A re-assessment of strong line metallicity conversions in the machine learning era
Hossen Teimoorinia, Mansoureh Jalilkhany, Jillian M. Scudder, Jaclyn Jensen, Sara L. Ellison
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed February 16, 2021 (MN L A TEX style file v2.2)
A re-assessment of strong line metallicity conversions in the machinelearning era.
Hossen Teimoorinia , , Mansoureh Jalilkhany , Jillian M. Scudder , Jaclyn Jensen ,Sara L. Ellison NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, BC, V9E 2E7, Canada Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada Department of Computer Science, University of Victoria, Victoria, BC, V8W 2Y2, Canada Department of Physics and Astronomy, Oberlin College, Oberlin, Ohio, OH 44074, USA
February 16, 2021
ABSTRACT
Strong line metallicity calibrations are widely used to determine the gas phase metallic-ities of individual HII regions and entire galaxies. Over a decade ago, based on the SloanDigital Sky Survey Data Release 4 (SDSS DR4), Kewley & Ellison published the coe ffi cientsof third-order polynomials that can be used to convert between di ff erent strong line metallic-ity calibrations for global galaxy spectra. Here, we update the work of Kewley & Ellison inthree ways. First, by using a newer data release (DR7), we approximately double the numberof galaxies used in polynomial fits, providing statistically improved polynomial coe ffi cients.Second, we include in the calibration suite five additional metallicity diagnostics that havebeen proposed in the last decade and were not included by Kewley & Ellison. Finally, wedevelop a new machine learning approach for converting between metallicity calibrations.The random forest algorithm is non-parametric and therefore more flexible than polynomialconversions, due to its ability to capture non-linear behaviour in the data. The random forestmethod yields the same accuracy as the (updated) polynomial conversions, but has the signif-icant advantage that a single model can be applied over a wide range of metallicities, withoutthe need to distinguish upper and lower branches in R calibrations. The trained randomforest is made publicly available for use in the community. Key words: galaxies: abundances — galaxies: fundamental parameters - methods: data anal-ysis - astronomical data bases: surveys - methods: statistical
The gas phase metallicity of a galaxy is a fundamental quantitythat captures the cumulative history of stellar chemical enrichment,outflows and delivery of pristine material from the intergalacticmedium. Commonly denoted as 12 + log(O / H), this metallicity isencoded into the emission line spectrum of an HII region or galaxy,and can be derived ‘directly’ by solving for the electron temper-ature and density. Electron densities are readily determined fromnebular doublets, such as [OII] λλ T e ) are similarly readily obtained, by measuring therelative strengths of emission lines of a given species that origi-nate from di ff erent upper energy levels, e.g. [OIII] λ λ T e limits this method to metallicities below ap-proximately 12 + log (O / H) ∼ T e ) metallicitymeasurements (e.g., Pettini & Pagel 2004; Marino et al. 2013; Curtiet al. 2017) and those that are calibrated against theoretical models(e.g., McGaugh 1991; Zaritsky et al. 1994; Kewley & Dopita 2002)Although the strong line metallicity calibrations show goodinternal consistency, there are strong systematic o ff sets betweenthem. Therefore, attempting to combine published measurementsof gas phase metallicity that have used di ff erent metallicity calibra-tions will result in error, with o ff sets as large as 0.7 dex (e.g., Kew-ley & Ellison 2008). In order to facilitate the combination of metal-licities derived using di ff erent methods, Kewley & Ellison (2008)published an extensive assessment of the major strong line metal-licity diagnostics of the time. Using metallicity data for ∼ © a r X i v : . [ a s t r o - ph . GA ] F e b galaxies in the Data Release 4 (DR4) of the Sloan Digital Sky Sur-vey (SDSS), Kewley & Ellison (2008) published tables of coef-ficients for third-order polynomials that could be used to convertbetween any pair of diagnostics used in their study.Although the metallicity conversions of Kewley & Ellison(2008) continue to be widely used, in the decade since their pub-lication several advances have been made in the community. First,the samples of star-forming galaxies that can be used to calibratemetallicities has grown considerably. Secondly, several new metal-licity calibrations have been introduced in the literature for whichno conversions (to other diagnostics) exist. Finally, the rapid growthof machine learning technologies means that new methods exist forcalibrating between di ff erent diagnostics. Such methods o ff er nu-merous advantages over the polynomial fitting approach of Kewley& Ellison (2008). For example, they can capture non-linearities inthe data, make no assumptions on the form of the conversion ( viz the third-order polynomial used in Kewley & Ellison 2008) andsimplify the process by avoiding di ff erent functional forms in dif-ferent regimes (e.g. upper and lower metallicity branches).Machine learning has already gained a solid foothold in as-tronomical data processing. For example, neural networks anddeep learning methods have been widely used for both imageclassification (Bottrell et al. 2019; Huertas-Company et al. 2019;´Ciprijanovi´c et al. 2020; Ferreira et al. 2020; Teimoorinia et al.2020a,b), ranking tasks (Teimoorinia et al. 2016; Bluck et al. 2019;Dey et al. 2019; Ellison et al. 2020), as well as for regression andpattern recognition applications (Ellison et al. 2016; Teimooriniaet al. 2017). In the realm of interstellar medium studies, neural net-works and other techniques, such as random forests, have also beenused to predict emission line fluxes (Teimoorinia & Ellison 2014)and metallicities from either broad band photometry (Acquavivaet al. 2015) or spectra (Ucci et al. 2018; Ho 2019).Given the advances made since the original work of Kewley& Ellison (2008), the goal of the work presented here is three-fold.First, using a larger sample of star-forming galaxies drawn fromthe SDSS Data Release 7 (DR7; Sec.2), in Sec. 3 we re-assessthe polynomial-based metallicity conversions presented byKewley& Ellison (2008). We show that there are small systematic errors(up to ∼ ffi cients aretabulated, providing a statistically improved set of functions for re-searchers desiring to use polynomial based conversions betweenmetallicity diagnostics (Sec. 3.1). Second, we extend the originalsuite of metallicity diagnostics with a further five calibrations thathave been presented in the literature since the original work ofKewley & Ellison (2008). Finally, in Sec. 4, we develop a randomforest alternative to the polynomial conversion approach. We em-phasize that our goal is not to develop new metallicity calibrations,but rather to facilitate the conversion between existing methods us-ing modern samples and methods. Our results are summarized inSec. 5. Despite its long presence in the literature, the DR7 remains one of themost widely used galaxy datasets for spectroscopic work in the nearby uni-verse, thanks to the public MPA / JHU database of physical properties, in-cluding emission line fluxes.
