Table Of ContentMon.Not.R.Astron.Soc.000,1–14(2015) Printed12January2016 (MNLATEXstylefilev2.2)
Quantifying correlations between galaxy emission lines and
stellar continua
Ro´bert Beck1,2∗, L´aszlo´ Dobos1, Ching-Wa Yip2,3,
Alexander S. Szalay2 and Istv´an Csabai1
6
1 1Department of Physics of Complex Systems, Eo¨tv¨os Lora´nd University, 1117 Budapest, Hungary
0 2Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
2 3Wolfram Research, Somerville, MA 02144, USA
n
a
J Accepted2015December22.Received2015November21;inoriginalform2015June4
1
1
ABSTRACT
]
A Weanalysethecorrelationsbetweencontinuumpropertiesandemissionlineequiv-
G alent widths of star-forming and active galaxies from the Sloan Digital Sky Survey.
Since upcoming large sky surveys will make broad-band observations only, including
.
h strongemissionlinesintotheoreticalmodellingofspectrawillbeessentialtoestimate
p physical properties of photometric galaxies. We show that emission line equivalent
- widths can be fairly well reconstructed from the stellar continuum using local mul-
o
tiple linear regression in the continuum principal component analysis (PCA) space.
r
t Line reconstruction is good for star-forming galaxies and reasonable for galaxies with
s
a active nuclei. We propose a practical method to combine stellar population synthesis
[ models with empirical modelling of emission lines. The technique will help generate
more accurate model spectra and mock catalogues of galaxies to fit observations of
1
the new surveys. More accurate modelling of emission lines is also expected to im-
v
prove template-based photometric redshift estimation methods. We also show that,
7
1 by combining PCA coefficients from the pure continuum and the emission lines, auto-
4 matic distinction between hosts of weak active galactic nuclei (AGNs) and quiescent
2 star-forming galaxies can be made. The classification method is based on a training
0 setconsistingofhigh-confidencestarburstgalaxiesandAGNs,andallowsforthesim-
. ilar separation of active and star-forming galaxies as the empirical curve found by
1
0 Kauffmann et al. We demonstrate the use of three important machine learning al-
6 gorithms in the paper: k-nearest neighbour finding, k-means clustering and support
1 vector machines
:
v Keywords: methods:dataanalysis–galaxies:active–galaxies:starburst–galaxies:
i stellar content.
X
r
a
1 INTRODUCTION not enough to account for observations made with broad-
bandphotometricfilters.Sincefuturelargeskysurveyswill
Stellar population synthesis models are very successful in
make photometric observations only, accurate modelling of
explainingthespectralenergydistributionofgalaxiesinthe
theemissionlineswillbeessentialtoestimatephysicalprop-
optical(Fioc&Rocca-Volmerange1997;Bruzual&Charlot
erties(includingphotometricredshifts)ofgalaxiesprecisely.
2003;Maraston&Stro¨mba¨ck2011;Vazdekisetal.2012)but
The purpose of this paper is to empirically quantify
theydonotaccountforthecharacteristicemissionlinesorig-
correlations between properties of the stellar continuum of
inating from the excited interstellar gas. Starburst galaxies
galaxyspectra,andthestrengthsofemissionlines.Wealso
and galaxies with an active nucleus can produce emission
propose a recipe for generating realistic emission lines in
lines so strong that can reach 60 per cent of the contin-
the optical regime for stellar continua taken from popula-
uumfluxincertainbands,orasmuchas1mag(Ateketal.
tionsynthesismodels.Moreover,wepresentanovelclassifi-
2011). As a result, pure population synthesis models are
cationmethodtodifferentiatebetweenstarburstandactive
galaxies.
∗ E-mail: [email protected], [email protected], Results presented in the paper are obtained with the
[email protected] helpofthreeimportant,widelyusedmachinelearningtech-
(cid:13)c 2015RAS
2 R. Beck, L. Dobos, C.W. Yip, A.S. Szalay and I. Csabai
niques that have just started to gain popularity in astro- can be characterized (Baldwin, Phillips & Terlevich 1981;
nomical data analysis. Local linear regression using nearest Kewley et al. 2001; Kauffmann et al. 2003b) The two pri-
neighbours(Csabaietal.2007;Kerekesetal.2013)hasbeen mary radiation sources are young, hot, massive stars and
usedforphysicalparameterestimationbasedonbroad-band activegalacticnuclei(AGNs).Theirdifferentspectra(ther-
photometry.k-meansclustering,anautomatic,unsupervised mal and power law, respectively) cause different ionization
classificationalgorithmhasbeenappliedsuccessfully,forin- states and ratios of the most common elements which, in
stance, to classify gamma-ray bursts (Chattopadhyay et al. turn, produce well measurable, strong, often broad emis-
2007;Veresetal.2010).Supportvectormachines(SVM),a sion lines: the Balmer series of hydrogen, [Oii], [Oiii], [Nii],
supervised classification algorithm has been used for star– [Sii], etc. Population synthesis models do not account for
galaxy separation (Kova´cs & Szapudi 2015) and transient the emission of the ISM.
detection (Wright et al. 2015). We will briefly introduce Photoionization models (Stasin´ska 1984; Ferland et al.
