictionsof[NII]λ6584(Figures6
and7).Again,thetrainingandvalidationsetsareidentical,demon-
8 Teimoorinia&Ellison

SN>8 SN>8
SN>5 SN>5
0.2 SN>3 0.2 SN>3
) )
x x
e0.1 e0.1
d d
( (
σ σ
0.05 0.05
Hα (SF) Hα (SF)
Training set Validation set
1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112
Run (index) Run (index)
Figure5.Thestandarddeviationofthedifferencebetweenestimatedand
observed Hα for the training (left) and validation (right) sets for star-
forminggalaxies.Thetwosetsshowsamebehavior,indicatingthatthenet-
workisrobust.
stratingthattheANN’sresultsarerobustandnotdependentonthe
particulargalaxiesusedinthetrainingset.Thestandarddeviations
tendtobehigheringeneralforthepredictionsof[NII]thanforHα.
Interestingly,thereisalmostnodependenceoftheaccuracyofthe
[NII] line prediction on the S/N of the training set. Although the
dependenceoftheANN’sperformanceislessdramaticallylinked
totheuseofHβforpredicting[NII]asitwasforHα,theHβstill
appearstobethemostimportantofthe3strongemissionlinesused
here([OII],[OIII]andHβ).Oftherunsusingasingleemissionline
(R2–R4),thelatter,whichusesHβhasthelowestscatter.Ofthe
runsthatcombinemassandasingleemissionline(R5–R7),again
itisthelatterwhichminimizesthescatter.Aswesawwasthecase
forthepredictionofHα,runR8,whichisattractiveforitspractical
combinationoffibrestellarmass,Hβand[OIII]λ5007,hasoneof
thesmallestscatters(∼ 0.07dex). Indeed, forboth Hαand[NII]
theaccuracyoftheANN’spredictionisimprovedlittlebyadding
[OII].
4.1 Ontheuseoffibrevs.totalstellarmass.
Upuntilnow,wehaveusedthefibrestellarmassundertheassump-
tionthatpredictingthespectrallinestrengthswithinagivenaper-
turewillbemostsensitivetothemassenclosedinthatsamearea.
However,whilsttotalgalaxystellarmassesmightbereadilyavail-
ablefrombroad-bandphotometry,determiningaperturemassesre-
quiresanadditionalanalysisstep.Therefore,inthespiritofuser-
friendliness, weinvestigatetheimpact of substitutingtotal stellar
massinsteadoffibremassintherunslistedinTable2.InFigure
8wecomparethestandarddeviationsofthe12runsfortheS/N>
5casewiththismass substitutionfor boththeHαand [NII]pre-
dictions(leftandrightpanelsrespectively).Sincesomeoftheruns
(R2,R3andR4)donotusemassasaninputparameterthesubsti-
tutionoftotalstellarmassforfibrestellarmasshasnoimpact.For
theotherruns,weseethat,asexpected,theuseoftotalstellarmass Figure6.TheANNpredicted[NII]λ6584luminositiescomparedtothe
observed values for the S/N>5 case for star-forming galaxies. The input
(redpoints)yieldsaslightlyhigherscatter,butthedifferenceistyp-
dataassociatedwitheachrunarelistedinTable2.Alltheluminositiesare
ically<0.01dexinthestandarddeviations.Weconcludethattotal
inunitsoferg/s.
stellar mass can be used as a convenient alternative for aperture
stellar mass, at least over the range of covering fractions probed
by the SDSS data (typically 0.1 – 0.4). Nonetheless, we investi-
gateexplicitlyinSection7.1thedependenceofourcalibrationson
coveringfraction.
ANN emissionlinecalibrations 9
Table4.ThenumberofgalaxiesineachrunwithdifferentS/Nthresholds
SN>8 SN>8 forthemixedgalaxies.
SN>5 SN>5
SN>3 SN>3
Runs Ngal(S/N>3) Ngal(S/N>5) Ngal(S/N>8)
0.2 )0.2
) x
x e R1 264792 264792 264792
e d
d ( R2 189738 123768 79780
σ ( 0.1 σ 0.1 R3 122333 67686 33030
R4 212757 174502 126286
NII (SF) NII (SF) R5 189738 123768 79780
0.05 Training Set 0.05 Validation Set R6 122333 67686 33030
1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 R7 212757 174502 126286
Run (index) Run (index) R8 166022 107014 65725
R9 120507 67306 32963
Figure7.Thestandarddeviationofthedifferencebetweenestimatedand R10 116334 65633 32528
observed [NII]λ6584 forthe training (left) andvalidation (right) sets for R11 67708 37242 19405
star-forminggalaxies.Thetwosetsshowsamebehavior,indicatingthatthe R12 67034 36858 19168
networkisrobust.