We refer the reader to more extensive treatises on metallicitymeasurements in galaxies for a full pedagogical description andtheoretical framework of this field (e.g., Pagel 1989; Osterbrock1989; Stasi´nska 2006; L´opez-S´anchez et al. 2012; P´erez-Montero& Amor´ın 2017; Kewley et al. 2019). Here, we review only the es-sential concepts, data selection and metallicity calibrations used inKewley & Ellison (2008) as context for our re-assessment of thatwork.Strong line metallicity diagnostics combine the fluxes fromtwo or more nebular emission lines, which can be input into (usu-ally, simple polynomial) functions that have been either empiricallyor theoretically calibrated in order to solve for the gas phase metal-licity, O / H. The function that converts line fluxes to O / H are not al-ways monotonic. One of the most well known examples of a strongline diagnostic that is double valued in O / H for a given line ratio, isthe family of R diagnostics, where R = [ OII ] λ + [ OIII ] λ + [ OIII ] λ H β (1)Additional constraints are required to break the R degener-acy in order to determine whether the galaxy (or HII region) lieson the upper or lower branch of the calibration. Kewley & Elli-son (2008) use the ratio of [NII] / [OII] to break the R degeneracy,since this ratio is not sensitive to ionization parameter and largelymonotonic with metallicity in the range of SDSS galaxy metallici-ties. Whilst this approach can work well for metal-rich and metal-poor galaxies, at intermediate metallicities the calibration can beambiguous and O / H is more uncertain.Kewley & Ellison (2008) select their galaxy sample from theSDSS DR4. A S / N of at least 8 is required in the [OII], [OIII],[NII], [SII] and Balmer lines used in the metallicity calibrations. g -band fibre covering fractions are required to be at least 20 percent, enforcing an e ff ective lower redshift cut z ∼ .
04. An upperredshift cut of z < . ∼ ff erent metallicitycalibrations: the direct (or T e ) method and nine di ff erent strong linemethods. T e based metallicities are not available for the majorityof individual SDSS galaxies, hence we consider it no further here(but see (Andrews & Martini 2013; Curti et al. 2017) for worksthat have derived T e based metallicities for SDSS spectral stacks).Of the nine strong line methods, we exclude two from our currentwork. First, we exclude the calibration of Denicol´o et al. (2002);this calibration is based on the ratio of [NII] / H α and has been su-perseded by several other calibrations used in this work. Second,we also exclude Pilyugin & Thuan (2005); as shown by Kewley& Ellison (2008) this diagnostic does not cross-calibrate well withother methods.Below we briefly review the remaining seven strong line meth-ods used by Kewley & Ellison (2008) that we will re-calibrate inthe work presented here: • McGaugh (1991). Hereafter M91, this calibration uses R andis calibrated using theoretical models. Kewley & Ellison (2008)calibrate the upper and lower branches of this diagnostic with © , 000–000 etallicity conversions separate polynomial fits, whose degeneracy is broken with the[NII] / [OII] ratio. • Zaritsky et al. (1994). Hereafter Z94, this is an average of threepreviously published theoretically calibrated R diagnostics and isapplicable only to the upper branch. • Kewley & Dopita (2002). Hereafter KD02, this diagnostic istheoretically calibrated. KD02 is applied in two regimes - at highmetallicities the calibration is based on the ratio [NII] / [OII], and atlow metallicities it uses an average of several R methods. • Kobulnicky & Kewley (2004). Hereafter KK04, this calibra-tion uses the same theoretical grids as KD02. Application of theKK04 calibration is a multi-step process that entails an assessmentof whether the galaxy is on the upper or lower R branch, follow-ing an initial estimate of the ionization parameter from the ratio of[OIII] / [OII]. The final metallicity and ionization parameter is de-termined through an iterative process based on R . • Pettini & Pagel (2004) N2 and O3N2. Hereafter PP04 N2and PP04 O3N2, these diagnostics are both empirically cali-brated against direct ( T e ) metallicities, and use either the ratio of[NII] / H α (PP04 N2) or additionally including [OIII] / H α (PP04O3N2). Specifically, the following indices are defined: N = log[([NII] λ / H α )] (2)and O N = log[([OIII] λ / H β ) / ([NII] λ / H α )] (3)The advantage of both of N2 and O3N2 calibrations is that theyare single valued and do not require dust correction of the emissionline fluxes, because the lines used in the ratios are close in wave-length. However, PP04 N2 (as with any N2 based calibration) hasthe limitation that it tends to saturate at higher metallicities. • Tremonti et al. (2004). Hereafter T04, this calibration uses abroader suite of emission lines, including [NII] and [SII] in additionto the commonly used oxygen and Balmer lines. Unlike the othercalibrations used by Kewley & Ellison (2008), T04 derive metallic-ities from a statistical assessment of a large suite of spectral synthe-sis models. The T04 metallicities are therefore not computed by us(nor were they by Kewley & Ellison 2008) from the raw emissionline fluxes, but rather are taken directly from the MPA / JHU cata-log. We note that the large number of galaxies with available T04metallicities, despite the large number of lines required, is becauseof the specific S / N requirements adopted in that work (which arenot applied to all of the listed lines).