thesemethodslaterinthepaper.Foradetailedintroduction 2013) yield accurate line ratios for any primary radiation
to the field, refer to Ivezi´c et al. (2014). spectrumandgascomposition.Tocouplestellarpopulation
As with all training set-based empirical methods, the synthesis with models of photoionization, shock-heating of
validityofourresultsislimitedtothetrainingset’scoverage the interstellar gas, emission of the dust etc., the star for-
of the parameter space (in our case the redshift, metallic- mationhistory,severalinteractionsbetweenthestellarpop-
ity, luminosity, continuum and line properties). Extrapola- ulations, the active nucleus, the dust and gas content need
tioncapabilitiesofempiricaltechniquestoparameterranges to be accounted for. For instance, AGNs are very likely to
outside the coverage is usually poor compared to theoreti- be responsible for quenching rapid star formation following
calmodels.Whilethiscertainlyconstrainstheapplicability starburst periods in the galaxy but they also emit ioniz-
of our results to strong emission line galaxies of the Sloan ingradiationthatexcitesgas,evaporatesdustandproduces
Digital Sky Survey (SDSS), the method itself can be eas- shock waves that heat the ISM. Also, starburst periods are
ily extended to galaxies outside the investigated sample by followed by high supernova activity that enriches the ISM
augmenting the training set. withmetals,leadingtosignificantchemicalevolutionwhich
The structure of the paper is as follows. In Sec. 2, we must be reflected in the models of emission lines. Addition-
explain the sample selection and data reduction methods. ally,arecentadvancementinstellarpopulationsynthesisis
Sec. 3 describes the line reconstruction methods we investi- the inclusion of stellar rotation and binary evolution effects
gated.Anempiricalmethodforstar-forming–activegalactic whichhavebeenshowntonoticeablyinfluencethestrengths
nucleus (AGN) separation is given in Sec. 4. We present a of some emission lines (Eldridge & Stanway 2012; Stanway
stochastic procedure to generate realistic emission lines for et al. 2014; Leitherer et al. 2014; Topping & Shull 2015).
continuummodelsinSec.5.Wesummarizeourfindingsand Taking everything into account is not possible without de-
outline future work in Sec. 6. tailed hydrodynamic simulation of the galaxies (Jonsson,
Wavelengths are generally quoted in vacuum. More in- Groves&Cox2010;Kewleyetal.2013)orwithoutmaking
formationonthedatausedforthisstudy,additionalfigures significant simplifications to the models. Various software,
and program source code are available on the web site of notably PE´GASE and BPASS (Fioc & Rocca-Volmerange
the paper1. Colour versions of the figures are available in 1997; Eldridge & Stanway 2012), can be used to generate
the online version of the paper. emissionlinesontopofstellarcontinuacomputedfromstel-
lar population synthesis. The photoionization part of these
softwares, however, introduces a large set of free parame-
1.1 Earlier work
ters that describe the distribution and composition of the
Thankstothelargeamountofflux-calibratedopticalgalaxy ISM. A frequently used way of reducing the number of free
spectraaccumulatedbytheSDSS,precisionofgalaxyspec- parametersistomaketheoreticalorempiricalassumptions.
trum modelling has been improved significantly during the Typical theoretical simplifications include the assumption
last decade. Many software tools and libraries exist to gen- of spherical symmetry or the use of a common ionization
eraterealisticstellarcontinuafromaprescribedstarforma- spectrum for all gas clouds (Stasin´ska 1984; Fioc & Rocca-
tionhistoryandvariouslibrariesofsinglestellarpopulation Volmerange 1997; Ferland et al. 2013). If no strict physical
spectra with a wide range of metallicities and initial mass considerations can be made, to generate realistic emission
function choices (Fioc & Rocca-Volmerange 1997; Bruzual linesontopofmodelledcontinuausinganyphotoionization
&Charlot2003;Maraston&Stro¨mba¨ck2011;Bressanetal. code,onehastoestimatethea priori distributionofmodel
2012;Vazdekisetal.2012).Modelshavealsobeenextended parameters by comparing large ensembles of models with
with descriptions of interstellar extinction, the UV–IR bal- observations. For instance, the code Le Phare (Ilbert et al.
ance(Silvaetal.1998;Charlot&Fall2000;daCunhaetal. 2006) uses the relations of Kennicutt (1998) to parametrize
2010)andthechemicalevolutionofthegasfromwhichstars emission lines.
can form (Dav´e, Finlator & Oppenheimer 2011). Anotherroutetotaketogeneraterealisticemissionlines
Emission lines of galaxy spectra carry a large amount is to work on an entirely empirical basis. Yip et al. (2004)
ofinformationabouttheabundanceandionizationstatesof demonstrated that stellar continua of SDSS galaxies form
elements in the interstellar gas. Based on ionization ratios a 1D sequence and thus, can be characterized by a single
of the various elements, the source of primary radiation re- numerical value, the eclass. The value of eclass for each
sponsiblefortheexcitationoftheinterstellarmedium(ISM) galaxyspectrumisobtainedbyexpressingthecontinuumon
a basis derived from principal component analysis (PCA).
Gyo˝ryetal.(2011)showedthatstrongcorrelationsbetween
1 http://www.vo.elte.hu/papers/2015/emissionlines the eclass (i.e. the stellar continua) of starburst and AGN
(cid:13)c 2015RAS,MNRAS000,1–14
Correlations of galaxy lines and continua 3
galaxies exist. They applied PCA to expand emission line ulationsyoungerthan10Myroratverylowmetallicitiesof
equivalent widths (EWs) of SDSS galaxies on a 3D basis Z ∼ 0.0001 (Molla´, Garc´ıa-Vargas & Bressan 2009), and
and correlate the principal components with the eclass of only about 0.5 per cent of our sample potentially fall into
the continua. We take their approach a step further: based this parameter range.
onthecorrelations,wegivearecipetoautomaticallygener- Due to discrepancies between continuum models and
ateemissionlineswithrealisticdistributionandrefinestar- SDSS spectra (Maraston et al. 2009), the continuum-
forming–AGN separation using the principal components subtractedspectrumconsistsofthreecomponents:theemis-
and SVM. sionlines,thenoiseandaslowlychangingbackgroundthat
originates from the imperfect models. Since the emission
linesandnoisearehigh-frequencycomponents,onecaneas-
ily eliminate the background by a high-pass filter. For this
2 DATA REDUCTION
purpose,weuseda50˚Awiderollingmedianfilter.Thiswas
We started with the entire spectroscopic galaxy sample of wideenoughtoleavebroadAGNlinesalmostintact,yetre-
the Sloan Digital Sky Survey Data Release 7 (?) which we moveanyresidualsoftheincorrectbackgroundsubtraction.
later filtered by signal-to-noise ratio and line strength. As Fig. 1 illustrates this procedure.
one of our goals was to accurately fit broad AGN lines, we Oncethelow-frequencybackgroundhasbeenremoved,
measured line parameters ourselves. linesarefittedusingatechniquewecallnoise-limitedfitting.