M M
fib fib SN>8 SN>8
Mtot Mtot SN>5 SN>5
0.2 SN>3 SN>3
0.2
ex) ex)0.2 x) x)0.2
d0.1 d e e
σ ( σ (0.1 σ (d0.1 σ (d
0.05 0.1
Hα (SF) NII (SF) 0.05 Hα NII
SN>5 SN>5
0.05 (mixed) (mixed)
1 2 3 4Ru5n 6(in7d8ex9)101112 1 2 3 4Ru5n 6(in7d8ex9)101112 1 2 3 4 5 6 7 8 9101112 0.05 1 2 3 4 5 6 7 8 9101112
Run (index) Run (index)
Figure8.Acomparisonbetweendifferentruns(1-12)withtotalandfibre
Figure9.Acomparisonbetweendifferentruns(1-12)withtheANNmeth-
massforS/N>5forstar-forminggalaxies.Usingthetotalstellarmassleads
odsfortheS/N>5mixedgalaxysample.Theleftandtherightpanelsare
toaslightlyenhancedscatter,althoughthedifferenceisprobablynegligible
relatedtoHαand[NII],respectively.
formostapplications.
5 THEPREDICTIONOF[NII]ANDHαLUMINOSITIES
different runs for the three S/N thresholds tested in the previous
FORTHEGENERALCASEOFEMISSIONLINE
section.
GALAXIES
ThescatterintheANNpredictedluminositiescomparedtothe
Intheprevioussection,wedemonstratedthatanANNcanbesuc- observedvalueforthemixedsampleisshowninFigure9,which
cessfullyusedtopredicttheHαand[NII]λ6584lineluminosities includestheoutput fromallthreeS/Ncuts.Here,again, thevali-
instarforminggalaxies.Inmanycases,theaccuracyofthelumi- dationandtrainingsetsshowthesamebehaviorsoweonlyshow
nositypredictioniswithin0.1dex.However,inpractice,onecan the results from validation set. As can be seen, the difference in
notalwaysassumethattheionizationoftheinterstellarmediumis scatters for different S/N cuts is more significant than that of the
dominatedbystars.Ionizationbyalternativesources,suchasplan- star-formingsample,forbothHα(Figure5)and[NII](Figure7).
etary nebulae, shocks or AGN will yield different line strengths. ThedifferenceismorenoticeableforHα.Thisshowsthatthemixed
ThisfactformsthebasisofdiagnosticdiagramssuchastheBPT. sample(addingtheAGNgalaxiestotheSFsample)cangenerate
Wethereforeinvestigatewhether theANNisabletopredict more complicated patterns in the dataset and these patterns have
lineluminositiesforemissionlinegalaxieswhosedominantsource moredependency onthechoiceof S/N.FromFigure9,however,
ofionizationisunknown. Inpractice,thetwomostlikelysources it seems that adding more information can give a more coherent
todominateinintegratedspectrasuchasthosefromtheSDSSare result.
stars and AGN. In order to define a general sample of emission Wecompareagaintheuseoffibreandtotalstellarmasses.As
linegalaxies,wethereforedropthefirstrequirementinSection4, waspreviouslyfoundforthestar-forminggalaxies,herealsototal
thatgalaxiesmustbeclassifiedasstar-forming.Thefullsampleof stellarmassperformsessentiallyaswellasthefibrestellarmass.A
galaxies is referred to as the ‘mixed’ sample, to reflect the com- directcomparisonoftheANN’sperformancebetweenusingonly
binationofstar-forming,AGNandotherminorityclasses,suchas star-forminggalaxiesandincludingotherclassesisshowninFigure
LINERs. We repeat all of the ANN runs listed in Table 2, with 10. Asexpected, thescatter ishigher when allgalaxy classesare
thethreeS/Nthresholds, and adoptingthesamestrategyof split- included,buttheANNcanstillreadilyyieldscatters<0.1dexfor
tingeachsampleintotrainingandvalidationsets.Table4liststhe both Hα and [NII] for several of the runs. Again, the maximum
numberofgalaxiesinthemixedsamplethatisusedineachofthe informationusedasinputdata(run12)yieldsthelowestscatter.