We include five new metallicity calibrations in the current studythat were not available to Kewley & Ellison (2008), but have be-come widely used in the literature since their publication. We re-view here their main properties, for comparison with the diagnos-tics used in Kewley & Ellison (2008), but refer the reader to theoriginal papers for full details. • Marino et al. (2013) N2 and O3N2. Hereafter M13 N2 andM13 O3N2, this study compiled over 600 HII regions with T e based metallicities. Two strong line diagnostics were calibrated A small number (6 / T e based metallicities used by PP04, we refer to this as an em-pirical method. based upon this large sample, [NII] / H α (i.e. the N2 index) and([OIII] / H β ) / ([NII] / H α ) (i.e. the O3N2 index). As with other N2 di-agnostics, saturation is a problem at higher metallicities, whereasO3N2 can be used up to 12 + log(O / H) ∼ • Dopita et al. (2016). Hereafter D16, this diagnostic uses aunique set of emission lines amongst those studied here. By com-bining [NII], H α and [SII], which are all located close togetherin wavelength space, this metallicity diagnostic o ff ers several ad-vantages. First, it is independent of extinction. Second, all of theemission lines can be covered in one spectral setting, even if theinstrumental coverage is relatively narrow. Finally, this diagnosticis characterised by a linear function up to super-solar metallicities(12 + log(O / H) ≈ • Curti et al. (2017) N2 and O3N2. Hereafter C17 N2 and C17O3N2, this study used over 110,000 star-forming galaxies drawnfrom the SDSS DR7 (the same dataset from which we will drawour sample) to derive an empirical metallicity calibration over anunprecedentedly broad range in O / H by stacking galaxies in nar-row bins of [OII] and [OIII] relative to H β . The very high S / N ra-tios achieved in the stacking process enable a detection of [OIII] λ T e based metallicities for two strongline ratios, N2 and O3N2.Table 1 summarizes the 12 diagnostics used in this paper;seven from the original Kewley & Ellison (2008) work and five newones. Table 1 also summarizes the metallicity regime over whichthe calibrations are valid and the emission lines used. Emission line fluxes used in this work are taken from the pub-licly available SDSS DR7 MPA / JHU catalog , e.g. Brinchmannet al. (2004). The emission lines in this catalog have been cor-rected for underlying stellar absorption and for Galactic extinction.We further correct for internal extinction by assuming an intrinsicH α / H β = .We begin by selecting all galaxies from the DR7 that are clas-sified as star-forming according to the Kau ff mann et al. (2003) def-inition. For this selection, we require that the S / N in the four emis-sion lines required for the star-forming classification have a S / N >
3. Approximately 159,000 galaxies are thus selected.In order to identify galaxies for which robust, global metallic-ities may be estimated, further cuts are required on emission lineS / N and fibre covering fraction. In order to facilitate an equitablecomparison with the conversions of Kewley & Ellison (2008), weadopt the same criteria as in that work, i.e., g -band fibre coveringfraction of at least 20 per cent, an upper redshift cut of 0.1 and aminimum S / N in the emission lines required for a given calibrationof 8. The application of the first two of these criteria result in asample that is reduced from ∼ ∼ / N re-quirement further reduces the sample to a size that depends on thespecific set of emission lines required for a given calibration. Thefinal column of Table 1 summarizes the number of galaxies selectedfor each calibration. The number of galaxies used in each pairwise https: // / SDSS / We test the influence of choosing a Milky Way type extinction curve andfind that it has negligible impact on the calculated metallicities. ©000
3. Approximately 159,000 galaxies are thus selected.In order to identify galaxies for which robust, global metallic-ities may be estimated, further cuts are required on emission lineS / N and fibre covering fraction. In order to facilitate an equitablecomparison with the conversions of Kewley & Ellison (2008), weadopt the same criteria as in that work, i.e., g -band fibre coveringfraction of at least 20 per cent, an upper redshift cut of 0.1 and aminimum S / N in the emission lines required for a given calibrationof 8. The application of the first two of these criteria result in asample that is reduced from ∼ ∼ / N re-quirement further reduces the sample to a size that depends on thespecific set of emission lines required for a given calibration. Thefinal column of Table 1 summarizes the number of galaxies selectedfor each calibration. The number of galaxies used in each pairwise https: // / SDSS / We test the influence of choosing a Milky Way type extinction curve andfind that it has negligible impact on the calculated metallicities. ©000 , 000–000 Table 1.
The list of metallicity calibrations used in this paper and number of galaxies in the DR7 dataset.Metallicity Reference Range of validity Lines required N galaxiesM91 McGaugh (1991) 7 . < + log(O / H) < . λ , [OII] λ , [OIII] λ , H β . (1994) 12 + log(O / H) > λ , [OII] λ , [OIII] λ , H β . < + log(O / H) < . λ , [OII] λ , [OIII] λ , H β . < + log(O / H) < . λ , [OII] λ , [OIII] λ , H β − . < N2 < − . λ , H α < . λ , H β, H α, [NII] λ . (2004) N / A [NII] λ , [OII] λ , [OIII] λ , [OI] 50774HeI , H β, H α, [SII] λλ , . (2013) − . < N2 < − . λ , H α . (2013) − . < O3N2 < . λ , H β, H α, [NII] λ . (2016) 7 . < + log(O / H) < .
05 [NII] λ , H α, [SII] λλ , . (2016) 7 . < + log(O / H) < .
85 [NII] λ , H α . (2016) 7 . < + log(O / H) < .
85 [OIII] λ , H β, H α, [NII] λ Figure 1.