To precisely fit all strong emission lines, including those of
activegalaxies,weusethreeincreasinglycomplexlinemod-
2.1 Continuum fitting and line measurements
els.
PCA is widely used to derive a representative basis from
• A single Gaussian:
opticalspectraofgalaxies.WhenperformingPCAonemis-
sion line galaxies, the eigenspectra are primarily sensitive F(λ)=A·e−(λ−σλ20)2
to the variations in emission line strengths and only sec-
ondlytocontinuumfeatures(Connollyetal.1995;Yipetal. • Two Gaussians centred on the same wavelength but
2004). Obviously, the slope of the continuum is correlated with different variance
wToithruenmPisCsioAnolinnetshebuptutrheecovanrtiiannucae, oofnethhealsinteosmisabskiggtehre. F(λ)=A·e−(λ−σλa20)2 +B·e−(λ−σλb20)2
regions of emission lines, or eliminate the lines completely
bysubtractinglinemodelsfromthemeasuredspectra.Line • Two Gaussians allowing for a small offset ∆λ < 5 ˚A
fits have to be precise enough so that the line-subtracted between the centres, different variance
ctiornetsineutaofcoSnDtaSiSnDmRin7imgaallarxeysisdpueacltsr.aWaeccroerpdrioncgestsoedthtehseeerne-- F(λ)=A·e−(λ−σλa2a)2 +B·e−(λ−σλb2b)2
quirements with our own implementation of the algorithm
While the first model is enough to fit emission lines with
detailed in this section.
typicalvelocitydispersion,thesecondmodelisnecessaryfor
One frequent method of fitting continua in the opti-
lines with broad wings and the third model for asymmetric
cal band is to express the spectrum as a non-negative lin-
lines. Our objective is to find the simplest, yet well-fitting
ear combination of template spectra (Tremonti et al. 2004)
model. Overlapping emission lines are – obviously – fitted
while also accounting for the intrinsic attenuation and ve-
together,butwedonotenforceanycorrelationontheEWs
locity dispersion. Although more advanced, Bayesian and
of lines from the same ion. Also, the velocity dispersions of
PCA-based methods exist (Kauffmann et al. 2003a; Chen
the lines, even of those from the same ion, are fitted inde-
et al. 2012) to derive physical properties from the contin-
pendently. First, we fit the lines with the simplest model,
uum,asweweremainlyinterestedintheemissionlines,we
subtract it from the measurement and compare the resid-
retained the former technique for continuum subtraction.
ual within the region of the emission line with the noise in
First, we corrected for galactic extinction, masked emission
wavelengthrangeswithoutlines.Ifrmsoftheresidualinside
lines and fitted the continuum using the templates from
the region of the line is at least two times than elsewhere,
Bruzual&Charlot(2003)byalsofittingthevelocitydisper-
werejectthemodelandattempttofitthelinewithamore
sion and intrinsic extinction in parallel. Intrinsic extinction
complex one. Fig 2 illustrates how this technique works on
was modelled following Charlot & Fall (2000). Metallicity
asymmetric broad AGN lines.
was taken into account by fitting four sets of templates of
Tab.2.1summarizesthefittedandsubtractedemission
differingmetallicitiesandchoosingtheonewithminimalre-
duced χ2. Thus, the fitted metallicity can take one of four lines.Linemodelfitparametersareavailableonline.Inpanel
(a)ofFig.3,weplottheBaldwin–Phillips–Terlevich(BPT)
values: Z =0.004, 0.008, 0.02 or 0.05. We did not take the
diagramofthesampleusingacolourcodingwearegoingto
nebularcontinuumemissionintoaccount,which,inthecase
use throughout the paper.
of young starburst galaxies, can contribute a non-negligible
flux to the near-infrared part of the spectrum (Leitherer &
Heckman 1995). Since the entire continuum was fitted with
2.2 Comparison with other work
stellar templates only, we expect a slight overestimation of
absorption lines, and therefore the overestimation of emis- It is interesting to compare our line fits to those of Brinch-
sion lines for starburst galaxies. On the other hand, within mann et al. (2004). In the cited work, the authors used a
thewavelengthcoverageofSDSSspectroscopy,nebularcon- simpler technique of fitting nebular emission lines of SDSS
tinuumemissionissignificantonlyinthecaseofstellarpop- galaxies with the primary focus on the signal-to-noise ratio
(cid:13)c 2015RAS,MNRAS000,1–14
4 R. Beck, L. Dobos, C.W. Yip, A.S. Szalay and I. Csabai
Fλ
model
Fλ,
F-λ
5000 5500 6000 6500 7000 5000 5500 6000 6500 7000
λ [Å] λ [Å]
Figure 1. Illustration of fitting the stellar continuum. In the top left panel the best non-negative least square fit from 10 Bruzual–
Charlot templates is plotted, the residual is visible in the bottom left panel. The effect of the low-pass filter on the residual is drawn
with a red curve in the bottom left panel; we subtract this curve from the noisy residual prior to fitting emission lines. The top right
panelillustratesthebest-fittingcontinuummodel,correctedfordiscrepanciesbyaddingbackthelow-pass-filteredresidualtothestellar
populationsynthesisspectrum.Thetoprightpanelshowsthehigh-pass-filteredresidualusedforfittingthelines.
Fλ
model
Fλ,
F-λ
4950 5000 5050 4950 5000 5050 4950 5000 5050
λ [Å] λ [Å] λ [Å]
Figure2.Illustrationofnoise-limitedfittingofasymmetricemissionlineswithincreasinglycomplexmodels.Thetoppanelsshow
the original continuum-subtracted spectrum in grey and the best-fitting models in black. The bottom panels show the residuals.
