diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzianb" "b/data_all_eng_slimpj/shuffled/split2/finalzzianb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzianb" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe abundance of heavy elements in the H~II regions of galaxies\nreflects the past history of star formation and the effects of\ninflows and outflows of gas. A characterization of the evolution\nof chemical abundances for galaxies of different masses is\ntherefore essential to a complete model of galaxy formation that\nincludes the physics of baryons \\citep{delucia04,finlator07}.\nImportant observational constraints for such models come from\ndetermining the scaling relations at different redshifts among\ngalaxy luminosity, stellar mass, and metallicity, which, for\nstar-forming galaxies, typically consists of the oxygen abundance.\nHowever, one of the key challenges is to take the observationally\nmeasured quantities, i.e. strong, rest-frame optical emission-line\nratios, and connect them with the physical quantity of interest,\ni.e. oxygen abundance.\n\n\n\n\\begin{deluxetable*}{lcccccccc}\n\\tablewidth{0pt} \\tabletypesize{\\footnotesize}\n\\tablecaption{Galaxies Observed with Keck~II\nNIRSPEC\\label{tab:obs}}\n\\tablehead{\n\\colhead{~~~~~~~~~~DEEP ID~~~~~~~~~~} &\n\\colhead{~~~~~R.A. (J2000)~~~~~} &\n\\colhead{~~~~~Dec. (J2000)~~~~~} &\n\\colhead{~~~~~$z_{{\\rm H}\\alpha}$~~~~~} &\n\\colhead{~~~~~$B$~~~~~} &\n\\colhead{~~~~~$R$~~~~~} &\n\\colhead{~~~~~$I$~~~~~} &\n\\colhead{~~~~~$M_B$~~~~~} &\n\\colhead{~~~~~$U-B$~~~~~}\n}\n\\startdata\n 42044579 \\dotfill & 02 30 43.46 & 00 42 43.60 & 1.0180 & 23.22 & 22.97 & 22.40 & -21.20 & 0.54 \\\\\n 22046630 \\dotfill & 16 50 13.83 & 35 02 01.78 & 1.0225 & 23.64 & 23.02 & 22.31 & -21.37 & 0.69 \\\\\n 22046748 \\dotfill & 16 50 14.55 & 35 02 04.31 & 1.0241 & 24.43 & 23.76 & 22.90 & -20.86 & 0.86 \\\\\n 42044575 \\dotfill & 02 30 44.85 & 00 42 51.33 & 1.0490 & 23.08 & 22.94 & 22.56 & -21.06 & 0.34 \\\\\n 42010638 \\dotfill & 02 29 08.74 & 00 23 28.40 & 1.3877 & 22.93 & 22.85 & 22.54 & -22.12 & 0.49 \\\\\n 42010637 \\dotfill & 02 29 08.74 & 00 23 32.87 & 1.3882 & 24.20 & 23.98 & 23.72 & -20.87 & 0.44 \\\\\n 42021412 \\dotfill & 02 30 44.55 & 00 30 50.73 & 1.3962 & 24.07 & 23.74 & 23.12 & -21.91 & 0.78 \\\\\n 42021652 \\dotfill & 02 30 44.70 & 00 30 46.19 & 1.3984 & 22.97 & 22.24 & 21.32 & -24.01 & 1.01 \\\\\n\\enddata\n\\tablecomments{Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes and arcseconds.}\n\\end{deluxetable*}\n\n\nIn the local universe, \\citet{tremonti04} have used a sample of\n$\\sim$53,000 emission-line galaxies from the Sloan Digital Sky\nSurvey (SDSS) to investigate the luminosity-metallicity ($L-Z$)\nand mass-metallicity ($M-Z$) relationships. For this sample,\nmetallicities were estimated on the observed spectra of several\nstrong emission lines, including [O II] $\\lambda\\lambda$3726,\n3729, H$\\beta$, [O III]$ \\lambda\\lambda$5007, 4959, H$\\alpha$, [N\nII] $\\lambda\\lambda$6548, 6584, and [S II] $\\lambda\\lambda$6717,\n6731. At increasing redshifts, as the strong rest-frame optical\nemission lines shift into the near-IR, metallicities are typically\nbased on smaller subsets of strong emission lines through the use\nof empirically calibrated abundance indicators\n\\citep[e.g.][]{pp04,pagel79}. Much progress has been made recently\nin assembling large samples of star-forming galaxies with\nabundance measurements at both intermediate redshift\n\\citep{kewley02,savaglio05} and at $z>2$ \\citep{erb06a}. However,\nwe have only begun to gather chemical abundance measurements for\ngalaxies at $z \\sim 1-2$ \\citep[][hereafter Paper\nI,]{maier06,shapley05}. In this work, we continue our efforts to\nfill in the gap of chemical abundance measurements during this\nimportant redshift range, which hosts the emergence of the Hubble\nsequence of disk and elliptical galaxies \\citep{dickinson00}, and\nthe buildup of a significant fraction of the stellar mass in the\nuniverse \\citep{drory05,dickinson03} prior to the decline in\nglobal star formation rate (SFR) density \\citep{madau96}.