Assessment of the DR4-derived metallicity conversions on the expanded DR7 dataset. In each panel, the DR7-derived PP04 O3N2 metallicity isshown on the x-axis. For the six other strong line methods calibrations with conversions by Kewley & Ellison (2008) that are used in this paper, the y-axisshows each metallicity diagnostic converted to PP04 O3N2 using the original DR4 polynomial coe ffi cients. The deviations of the data from the diagonal 1:1dashed line demonstrate that the DR4 conversions are not optimized for the DR7 dataset. cross-calibration (which requires a S / N > Derived from a sample of ∼ ffi -cients for a third order polynomial that could be used to convert be-tween di ff erent metallicity calibrations. We begin by re-assessing the validity of these metallicity conversions as applied to our largerDR7 dataset. We will then extend these polynomial based conver-sions to include the five additional diagnostics summarized in Sec-tion 2.2. Of the ten original metallicity diagnostics explored in Kewley &Ellison (2008), conversions between eight of the strong line metal-licity calibrations were presented in Kewley & Ellison (2008). Of © , 000–000 etallicity conversions Figure 2.
The same plot as Fig1, but with the newly determined polynomial coe ffi cients derived from the DR7 dataset. The systematic o ff sets found in Fig1using the DR4 coe ffi cients have now been removed and the converted metallicities perform well. these eight, we have removed Denicolo et al. (2002), for the reasonsdescribed in Sec. 2.1, leaving the following seven strong line diag-nostics: M91, Z94, KD02, KK04, PP04 N2, PP04 O3N2 and T04.Gas phase metallicities are computed for the first six of these di-agnostics (the T04 metallicity is taken directly from the MPA / JHUcatalog) for our DR7 sample of galaxies (see the last column ofTable 1 for the number of galaxies available for each diagnostic).Fig. 1 shows some examples of the Kewley & Ellison (2008)conversion coe ffi cients (derived for DR4) as applied to our DR7dataset. For this exemplar, we have selected the PP04 O3N2 as thetarget diagnostic, which is shown on the x-axis of each panel ofFig. 1 for the DR7 dataset. Each of the other six diagnostics is thenconverted onto the PP04 O3N2 system using the polynomial co-e ffi cients derived by Kewley & Ellison (2008). If the conversionsworked perfectly, then the converted version of a given calibration(y-axis) should be identical to its directly measured value (x-axis),and the points will line up along the diagonal 1:1 dashed line.The results shown in Fig. 1 (and a thorough assessment of allthe combinations of targets and conversions, not shown here forbrevity) confirm that the Kewley & Ellison (2008) DR4-derivedcoe ffi cients are not optimized for the DR7 dataset. The o ff setsbetween the converted and target metallicities are typically quitesmall, < ff ects arestatistically significant, and depending on the calibration in ques-tion, can a ff ect a significant fraction of galaxies.We determine that the origin of these o ff sets and biases islikely due to changes in the emission line flux values, which aretypically higher in the DR7 than in the DR4. While uniform en-hancements in the line fluxes should not produce systematic biasesin metallicity values, we find that not all emission lines are stronger in the DR7 by the same percentage, which will present a bias whencomparing metallicities based on di ff erent emission line ratios. Fig.1 shows conversions between calibrations which use di ff erent emis-sion line ratios, and are therefore strongly a ff ected by this change inline strengths. Conversions between calibrations based on the sameemission lines are less discrepant between the two data sets.We therefore re-compute improved coe ffi cients between theseven metallicity diagnostics in common between our sample andthat of Kewley & Ellison (2008), using an identical functional formof a third order polynomial. The new coe ffi cients are tabulated inTables A1 – A7. Fig. 2 shows the same combination of conversionsas Fig.1, but using these updated polynomial coe ffi cients. The sys-tematic o ff sets and biases that were present in Fig.1 have now beenlargely eliminated. ffi cients for converting five additionaldiagnostics In addition to the diagnostics previously presented in Kewley & El-lison (2008), we have additionally derived conversions, using thepolynomial method, for the five additional diagnostics summarizedin Sec 2.2 (M13 N2, M13 O3N2, D16, C17 N2 and C17 O3N2). Weagain use a third-order polynomial, in order to be consistent withthe other diagnostic conversions. The coe ffi cients for these new di-agnostic conversions are presented in Tables A8 - A12.In Fig. 3, we show an example of these new conversions, usingC17 O3N2 as the target metallicity diagnostic. Similar to Figs 1 and2, the C17 O3N2 metallicities (derived directly from the SDSS DR7spectra) in a given calibration are plotted on the x-axis, with eachof the other diagnostics, converted into C17 O3N2 via their poly- ©000
The same plot as Fig1, but with the newly determined polynomial coe ffi cients derived from the DR7 dataset. The systematic o ff sets found in Fig1using the DR4 coe ffi cients have now been removed and the converted metallicities perform well. these eight, we have removed Denicolo et al. (2002), for the reasonsdescribed in Sec. 2.1, leaving the following seven strong line diag-nostics: M91, Z94, KD02, KK04, PP04 N2, PP04 O3N2 and T04.Gas phase metallicities are computed for the first six of these di-agnostics (the T04 metallicity is taken directly from the MPA / JHUcatalog) for our DR7 sample of galaxies (see the last column ofTable 1 for the number of galaxies available for each diagnostic).Fig. 1 shows some examples of the Kewley & Ellison (2008)conversion coe ffi cients (derived for DR4) as applied to our DR7dataset. For this exemplar, we have selected the PP04 O3N2 as thetarget diagnostic, which is shown on the x-axis of each panel ofFig. 