Theleft-handpanelcorrespondstoasingleGaussianfit,themiddlepaneltotwoGaussianscentredonthesamemeanwavelength
but with different variance while the right-hand panel shows the results from fitting two Gaussians with slightly different centre
wavelengths. In this case, the most complex model is accepted as the line residuals are higher than the average noise for both
simplermodels,whereasthelineresidualiscomparabletotheaveragenoiseinthethirdcase.
of line measurements and not on the minimization of the we estimate weak emission lines significantly higher. This
residuals after line subtraction. As a result, their line mod- is very likely due to the high-pass filtering applied to the
els cannot directly be used to get a pure continuum due to continuum-subtracted spectrum, cf. Sec. 2.1. Yet, between
the high residuals of the fitting. 97.9% and 99.1% of our line, EWs are within 3σ of Brinch-
In Fig. 4, we compare the EWs of the most promi- mannetal.(2004),withtheexceptionofHαandHβ where
nent emission lines as derived with our technique and with only93.9%and87.7%,respectively,ofthemeasurementsare
the method of Brinchmann et al. (2004). In the case of within 3σ. Also, Brinchmann et al. (2004) measured weak
strongemissionlines,ourmeasurementsoflinestrengthsare linesbyfittingthemtogetherwithstrongerlinesofthesame
very similar to the results of Brinchmann et al. (2004), but ion, imposing a constraint on line ratios, whereas we fitted
(cid:13)c 2015RAS,MNRAS000,1–14
Correlations of galaxy lines and continua 5
Line λvac (˚A) Line λvac (˚A) Line λvac (˚A) The sample size was further limited to an easily man-
Oii 3727.09 Hγ 4341.68 Oi 6365.54 ageable number by choosing a section of the sky (right as-
Oii 3729.88 Oiii 4364.44 Ni 6529.03 cension between 220◦ and 230◦).
Hθ 3798.98 Hβ 4862.68 Nii 6549.86 The requirement that all 11 emission lines should be
Hη 3836.47 Oiii 4932.60 Hα 6564.61 measurableresultsinasamplecontaininggalaxieswithon-
Hζ 3890.16 Oiii 4960.30 Nii 6585.27
going star formation or possessing an active nucleus only.
H(cid:15) 3971.20 Oiii 5008.24 Sii 6718.29
Fig. 5 shows the selection effects on the distribution of the
Sii 4072.30 Hei 5877.65 Sii 6732.67
apparent andabsoluter-bandmagnitudes,the redshiftand
Hδ 4102.89 Oi 6302.05
themetallicity.Whilethecutinsignal-to-noiseratiodidpro-
Table 1.Listofthefittednebularemissionlines. duce a cutoff around r=19 apparent magnitude and a rel-
ativeincreaseofobjectstowardssmallerredshifts,theabso-
lutemagnitudedistributionshowsthatourselectionmethod
prefers fainter, smaller and younger galaxies. Galaxies with
lower metallicities are also selected with higher probability,
presumablyduetocorrelationsbetweenongoingstrongstar
formation, metallicity and age. Nevertheless, galaxies with
solar and above solar metallicities are still present in the
sample.
The SDSS DR7 main galaxy sample, which makes up
the majority of our training set, was not selected for mor-
phological type or colour, and thus includes a wide variety
of galaxies (Strauss et al. 2002). At larger redshifts, how-
ever,differentenvironments,e.g.harderradiationfieldsand
higher ionization parameters (Steidel et al. 2014), can lead
to significantly different emission line characteristics. Cer-
tainly,thevalidityofourresultsisconstrainedbyparameter
rangescoveredbythesample.Byusingadatasetthatgoes
beyondthetypesofgalaxiesobservedbytheSDSS,onecan
easily apply our method to a broader range of galaxies.
2.4 Continuum principal components
Figure 3. Original and reconstructed BPT diagrams of strong
Principalcomponentsofthestellarcontinuumwerederived
emission line galaxies sampled from SDSS DR7. In each panel,
from the fitted model spectra instead of the measurements
the dashed curved shows the empirical segregation line between
directly. Although the precise line modelling would make it
star-forming galaxies and AGN as defined by Kauffmann et al.
possibletosubtractemissionlinesfromtheoriginalspectra
(2003b). Panel (a) is plotted from directly measured line EWs.
Galaxies are colour coded based on their loci in the BPT plane: or run PCA directly on the measurements by masking out
bluegalaxiesarestar-forming,greenonesareAGNsandredones emission lines, due to the limited size of the sample which
aretheintermediateweakAGNsinthebottomcornerofthedis- wouldmakeeigenspectranoisy,wechoosetousethemodels
tribution.Thiscolourcodingbasedondirectlymeasuredemission instead.Fittedcontinuummodelsweretakenatrestframe,
linesisusedinallBPTsthroughoutthepaper.Panel(b)displays convolvedwiththebest-fittingvelocitydispersionkerneland
the BPT of line log EWs reconstructed from continuum princi- normalized to have equal flux in the following featureless
palcomponentsusingthelocallinearregressionmethodwiththe rest-frame wavelength ranges: 4250–4300 ˚A, 4600–4800 ˚A,
30 nearest neighbours in PCA space. Panels (c) and (d) show
5400–5500 ˚A, 5600–5800 ˚A. PCA was done in the 3722–
galaxiesonlythatwereoriginallyclassifiedas(c)AGNs,(d)star-
6761 ˚A range with 0.6 ˚A binning. The average continuum
formingusingdirectlymeasuredlineEWs.Whilelinesofstrong
was subtracted from the individual spectra prior to calcu-
AGNsandextremestarburstgalaxiescanbereconstructedwell,
thereissignificant‘cross-talk’inthequiescentregion. lating the covariance matrix.
EigenspectraweredeterminedusingtheLanczossingu-
lar value decomposition (SVD) algorithm from PROPACK
these lines independently. Weaker lines can easily become (Larsen2005).Thealgorithmcalculatesonlyagivennumber
undetectable in noisy regions, hence our fitting method in- of singular vectors with the largest corresponding singular
troduces some selection bias. values. This was very useful in our case as the spectra con-
sistedof5065datapoints,whereaswewereinterestedinthe
first five principal components only.