10 Teimoorinia& Ellison

SF SF ANN ANN
Mixed Mixed f(x) f(x)
0.2 0.2
x) x)0.2 x) x)0.2
e e e e
d0.1 d d0.1 d
( ( ( (
σ σ σ σ
0.1 0.1
0.05 0.05
Hα NII Hα NII
SN>5 0.05 SN>5 (mixed) 0.05 (mixed)
1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112
Run (index) Run (index) Run (index) Run (index)
Figure10.Acomparisonbetweendifferentruns(1-12)withtheANNfor Figure12.Acomparisonbetweendifferentruns(1-12)withANNandf(x)
theresults obtained forSF(bluepoints)andmixed(redpoints)samples. methodsforS/N> 5.Theleftandtherightpanels arerelated toHαand
TheleftandtherightpanelsarerelatedtoHαand[NII],respectively. [NII],respectively.Ingeneral,thebasisfunctioncanaccuratelyreproduce
theANNresults.

ANN ANN
metallicity)ofthesample.Finally,wetestwhetherthecalibrations
f(x) f(x)
derivedfortheSDSSdataworkwellwhenappliedtoanindepen-
0.2
dentdataset.
x) x)0.2
e0.1 e
d d
( (
σ σ 7.1 Sizeandsamplingofthetrainingset
0.05 0.1
AfirstconsiderationintheANNapplicationisthenumberofneu-
Hα (SF) NII (SF)
ronsused.Forexample,usingalargenumber oftheneuronsina
SN>5 SN>5
0.05 networkcanincreaseitspower,butitcanalsoresultinover-fitting.
1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 Toprevent this, the networks’ parameters can be tuned insuch a
Run (index) Run (index)
waythatthetrainingandvalidationsetsshowthesamescatter.In
theworkpresentedhere,wefindthat8-10neuronsworkswellwith
Figure11.Acomparisonbetweendifferentruns(1-12)withANNandf(x)
methodsforS/N>5inthestar-formingsample.Theleftandtherightpanels ourdataset.Wealsocheckseveraloptimizationalgorithmsandfind
arerelatedtoHαand[NII],respectively. thattheLevenberg-MarquardtandBayesianregulationbackprop-
agationprovide thesameresultsasoneother,andaresuperior to
otheroptimizationalgorithmssuchasagradientdescentbackprop-
6 DERIVATIONOFBASISFUNCTIONCOEFFICIENTS
agation.
Wenow investigatehow effectivelytheselectedpolynomial basis The ANN will only work well if the training set is a good
function (i.e., Eq. 6) can predict the results of the ANN. In Fig- representationofthedataforwhichthepredictionisdesired.The
ure11 weshow acomparison betweendifferent runs(1-12) with scatterinthetrainingandvalidationsetsalsodependsonthesize
theANNandf(x)methodsforS/N>5inthestar-formingsample. ofthetrainingset.Asmentionedpreviously,werepeateachrun20
Theleft andthe right panelsare relatedtoHαand [NII],respec- times,withdifferentinitialization.Forhighernumbersofgalaxies
tively.Thereisanexcellentmatchbetweentheperformanceofthe selectedfortrainingANNtheoutputsofeachruncoverthesame
twomethods, indicating that noaccuracy islost inthe useof the range. In other words, here, there is a stability. When we reduce
polynomial. the number of galaxies for training the output of the runs are in
Werepeat thisexercisefor thegeneral caseof emissionline significantlydifferentranges. Wefindthat atrainingsampleof >
galaxies,whichrepresentsamixofstar-formingandAGNclasses, 3000galaxiescanpredictanacceptableresultinwhichthebehavior
withtheresultsshowninFigure12.Theleftpanel,forestimating ofthetrainingandvalidationsetsarethesame.
Hα,showsthatthepolynomial onceagainexcellentlyreproduces
theresultsoftheANN.However,thescatterinthe[NII]linelumi-
nositypredictionsisslightlyhigherwhenthepolynomial isused, 7.2 Theeffectofthephysicalparametersonestimated
comparedtotheANN,buttheeffectisverysmall,<0.01dex. parameters
In Appendix A wepresent all coefficients for estimatingthe
In Figures 13 and 14 we test the performance of the ANN as a
Hαand[NII]luminositiesofstar-formingand‘mixed’galaxies.
functionofvariousphysicalandobservationalparameters,forstar-
formingandmixedgalaxies,respectively.Ineachfigurewecalcu-
late the offset of the line luminosity predictions (of either Hα or
7 TESTINGTHEANNCALIBRATION [NII])relativetotheobservedluminositiesas
In this section we test the robustness of the ANN calibration in
∆=log(L /L ). (8)
pred obs
threeways.First,weinvestigatetheimportanceoftheselectedpa-
rameters used in ANN and the size and sampling of the training ThebluelinesinFigures13and 14show theoffsetsforHα
set. Second, we examine whether the scatter inpredicted linelu- andtheredlinesshowoffsetsfor[NII].Inallpanelsofthesetwo
minositiesdepends onthephysical parameters(suchasmassand figures, weadopt theresultsof run R8withaS/N>5for demon-