1 for the DR7 dataset. Each of the other six diagnostics is thenconverted onto the PP04 O3N2 system using the polynomial co-e ffi cients derived by Kewley & Ellison (2008). If the conversionsworked perfectly, then the converted version of a given calibration(y-axis) should be identical to its directly measured value (x-axis),and the points will line up along the diagonal 1:1 dashed line.The results shown in Fig. 1 (and a thorough assessment of allthe combinations of targets and conversions, not shown here forbrevity) confirm that the Kewley & Ellison (2008) DR4-derivedcoe ffi cients are not optimized for the DR7 dataset. The o ff setsbetween the converted and target metallicities are typically quitesmall, < ff ects arestatistically significant, and depending on the calibration in ques-tion, can a ff ect a significant fraction of galaxies.We determine that the origin of these o ff sets and biases islikely due to changes in the emission line flux values, which aretypically higher in the DR7 than in the DR4. While uniform en-hancements in the line fluxes should not produce systematic biasesin metallicity values, we find that not all emission lines are stronger in the DR7 by the same percentage, which will present a bias whencomparing metallicities based on di ff erent emission line ratios. Fig.1 shows conversions between calibrations which use di ff erent emis-sion line ratios, and are therefore strongly a ff ected by this change inline strengths. Conversions between calibrations based on the sameemission lines are less discrepant between the two data sets.We therefore re-compute improved coe ffi cients between theseven metallicity diagnostics in common between our sample andthat of Kewley & Ellison (2008), using an identical functional formof a third order polynomial. The new coe ffi cients are tabulated inTables A1 – A7. Fig. 2 shows the same combination of conversionsas Fig.1, but using these updated polynomial coe ffi cients. The sys-tematic o ff sets and biases that were present in Fig.1 have now beenlargely eliminated. ffi cients for converting five additionaldiagnostics In addition to the diagnostics previously presented in Kewley & El-lison (2008), we have additionally derived conversions, using thepolynomial method, for the five additional diagnostics summarizedin Sec 2.2 (M13 N2, M13 O3N2, D16, C17 N2 and C17 O3N2). Weagain use a third-order polynomial, in order to be consistent withthe other diagnostic conversions. The coe ffi cients for these new di-agnostic conversions are presented in Tables A8 - A12.In Fig. 3, we show an example of these new conversions, usingC17 O3N2 as the target metallicity diagnostic. Similar to Figs 1 and2, the C17 O3N2 metallicities (derived directly from the SDSS DR7spectra) in a given calibration are plotted on the x-axis, with eachof the other diagnostics, converted into C17 O3N2 via their poly- ©000 , 000–000 Figure 3.
Metallicity conversions using the polynomial method and DR7 data for an expanded set of diagnostics. In this example, C17 O3N2 is the targetmetallicity diagnostic and the other 11 calibrations have been converted to it using the coe ffi cients in the Appendix tables. nomial fit, are plotted on the y-axis. The new conversions show nosystematic o ff sets or skewed behaviour, confirming that the third-order polynomial fit between pairs of calibrations is a good repre-sentation of the data.Taken together, the updated coe ffi cients in Tables A1 – A12represent an improvement in both accuracy and available diag-nostics over the original work on this topic by Kewley & Elli-son (2008). Given the cumbersome nature of the table format, wehave also developed a Graphical User Interface (GUI) into whichthese coe ffi cients are coded, to facilitate the use of the polyno-mial functions within the community. We note for transparency thatthe metallicities output by this GUI are rounded to three decimalplaces. The GUI can be downloaded here: Polynomial-link. Figures 2 and 3 show that, for the 12 strong line metallicity calibra-tions presented here, a third-order polynomial can be used to con-vert between diagnostics with no systematic o ff sets. None the less,there are several reasons to motivate an investigation of non-linearsolutions to these conversions. First, since a polynomial function isa linear model based on coe ffi cients, it is less able to capture anynonlinear features in the correlations between calibrations than arandom forest model which is nonlinear by construction. A further potential advantage of a machine learning approach is that degen-eracy breaking decisions (such as upper and lower branch of R )are naturally included in the model (see also the discussion in Ucciet al. 2018; Ho et al. 2019). A random forest (RF) is a supervised and non-parametric modelused for regression and classification problems. In a non-parametricmodel, there is no internal parameter to train and learn. Since an RFis a supervised method, it implies that for a vector of input parame-ters, there is a target vector that we want to predict. An RF consistsof several similar decision trees - the building blocks of the ran-dom forest model. A decision tree is a system of yes / no options(see Fig.4 for a schematic of the process). This system is a recur-sive partitioning model in which, from the root node, the input datais repeatedly divided and sub-divided. First, the root node is splitinto two (internal) nodes based on the binary decisions on an inputvalue of the data under study. Then the procedure is repeated for thetwo nodes (i.e., in layer-2). In this way, the depth can be increased.The depth of a tree is a parameter that is chosen by the user. Wecan stop the tree from growing according to some criteria such asthe maximum depth or the minimum samples in a node. The endresult is a decision tree with final nodes that are called leaves. Each © , 000–000 etallicity conversions Figure 4.