2.3 Galaxy sample selection The average spectrum and the resulting eigenspectra
are plotted in Fig. 6. As the average was subtracted, the
We selected a smaller sample of N = 13788 galaxies from
first eigenspectrum corresponds to the colour of the galaxy.
the entire set of continuum and line-fitted spectra that met
The following two basis vectors are rather similar at first
the following criteria:
sight but the third one shows more prominent absorption
• observed at a signal-to-noise ratio S/N >5, lines. They are together very likely to determine the age
• all 11 emission lines listed in Tab. 2 are measured and and metallicity of the galaxy as the 4000 ˚A-break is very
non-zero.TheselinesarethesameasinGyo˝ryetal.(2011). strong in both of them. The fourth vector probably corre-
(cid:13)c 2015RAS,MNRAS000,1–14
6 R. Beck, L. Dobos, C.W. Yip, A.S. Szalay and I. Csabai
Figure 4.ComparisonofemissionlineequivalentwidthofBrinchmannetal.(2004)(y-axis)withours(x-axis).EWsareexpressedin
angstroms;scalesonbothaxesarethesame.DensityplotsarenormalizedforEWbinsinsuchawaythatstrongerlinesarealsovisible.
Our estimate on weak lines is systematically higher, which appears more pronounced due to the log scale, but is not that significant
comparedtoerrors.
0.04
ensity 0.03 0.08
d 0.06
ility pobabr 00..0012 00..0024 Fλ average
0.00 -24 -22 -20 -18 -16 15 17 19 0.00 Fλ s1 = 965 eig1
absolute magnitude r apparent magnitude r
s2 = 198
0.10 0.40 Fλ eig2
y
ensit 0.08 0.30 s3 = 110
ility dab 00..0046 0.20 Fλ eig3
b
por 00..0002 00..0100 Fλ s4 = 77.4 eig4
0.0 0.1 0.2 0.3 0.4 0.004 0.008 0.02 0.05
redshift metallicity Fλ s5 = 60.6 eig5
Figure5.Normalizedhistogramsofgalaxyproperties.Oursam- 4000 4500 5000 5500 6000 6500
pleisplottedinblue,thegreylinescorrespondtotheentireDR7
λ [Å]
spectroscopicgalaxysample,whiletheblacklinesshowtheDR7
sampleexcludingthedeeperLRGsub-sample.Thelatterprovides
Figure6.Theaverageandthefirstfiveeigenvectorsoftheprin-
a better comparison to our data set, since we selected predomi-
cipalcomponentanalysisofgalaxycontinua,orderedbythecor-
nantlyfromthemaingalaxysample.
responding singular values (as displayed in each panel). See the
textforthephysicalinterpretationoftheeigenspectra.
spondstothewidthofabsorptionlinesthuscorrelateswith
velocity dispersion. The magnitude of the fourth and fifth
line ratios instead of absolute line strengths and uses lin-
eigenvalues is similar, and they already mark the start of
earmethodsfortheanalysis.Wehavetomention,however,
theplateauinthedistributionofeigenvalues,thereforetak-
that using the logarithm of the EWs also means that the
ing more eigenspectra into account does not significantly
resultspresentedintherestofthepaperwillbevalidinthe
increase the variance explained by them.
logarithmic sense only.
2.5 Emission line principal components
3 RECONSTRUCTING EMISSION LINES
In contrast to what was done by Gyo˝ry et al. (2011), we
calculateprincipal components ofthelogarithm ofemission OurgoalwastoempiricallyestimateemissionlineEWsfrom
line EWs. Fig. 7 shows the resulting singular vectors. Tak- continuum principal components. If there exists any corre-
ing the logarithm is more useful when one is interested in lation between the continuum and emission lines of galaxy
(cid:13)c 2015RAS,MNRAS000,1–14
Correlations of galaxy lines and continua 7
is similar if r are vectors instead of scalars but the a coef-
i
[OII][OII] Hγ Hβ[OIII][OIII] [NII]Hα[NII] [SII][SII] fiofcidentscabnecboemiencmoraptroircaetse.dEirnrotorstihnerviaalunedotfhewc.omSipmoinlaernltys,
i i
neighbours in NN can be ordered by distance from d and
s1 = 696 0.5 i
the inverse of (the square of) the distance can be used as a
0.0
weight in Eq. 2.
-0.5
Locallinearregressionhasmanyadvantagesoverglobal
non-linear modelling. First of all, global models are usually
s2 = 160 0.5 either too simple to describe the data or prone to overfit-
0.0 ting.Locallinearmodels,ontheotherhand,aresimpleand
W -0.5 can be used to characterize the local estimation errors. For
E
log 10 s3 = 90.1 0.5 irnisbtayncthe,eoχn2ecoafnthmeealoscuarleltihneeagroofidt.neTssheofcthhaelleenstgiemaintioloncoafl
linear fitting is to find the k-nearest neighbours quickly in
0.0
largedatasets.Spatialindexing,mostoftenakD-treeindex
-0.5
is used for this purpose (Csabai et al. 2007).
s4 = 60.3 0.5
0.0 3.2 Emission line reconstruction from the
-0.5 continuum
Weappliedthelocallinearregressiontechniquetoestimate
4000 4500 5000 5500 6000 6500
emissionlineEWsfromthestellarcontinuumprincipalcom-
λ [Å]
ponents. To test whether continuum PCs carry more infor-
mation regarding the emission lines than broad-band SDSS
Figure 7. The first four singular vectors of the correlation ma-
magnitudes,wewillalsoperformtheregressionanalysisdi-
trix of the logarithm of emission line equivalent widths, ordered
rectly on the photometric magnitudes in Sec. 3.3. Further
decreasingly by the corresponding singular values (as displayed
ineachpanel).Thefourthvectorisverylikelytobejustnoiseas tests are done with randomized samples (Sec. 3.5) to get a
[Oii]linesshouldnothavedifferentsigns. picture of the performance of our method.