A schematic diagram of the structure of a decision tree. A randomforest is comprised of many individual decision trees, which mitigates theproblem of over-fitting the data. leaf consists of a set of arranged input parameters and the corre-sponding target value. In a regression problem, the average valueof the targets in a leaf will be predicted in that leaf. A single treehas the potential for overfitting as we increase the depth of the tree.To tackle this problem, we can extend the use of a single decisiontree to a ‘forest’ (Tin Kam Ho 1998).In a RF model, a set (e.g., 100) of similar trees are made (i.e.,with the same depth and the same decision procedure). Input to thefirst tree is a portion (e.g., 20%) of the primary input data, which israndomly selected (with replacement) and then fed to the tree. Thisprocedure is repeated for all trees one by one. In this way, eachtree’s prediction will be a slightly di ff erent from other trees dueto the randomization procedure. The final regression prediction isthe average of all predicted results of all similar trees. This methodprevents overfitting and generally gives better and more accurate re-sults. One advantage of using an RF model is that for a two-branchproblem, such as R metallicity conversions in Kewley & Ellison(2008), one single RF model can predict the metallicity values inbranches, without the need for a user-defined break point betweenthe branches. An RF approach therefore not only reduces the num-ber of required models, but also removes the need for subjectiveboundaries. In this work, we use the RF package from scikit-learn(Pedregosa et al. 2011). We use an RF with 100 estimators (trees)and increase the depth of the trees to have at least 10 members(metallicities) in leaves, which is found to be the optimal numberfor the present work. Fig. 5 has the same format as Fig. 3, i.e. using C17 O3N2 as thetarget and converting from the 11 other strong line methods. Fig.5 shows that the RF models have successfully converted from thebase to the target metallicity diagnostic with no systematic o ff setsor skewness.Visually, Figs. 5 (RF conversions with C17 O3N2 as the tar-get) and 2 (DR7 polynomial fits with C17 O3N2 as the target) lookvery similar. In Fig. 6 we compare the two methods quantitativelyby plotting the scatter within our new polynomial (blue circles)and random forest (red squares) conversions. In this example, C17O3N2 is again the target metallicity and the scatter is shown foreach of the other 11 metallicity calibrations used in the conver-sion. From Fig. 6, it can be seen that for any given base calibration,when converting to the C17 O3N2 calibration, the scatter for thetwo techniques is almost identical. We have found this to be gener-ally true for any combination of metallicity diagnostics, indicating that in terms of overall performance, the polynomial and RF meth-ods are equally strong. Therefore, the random forest approach doesnot o ff er any performance advantage over the polynomial method .However, operationally, the random forest is simpler for the user,requiring no subjective assessment of degeneracies or iterative so-lution for ionization parameter. We caution that this RF model hasbeen trained and tested on the DR7 data set only, and may not trans-fer well to other data sets.In order to facilitate the application of our RF model withinthe community, we have developed a GUI for all metallicity con-versions in this paper. The interface uses 132 di ff erent RF modelsfor the conversions. As with the polynomial GUI, the metallicitiesoutput by this RF GUI are rounded to 3 decimal places. The codecan be downloaded from (RF-link) and uses the same format as thepolynomial GUI also provided as a companion to this paper.Kewley & Ellison (2008) demonstrated (their Fig. 2) that themass-metallicity relation shows a range of 0.7 dex in metallicity(at fixed stellar mass) for the suite of calibrations used in that pa-per. Kewley & Ellison (2008) also demonstrated (their Fig. 4) thatafter the application of their metallicity conversions, these varia-tions were e ff ectively removed. In Fig. 7 and Fig. 8 we repeat thisdemonstration with the full suite of 12 calibrations used in this pa-per and the application of the random forest conversions. In Fig.7, we show a third order polynomial fit to the mass-metallicityrelation for the 12 calibrations in the original metallicity sample,prior to any conversions, described in Table 1. The stellar massescome from the MPA-JHU data release of Brinchmann et al. (2004).Broadly consistent with the results of Kewley & Ellison (2008),we find a 0.6 dex range in metallicity between calibrations at fixedstellar mass. This is slightly less than the range found by Kewley &Ellison (2008) because we have excluded the most deviant calibra-tion (Pilyugin & Thuan 2005) from our suite. In Fig. 8, we show theimprovement to the mass-metallicity relation for the DR7 sampleafter the application of the RF metallicity conversions. In each ofthe 12 panels, one of the metallicity calibrations is selected in turnas the target. The other 11 diagnostics are then converted to thistarget using our RF model. For each of the 12 metallicity calibra-tions, it can be seen that after the RF conversion has been applied,the mass-metallicity relation is invariant to the choice of metallicitydiagnostic. We have presented a three step re-assessment of the metallicitycalibration conversions originally presented in Kewley & Ellison(2008). First, we have approximately doubled the number of star-forming galaxies in the sample, by extending the original DR4 sam-ple used by Kewley & Ellison (2008) to the DR7. Our completestar-forming galaxy sample, with cuts on covering fraction and red-shift, contains ∼ / N) typically contain ∼ Because of the discrete nature of the RF ‘leaf’, there may be more clus-tering around particular target metallicities in the RF model, compared tothe polynomial conversion. The preservation of equal magnitudes of scatterindicates that this is not causing major biases in the conversion process. ©000