By using the notation of Sec. 3.1, d became the first
i
five continuum principal components and r became the
i
spectra, it is clearly non-linear. Global linear methods to log EWs. Emission line log EWs were fitted individually
analyse the correlations are not useful in this case, yet lo- basedonthelogEWsofthek=30nearestneighbourgalax-
cally linear methods still can be used. ies in the continuum PCA space. The χ2 of the fitting was
weighted by the inverse-square distance of the neighbours
from the query point. The value of k =30 was chosen as a
3.1 Local linear regression ruleofthumb:wearefitting5+1parametersandthenum-
ber of data points must be large enough to adequately de-
Let us consider an ensemble of measurements where mea-
terminethatmanyparametersbutsmallenoughtopreserve
suredvaluesaresplitintotwosetsD={d }andR={r },
i i
locality. Modifying this parameter within reasonable limits
iindexingtheindividualmeasurements.Forthesakeofsim-
(e.g. 25−40) does not significantly impact the results.
plicity,r aretakentobescalarswhereasd arevectors,thus
i i
Fig. 8 shows the reconstructed log EWs of emission
DformsametricspaceofdimensionN.TheEuclideanmet-
linesasfunctionsofthedirectlymeasuredEWs.EWsrecon-
ric is often used to measure distances among data vectors
structed from the continuum are in reasonably good agree-
of D even though it might lack any physical interpretation.
ment with directly measured log EWs. The relative flux er-
Ourobjectiveistocharacterizeknown,orpredictunknown
ror σ of the line reconstruction is Gaussian but a system-
r from the always known d vectors. To estimate r from r
i i i atic shift δ is visible in the case of [Oii], [Oiii] and [Nii]
d ,firstwefindthek-nearestneighboursofd inD.Letus
i i
(δ ≈ 0.1,0.15,0.15, respectively). The typical value of the
denote the set of indices of these nearest neighbours with
relativeerrorisσ ≈0.3forhydrogenandsulfur,σ ≈35%
NN(d ,D,k),wherei∈/ NN bydefinition.Thenweexpress r r
i for [Oii] and [Nii], and σ ≈ 45% for [Oiii]. Col. 3-4 of
r in the following form r
i
Tab.2listtheoutcomeofthecorrelationanalysisforthe11
ri ≈ci+aidi. (1) investigatedlinesusingthelocallinearregressiontechnique.
Pearson’s product-moment correlation coefficient ρ and the
Note, that both a and d are vectors and their dot prod-
i i rms error σ were calculated for each line. These numbers
uct is taken in the formula above. The c constants and
i also show that fits are most accurate for the hydrogen and
the a coefficients need to be determined individually for
i sulfur lines (ρ>0.8) whereas oxygen and nitrogen lines are
every (d ,r ) measurement using standard linear regression
i i significantly less correlated with direct line measurements.
by minimizing
Fig.10showsthedependenceoftheerroroftherecon-
χ2 = (cid:88) (rj−ci−aidj)2, (2) struction on galaxy properties (cf. Fig. 5 for histograms of
i w these) for select lines. The brighter and higher metallicity
j
j∈NN galaxies have a larger fraction of AGNs and are estimated
where i is still the index of the measurement, j runs on the with higher errors, especially in the case of oxygen and sul-
nearestneighboursandw isaweight.Theexpressionofχ2 fur lines. Objects at higher redshifts generally exhibit de-
j
(cid:13)c 2015RAS,MNRAS000,1–14
8 R. Beck, L. Dobos, C.W. Yip, A.S. Szalay and I. Csabai
[OIII] [NII] [SII] Hα
1.00
error 0.75
MS 0.50
R
0.25
-23 -21 -19 -17 15 17 19
absolute magnitude r apparent magnitude r
1.00
error 0.75
MS 0.50
R
0.25
0.0 0.1 0.2 0.3 0.004 0.008 0.02 0.05
redshift metallicity
Figure8.ReconstructedlogEWsfromcontinuumprincipalcom- Figure10.ThermserrorofemissionlinelogEWreconstruction
ponents. Estimated log EWs are plotted as functions of the di- as the function of various galaxy properties. The colours corre-
rectlymeasuredlogEWsforthe11emissionlinesweused.Colour spondtothefollowingemissionlines:red–[Oiii]5008˚A,black–
coding of data points is the same as in panel (a) of Fig. 3 and [Nii]6585˚A,grey–[Sii]6718˚A,andblue–Hα.Seethetextfor
reflectstheactivityclassofgalaxies. adiscussion.
tion and the ISM: young massive stars are responsible for
the excitation of interstellar gas clouds. Nevertheless, [Oii]
and [Nii] lines show a significant scatter even in the star-
formingcase.Interestingly,sulfurlinescanbereconstructed
much better.
Oneintriguingresultisthat,while[Oiii]isanimportant
indicatorofnuclearactivity,itsreconstructionfromcontin-
uum properties in case of AGNs seems rather problematic.
It is understandable as AGN activity correlates much less
with the properties of the stellar populations than in the
star-forming case. Yet, some connections exist as it is visi-
ble from [Nii] and the hydrogen lines.
3.3 Emission line reconstruction from broad-band
magnitudes
To see if continuum PCA is any better than directly esti-
matingemissionlinesfrombroad-bandmagnitudes,weper-
formedtheaboveanalysisusingtheSDSSphotometricmag-
nitudes instead of the principal components. For this pur-
Figure9.ReconstructedlogEWsfrombroad-bandSDSSmagni- pose, we used dereddened model magnitudes without any
tudes.EstimatedlogEWsareplottedasfunctionsofthedirectly K-correction. The lack of K-correction is not supposed to
measuredlogEWsforthe11emissionlinesweused.Colourcod-
significantlyaffecttheprocedureastheredshiftdistribution
ingofdatapointsisthesameasinpanel(a)ofFig.3andreflects
of the sampled galaxies is rather sharp.
theactivityclassofgalaxies.
While broad-band magnitudes are strongly correlated
with continuum principal components, it is still interesting
creasing accuracy. The error of line reconstruction visibly to see how lines are reconstructed from them. First of all,
increases towards the limits of our training set. This is due magnitudes are highly correlated with each other, whereas
to the fact that near the edges of the training set there are PCA eliminates covariance. Also, observed magnitudes are
fewer galaxies and the nearest neighbours used to estimate already ‘contaminated’ with emission lines which might re-
theemissionlinesaregenerallylesssimilartoeachotherand sultinstrongercorrelationswithEWs.Resultsoflinerecon-
to the galaxy whose lines are being fitted. struction from magnitudes are plotted in Fig. 9. Compared
As we expected, emission lines can be much better re- with line reconstruction from PCA as plotted in Fig. 8, no
constructedfromthecontinuumofstar-forminggalaxiesdue cleardifferencecanbeseenintermsofscatter,perhapswith
to the strong connection between the young stellar popula- the exception of more outliers being visible in the photo-
(cid:13)c 2015RAS,MNRAS000,1–14
Correlations of galaxy lines and continua 9
Principalcomponents Magnitudes Principalcomponents Randomized
30NN galaxies 30NN galaxies 30randomgalaxies 30NN galaxies
line λ[˚A] ρ σ ρ σ ρ σ ρ×102 σ
Hα 6565 0.898 0.388 0.842 0.481 0.803 0.561 -1.46 0.961
Hβ 4863 0.882 0.369 0.840 0.430 0.795 0.535 -1.52 0.854
Sii 6718 0.839 0.416 0.773 0.492 0.798 0.484 -1.23 0.832
Sii 6733 0.827 0.433 0.751 0.516 0.754 0.527 -1.36 0.840
Hγ 4342 0.816 0.418 0.773 0.465 0.698 0.772 -2.81 0.790
Oii 3727 0.749 0.498 0.700 0.547 0.556 0.716 -0.710 0.817
Oiii 5008 0.743 0.784 0.673 0.884 0.673 0.877 -1.10 1.268
Oiii 4960 0.721 0.773 0.659 0.858 0.628 0.890 -1.35 1.208
Nii 6585 0.680 0.514 0.677 0.527 0.411 0.815 -0.318 0.757
Nii 6550 0.664 0.570 0.669 0.579 0.367 0.880 0.306 0.820
Table 2. Numerical properties of the various line reconstruction methods, for all 11 emission lines. The four methods are as follows:
(1) from continuum principal components, fitting the 30 nearest neighbours, (2) from broad-band magnitudes, fitting the 30 nearest
neighbours,(3)fromcontinuumprincipalcomponents,butinsteadofusingthe30nearestneighboursweused30randomgalaxies,and
(4) from continuum principal components, but with a randomized sample (as a cross-test). For each reconstruction, we calculated the
Pearsonproduct-momentcorrelationcoefficientρandrmserrorσ.Emissionlinesareorderedbyreconstructabilityusingthefirstmethod.
metriccase.Thus,logEWscanbereconstructedfrommag- 3.6 Reconstructing the BPT
nitudes almost as well as from the principal components.
As the BPT diagram is generally used to classify emission
Quantitative results are listed in Col. 5-6 of Tab. 2. We
line galaxies into star-forming and AGN, it is very infor-
have to emphasize here that our sample contained strong
mative to see how well the various methods can recover it
emission line galaxies only, thus the strong correlation be-
solelyfromthecontinuum.Thedifferenceamongthelinees-
tween magnitudes and log EWs exists only for our sample
timationmethodsintroducedaboveisobviousoncetheBPT
and cannot be generalized to all galaxies.
diagramisplottedfromthereconstructedlogEWs,asitwas
doneinFig.11.LocallinearfittingofEWsusingthenearest
neighboursandreconstructinglinesfrombroad-bandmagni-
3.4 Non-local line reconstruction from the tudesgivesimilarlyfair,qualitativelycorrectBPTdiagrams
continuum whiletherandomizationofthesampledisruptsthediagram
entirely.
To test whether a single global linear model is sufficient to
To further analyse the properties of a reconstructed
reproducethelines,werepeatedtheprocedureofestimating
BPT diagram, we will stick to local linear regression based
log EWs from the continuum principal components as de-
on the continuum principal components. In Fig. 3, we plot
scribedinSec.3.2butinsteadofusingthe30nearestneigh-
the original BPT for reference, the reconstructed diagram
bour galaxies, we randomly selected 30 galaxies from the
for all galaxies, and two diagrams showing AGNs and star-
entire sample. Another difference was that the χ2 of the fit
forming galaxies only, as classified by Kauffmann et al.
wasnotweightedbytheinverse-squareofthedistancefrom
(2003b).
the query point to relax the effect of locality. By looking at
The first thing to see in panels (c) and (d) of Fig. 3 is
Col. 7-8 of Tab. 2, it is somewhat surprising that correla-
themixingofweakstar-forminggalaxieswithweakAGNsin
tion coefficients and rms errors of the individual lines did
the bottom corner of the reconstructed BPT diagram. The
not get much worse. By looking at panel (c) of Fig. 11, one mixing is caused by the bad reconstructability of the [Oiii]
can clearly see, however, that that the star-forming branch
line which is most likely due to lack of a strong correlation
of the BPT diagram cannot be reconstructed this way, and
between AGN activity level and the stellar continuum.
theAGNsequenceisalsogreatlydistorted.Theconclusionis
thatemissionlinelogEWscannotbeexplainedbyasimple,
global linear relationship with continuum principal compo-
4 REVISITING STAR-FORMING/AGN
nents. Thus, local fitting from nearest neighbours is neces-
SEPARATION
sarytoreconstructtheBPTfromeithercontinuumprincipal
components or broad-band magnitudes. The mixed, low activity – low star formation rate region
is located at the bottom corner of the [Nii]/Hα–[Oiii]/Hβ
BPT diagram. Empirically drawn BPT diagrams are noisy
enoughtosmearpurestar-forminggalaxiesandmixedstar-
3.5 Cross-tests with randomized data
forming/AGNs together in this part of the BPT so it is
Asanothertest,weshuffledthesampleandrandomlypaired an interesting question whether it is possible to segregate
continuum principal components with emission line vectors galaxiesintotwodistinctclassesornotbyincorporatingin-
of other galaxies to break the locality in the PCA space. formationonthestellarcontinuumintoclassificationmodel.