A schematic diagram of the structure of a decision tree. A randomforest is comprised of many individual decision trees, which mitigates theproblem of over-fitting the data. leaf consists of a set of arranged input parameters and the corre-sponding target value. In a regression problem, the average valueof the targets in a leaf will be predicted in that leaf. A single treehas the potential for overfitting as we increase the depth of the tree.To tackle this problem, we can extend the use of a single decisiontree to a ‘forest’ (Tin Kam Ho 1998).In a RF model, a set (e.g., 100) of similar trees are made (i.e.,with the same depth and the same decision procedure). Input to thefirst tree is a portion (e.g., 20%) of the primary input data, which israndomly selected (with replacement) and then fed to the tree. Thisprocedure is repeated for all trees one by one. In this way, eachtree’s prediction will be a slightly di ff erent from other trees dueto the randomization procedure. The final regression prediction isthe average of all predicted results of all similar trees. This methodprevents overfitting and generally gives better and more accurate re-sults. One advantage of using an RF model is that for a two-branchproblem, such as R metallicity conversions in Kewley & Ellison(2008), one single RF model can predict the metallicity values inbranches, without the need for a user-defined break point betweenthe branches. An RF approach therefore not only reduces the num-ber of required models, but also removes the need for subjectiveboundaries. In this work, we use the RF package from scikit-learn(Pedregosa et al. 2011). We use an RF with 100 estimators (trees)and increase the depth of the trees to have at least 10 members(metallicities) in leaves, which is found to be the optimal numberfor the present work. Fig. 5 has the same format as Fig. 3, i.e. using C17 O3N2 as thetarget and converting from the 11 other strong line methods. Fig.5 shows that the RF models have successfully converted from thebase to the target metallicity diagnostic with no systematic o ff setsor skewness.Visually, Figs. 5 (RF conversions with C17 O3N2 as the tar-get) and 2 (DR7 polynomial fits with C17 O3N2 as the target) lookvery similar. In Fig. 6 we compare the two methods quantitativelyby plotting the scatter within our new polynomial (blue circles)and random forest (red squares) conversions. In this example, C17O3N2 is again the target metallicity and the scatter is shown foreach of the other 11 metallicity calibrations used in the conver-sion. From Fig. 6, it can be seen that for any given base calibration,when converting to the C17 O3N2 calibration, the scatter for thetwo techniques is almost identical. We have found this to be gener-ally true for any combination of metallicity diagnostics, indicating that in terms of overall performance, the polynomial and RF meth-ods are equally strong. Therefore, the random forest approach doesnot o ff er any performance advantage over the polynomial method .However, operationally, the random forest is simpler for the user,requiring no subjective assessment of degeneracies or iterative so-lution for ionization parameter. We caution that this RF model hasbeen trained and tested on the DR7 data set only, and may not trans-fer well to other data sets.In order to facilitate the application of our RF model withinthe community, we have developed a GUI for all metallicity con-versions in this paper. The interface uses 132 di ff erent RF modelsfor the conversions. As with the polynomial GUI, the metallicitiesoutput by this RF GUI are rounded to 3 decimal places. The codecan be downloaded from (RF-link) and uses the same format as thepolynomial GUI also provided as a companion to this paper.Kewley & Ellison (2008) demonstrated (their Fig. 2) that themass-metallicity relation shows a range of 0.7 dex in metallicity(at fixed stellar mass) for the suite of calibrations used in that pa-per. Kewley & Ellison (2008) also demonstrated (their Fig. 4) thatafter the application of their metallicity conversions, these varia-tions were e ff ectively removed. In Fig. 7 and Fig. 8 we repeat thisdemonstration with the full suite of 12 calibrations used in this pa-per and the application of the random forest conversions. In Fig.7, we show a third order polynomial fit to the mass-metallicityrelation for the 12 calibrations in the original metallicity sample,prior to any conversions, described in Table 1. The stellar massescome from the MPA-JHU data release of Brinchmann et al. (2004).Broadly consistent with the results of Kewley & Ellison (2008),we find a 0.6 dex range in metallicity between calibrations at fixedstellar mass. This is slightly less than the range found by Kewley &Ellison (2008) because we have excluded the most deviant calibra-tion (Pilyugin & Thuan 2005) from our suite. In Fig. 8, we show theimprovement to the mass-metallicity relation for the DR7 sampleafter the application of the RF metallicity conversions. In each ofthe 12 panels, one of the metallicity calibrations is selected in turnas the target. The other 11 diagnostics are then converted to thistarget using our RF model. For each of the 12 metallicity calibra-tions, it can be seen that after the RF conversion has been applied,the mass-metallicity relation is invariant to the choice of metallicitydiagnostic. We have presented a three step re-assessment of the metallicitycalibration conversions originally presented in Kewley & Ellison(2008). First, we have approximately doubled the number of star-forming galaxies in the sample, by extending the original DR4 sam-ple used by Kewley & Ellison (2008) to the DR7. Our completestar-forming galaxy sample, with cuts on covering fraction and red-shift, contains ∼ / N) typically contain ∼ Because of the discrete nature of the RF ‘leaf’, there may be more clus-tering around particular target metallicities in the RF model, compared tothe polynomial conversion. The preservation of equal magnitudes of scatterindicates that this is not causing major biases in the conversion process. ©000 , 000–000 Figure 5.
The same as Fig. 3 but using a random forest method for calculating the metallicity conversions.
Figure 6.
A comparison between the scatter obtained by the new polyno-mial equations and the RF results. The results do not show a significantdi ff erence. The title shows the base, and on the ‘ x ’ axis the input is shown. is therefore to repeat the third order polynomial fitting procedureused by Kewley & Ellison (2008) to derive new coe ffi cients (Ta-bles A1 - A7) which better represent the DR7 data (Fig. 2). We alsoderive coe ffi cients for five additional strong line metallicity diag-nostics that have been presented in the literature since the original Figure 7.
Third order polynomial fits to the mass-metallicity relations forthe original metallicity sample as recalculated for the DR7, for each of the12 calibrations, described in Table 1. work of Kewley & Ellison (2008) (Fig. 3 and Tables A8 - A12).Taken together, the coe ffi cients presented in Table A1 - A12 rep-resent the most complete polynomial based conversion factors forglobal galaxy metallicities.The third stage of our re-assessment of metallicity conversionsfocuses on the potential improvement over polynomial fits that ma-chine learning methods may provide. Such methods can potentiallybetter capture complexity in the data, and have the specific advan- © , 000–000 etallicity conversions Figure 8.