Results are listed in Col. 9-10 of Tab. 2. It is not surprising By visually inspecting the projections of the 5D continuum
that correlations almost entirely disappear in the shuffled PCA space, one can see that while AGNs and star-forming
case(notethatρvaluesaremultipliedby102).Thissupports galaxies occupy different loci, they cannot be clearly sepa-
thatinformationabouttheemissionlinesisindeedencoded rated into two disjoint sets by cuts in any principal compo-
in the continuum of a galaxy spectrum. nentdimensions,northedistributionofgalaxiesisbimodal.
(cid:13)c 2015RAS,MNRAS000,1–14
10 R. Beck, L. Dobos, C.W. Yip, A.S. Szalay and I. Csabai
sified with high confidence either as pure star-forming or
AGN.Toselecthigh-confidenceAGNsonly,wepickedgalax-
iesabovethetheoreticalmaximumstarburstlineofKewley
et al. (2001):
(cid:18)[Oiii](cid:19) (cid:20) (cid:18)[Nii](cid:19) (cid:21)−1
log >0.61 log −0.47 +1.19.
10 Hβ 10 Hα
Star-forming galaxies were selected to fall below the empir-
ical starburst line of Kauffmann et al. (2003b):
(cid:18)[Oiii](cid:19) (cid:20) (cid:18)[Nii](cid:19) (cid:21)−1
log <0.61 log −0.05 +1.3,
10 Hβ 10 Hα
andatthesametimebeabovethefollowingline,definedby
us:
(cid:18)[Oiii](cid:19) (cid:18)[Nii](cid:19)
log >3log +1.55.
10 Hβ 10 Hα
Thelinewasdrawnempiricallytocutoutthemostreliably
identifiable part of the star-forming population. Curves on
Fig. 12 illustrate these cuts.
Figure11.BPTdiagramswithlogEWsreconstructedfromthe We used the first five continuum principal components
stellar continuum using different methods and cross-tests. Panel andthefirstfourlogEWprincipalcomponentsofthetrain-
(a)showsreconstructedlogEWsfromcontinuumprincipalcom- ingsetgalaxiesasinputdatavectorstoSVM.Bycombining
ponents using local linear regression from the 30 nearest neigh-
informationfromthecontinuumintothetraining,wemight
bours. Panel (b) is the reconstruction of lines from broad-band
hopeabetterseparationofthetwogalaxytypesinthemixed
magnitudes by local linear regression from the 30 nearest neigh-
lowercorneroftheBPTthansimplyfromtheemissionlines.
bours.Panel(c)wasdrawnfromlinesestimatedusingtheglobal
AsSVMisastrictlyempiricalmodel,weshallnot,however,
linear regression technique from 30 randomly selected galaxies.
Panel (d) is the cross-test using local linear regression but with draw far-reaching theoretical conclusions from its outcome.
shuffled continuum principal components. The colour coding of Since our training set was not containing the mixed region,
the data points is based on the original BPT as in panel (a) of itwasdirectlyseparableintotwodisjointclassesbyalinear
Fig.3.Seethetextfordiscussion. cut.Consequently,datapointsdidnotneedtobeprojected
into any higherdimensionalspacebya kernel function, like
in most applications of SVM, we simply ran it on the 5+4
WeturnedtoSVM,amachinelearningalgorithm,todeter-
dimensional vectors of the continuum + line PCA space.
mineanempiricalsegregationplanebetweenthetwoclasses
We plot the results of the SVM-based classification in
in the continuum-PCA–line-PCA space.
Fig.12.Panels(a)and(b)showthetwotrainingsetclasses
(star-forming and AGN, respectively) with log EWs of the
4.1 Support vector machines originally measured emission lines. Panels (c) and (d) show
the outcome of the SVM classification. Even though the
SVM are supervised learning algorithms which can be
mixed region was not included in the training set at all,
trainedtoautomaticallyclassifymultidimensionaldatavec-
SVM reproduced the empirical segregation line of Kauff-
torsintotwodisjointsets(Vapnik1998;Karatzoglou,Meyer
mannetal.(2003b)surprisinglywell,withonlyabout6per
& Hornik 2006). The training phase starts with compiling
cent of the sample scattered into the opposite region.
a training set of data vectors that are tagged as either be-
longing to class A or class B. During learning, the model
will find a hyperplane in the space of data vectors which
separates the elements of A and B with the largest possi- 5 GENERATING REALISTIC RANDOM
ble margin2. Once the model is trained, it can be used to EMISSION LINES
classify any query point into one of the two classes.
5.1 The stochastic recipe
Basedonourfindings,weproposeasimplestochasticrecipe
4.2 Automatic star-forming/AGN classification
togeneratearealisticdistributionofemissionlinesforstel-
We compiled the training set from our emission line galaxy larpopulationsynthesismodelsthatprovidethecontinuum
samplebyselectinggalaxiesontheBPTthatcouldbeclas- only. The algorithm works by expressing the model contin-
uum as a linear combination of the basis vectors derived
from PCA of the continua of SDSS galaxies. According to
2 Since this is often not possible in the original space of data
these principal components, the model spectrum is classi-
vectorsbecausethedistributionsofthetwohalvesofthetraining
fied into one of the 60 continuum classes. We used k-means
setarenon-convex,kernelfunctionsareusedtomaptrainingset
clustering to define the continuum classes, as described in
vectors into a higher dimensional space where linear segregation
ispossible(Sch¨olkopfetal.2000).Anotheroptiontohandlenon- Sec. 5.3.
convexsituationsistofindabestpossiblesegregationplanewhich Let us denote the average continuum vector with e0,λ
minimizestheoverlap. and the PCA basis vectors with ei,λ, where i indexes the
(cid:13)c 2015RAS,MNRAS000,1–14