The mass-metallicity relations derived from the DR7 data after application of the random forest conversions. In each panel, one of the 12 metallicitycalibrations is selected as the target (top left legend) and the other 11 diagnostics are converted to this target. The curves are all in excellent agreement, showingthat the conversions have successfully re-calibrated each diagnostic to a common scale. ©000
The mass-metallicity relations derived from the DR7 data after application of the random forest conversions. In each panel, one of the 12 metallicitycalibrations is selected as the target (top left legend) and the other 11 diagnostics are converted to this target. The curves are all in excellent agreement, showingthat the conversions have successfully re-calibrated each diagnostic to a common scale. ©000 , 000–000 tage in the realm of metallicity determinations of not requiring sep-arate functions for upper and lower branch R calibrations. We suc-cessfully model the metallicity conversions between each of the 12diagnostics presented in this work (Fig 5). Despite the additionalpotential of the random forest compared to a simple third orderpolynomial fit, the two methods perform very similarly. The pre-dicted metallicity conversions are very similar (e.g. comparing Fig.3 and Fig. 5) and the uncertainties of the two methods are almostidentical (Fig. 6). Nonetheless, the RF o ff ers the afore-mentionedadvantage of avoiding uncertainty in decisions concerning upperand lower branch R branches, and therefore may be more robustfor metallicities in the turnover regime at intermediate metallici-ties. We have made our random forest models publicly available ina user-friendly format for use by the community. ACKNOWLEDGEMENTS
SLE gratefully acknowledges the receipt of an NSERC DiscoveryGrant.We are grateful to the MPA / JHU groups for making theirSDSS catalogs public. The work presented here would not havebeen possible without this resource.Funding for the SDSS and SDSS-II has been provided bythe Alfred P. Sloan Foundation, the Participating Institutions, theNational Science Foundation, the U.S. Department of Energy,the National Aeronautics and Space Administration, the JapaneseMonbukagakusho, the Max Planck Society, and the Higher Ed-ucation Funding Council for England. The SDSS Web Site ishttp: // / .The SDSS is managed by the Astrophysical Research Con-sortium for the Participating Institutions. The Participating Institu-tions are the American Museum of Natural History, AstrophysicalInstitute Potsdam, University of Basel, University of Cambridge,Case Western Reserve University, University of Chicago, DrexelUniversity, Fermilab, the Institute for Advanced Study, the JapanParticipation Group, Johns Hopkins University, the Joint Institutefor Nuclear Astrophysics, the Kavli Institute for Particle Astro-physics and Cosmology, the Korean Scientist Group, the ChineseAcademy of Sciences (LAMOST), Los Alamos National Labora-tory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State Uni-versity, Ohio State University, University of Pittsburgh, Universityof Portsmouth, Princeton University, the United States Naval Ob-servatory, and the University of Washington. We have developed two GUIs for all polynomial and RF metallicityconversions in this paper. Each interface uses 132 di ff erent set ofcoe ffi cients and models for the conversions. The data underlyingthis article are available in:Polynomial: (Polynomial-link).Random Forest: (RF-link) References
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APPENDIX A: POLYNOMIAL COEFFICIENTS FROMDR7 DATA
As shown in Fig. 1, the Kewley & Ellison (2008) polynomial coe ffi -cients derived from DR4 data are not optimized for the DR7 datasetused herein. In Tables A1 – A7 we provide updated coe ffi cients forsix of the metallicity diagnostics used in Kewley & Ellison (2008)for the DR7 dataset, using the same polynomial form, i.e., y = a + bx + cx + dx (A1)y is the target calibration (i.e. what you want to convert to), and xis the metallicity that we want to convert from. The coe ffi cients a,b, c , and d can be found for di ff erent targets. For example, supposethat we have an estimated metallicity (x) that has been obtainedby method M91. If we want to convert it to (target) Z94, then weshould use the coe ffi cients in the first row of Table A2.Furthermore, in Tables A8 – A12 we present coe ffi cients forfive more recent metallicity diagnostics included in this paper thatwere not available to Kewley & Ellison (2008). ©000
As shown in Fig. 1, the Kewley & Ellison (2008) polynomial coe ffi -cients derived from DR4 data are not optimized for the DR7 datasetused herein. In Tables A1 – A7 we provide updated coe ffi cients forsix of the metallicity diagnostics used in Kewley & Ellison (2008)for the DR7 dataset, using the same polynomial form, i.e., y = a + bx + cx + dx (A1)y is the target calibration (i.e. what you want to convert to), and xis the metallicity that we want to convert from. The coe ffi cients a,b, c , and d can be found for di ff erent targets. For example, supposethat we have an estimated metallicity (x) that has been obtainedby method M91. If we want to convert it to (target) Z94, then weshould use the coe ffi cients in the first row of Table A2.Furthermore, in Tables A8 – A12 we present coe ffi cients forfive more recent metallicity diagnostics included in this paper thatwere not available to Kewley & Ellison (2008). ©000 , 000–000 Table A1.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = M91 x x-range N -Galaxies a b c d Z94 8 . − . (U) . − . (L) . − . . − .
15 32102 -2193.9176 751.0972 -85.452 3.2434PP04 N2 (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . Table A2.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = Z94 x x-range N -Galaxies a b c d M91 8 . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . Table A3.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = KD02 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . (U) . − . (L) . − .
25 174 -61.3755 16.8597 -1.0199 0 © , 000–000 etallicity conversions Table A4.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = KK04 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − .
25 425 721.01897 -260.4697 31.681 -1.28229Z94 8 . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . Table A5.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = PP04 N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . Table A6.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = PP04 O3N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . ©000
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = PP04 O3N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . ©000 , 000–000 Table A7.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = T04 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . . − . . − . Table A8.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = M13 N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . . − . . − . . − . . − . Table A9.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = M13 O3N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . . − .
65 44082 6083.3786 -2158.8367 255.5638 -10.0782D16 7 . − . . − . . − . © , 000–000 etallicity conversions Table A10.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = D16 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . . − . . − .
65 41348 -718.1404 264.3618 -32.295 1.3239C17 N2 8 . − . . − . Table A11.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = C17 N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . (U) . − . (L) . − . (U) . − . (L) . − . . − . . − . (U) . − . (L) . − . . − . . − .
65 42099 1256.816 -457.1278 55.6542 -2.2528D16 7 . − . . − . Table A12.
Metallicity calibration conversion constants for di ff erent inputs (x). Target (y) = C17 O3N2 x x-range N -Galaxies a b c d M91 (U) . − . (L) . − . . − . (U) . − . (L) . − . . − . . − . . − . (U) . − . (L) . − . . − .
65 44930 1479.4498 -528.8548 63.2182 -2.5125M13 O3N2 8 . − . . − . . − . ©000