\n\n\n\n\n\nChemical abundances for high-redshift galaxies are commonly\nestimated using locally calibrated empirical indicators. Yet it is\ncrucial to recognize the fact that a considerable fraction of the\n$z \\sim 1$ and $2$ galaxies with measurements of multiple\nrest-frame optical emission lines do not follow the local\nexcitation sequence described by nearby H~II regions and\nstar-forming galaxies in the diagnostic diagram featuring the [O\nIII] $\\lambda$5007\/H$\\beta$ and [N II] $\\lambda$6584\/H$\\alpha$\nemission-line ratios \\citep[Paper I;][]{erb06a}. On average, the\ndistant galaxies lie offset towards higher [O III]\n$\\lambda$5007\/H$\\beta$ and [N II] $\\lambda$6584\/H$\\alpha$,\nrelative to local galaxies. As discussed in Paper I and\n\\citet{groves06}, several causes may account for this offset, in\nterms of the prevailing physical conditions in the H~II regions of\nhigh-redshift galaxies. The relevant conditions are H~II region\nelectron density, hardness of the ionizing spectrum, ionization\nparameter, the effects of shock excitation, and contributions from\nan active galactic nucleus (AGN). It is still unclear which of\nthese are most important for determining the emergent spectra of\nhigh- redshift galaxies. Understanding this offset in\nemission-line ratios is important, not only because it provides\nevidence that physical conditions in the high redshift universe\nare different from the local ones, but also because the\napplication of an empirically calibrated abundance indicator to a\nset of H~II regions or star-forming galaxies rests on the\nassumption that these objects are similar, on average, to those on\nwhich the calibration is based.\n\nIn this sense, the current work is also motivated by the\ninterpretation of the offset in emission-line ratios among distant\ngalaxies, and an assessment of the reliability of using local\nabundance calibrations for high-redshift star-forming galaxies.\nInstead of focusing on high-redshift objects, another approach is\nto study the properties in a class of nearby objects, which\nexhibit similar offset behavior on the emission-line diagnostic\ndiagram. Unravelling the relations between the physical conditions\nand unusual diagnostic line ratios for such objects aids the\nunderstanding of high-redshift galaxies. The SDSS, with its rich\nset of photometric and spectroscopic information, provides an\nideal local comparison sample.\n\n\nIn this paper we expand on the analysis presented in Paper I, with\nan enlarged sample of DEEP2 star-forming galaxies observed with\nNIRSPEC on the Keck II telescope. The larger number of DEEP2\nobjects with near-IR observations enforces the conclusions drawn\nin the previous work. Furthermore, our detailed study of nearby\nSDSS objects with similar emission-line diagnostic ratios leads to\na clearer physical explanation of the observed properties of the\nDEEP2 galaxies. The DEEP2 sample, near-IR spectroscopic\nobservations, data reduction, and measurements are described in \\S\n2. We present the oxygen abundances derived from measurements of\n[O III], H$\\beta$, H$\\alpha$, and [N II] emission lines in both\nindividual as well as composite spectra in \\S 3. The\nmass-metallicity relationship and its evolution through cosmic\ntime are discussed in \\S 4. In \\S 5 we investigate differences in\n$z \\sim 1.0-1.5$ H II region physical conditions with respect to\nlocal samples by examining nearby SDSS galaxies with similar\nemission-line diagnostic ratios. Finally, we summarize our main\nconclusions in \\S 6. A cosmology with $\\Omega_m = 0.3$,\n$\\Omega_{\\Lambda} = 0.7$, and $h = 0.7$ is assumed throughout.\n\n\\section{DEEP2 Sample, Observations, and Data Reduction}\n\n\\subsection{DEEP2 Target Sample and Near-IR Spectroscopy}\n\nThe high-redshift galaxies presented in this paper are drawn from\nthe DEEP2 Galaxy Redshift Survey \\citep[hereafter\nDEEP2;][]{davis03,faber05}, which contains $>30,000$ galaxies with\nhigh-confidence redshifts at $0.7 \\leq z \\leq 1.5$ down to a\nlimiting magnitude of $R_{AB}=24.1$. The motivation for our\nfollow-up near-infrared spectroscopic program, along with detailed\ndescriptions of the sample selection, optical and near-IR\nphotometry, and spectroscopy are presented in Paper I. Only a\nbrief overview is given here.\n\n\n\nThe new sample contains four galaxies at $z \\sim 1.0$ and four\ngalaxies at $z \\sim 1.4$, which, in combination with the pilot\nprogram presented in Paper I, leads to a sample of 20 galaxies in\ntotal. These galaxies are located in fields 2, 3, and 4 of the\nDEEP2 survey, at 16, 23, and 2 hr right ascension, respectively.\nTo probe chemical abundances and H~II region physical conditions,\nobservations of several strong H~II region emission lines are\nrequired, ideally at least [O II], H$\\beta$, [O III], H$\\alpha$,\nand [N II]. At $z \\geq 0.85$, however, the only strong H~II region\nemission feature contained in the DEEP2 DEIMOS spectroscopic data\nis the [O II] doublet. Therefore, near-IR spectroscopy is needed\nto measure longer wavelength H~II region emission lines at $z \\geq\n1$. We target two narrow redshift windows within the larger DEEP2\nredshift distribution: $0.96 \\leq z \\leq 1.05$ and $1.36 \\leq z\n\\leq 1.50$, within which it is possible to measure the full set of\nH$\\beta$, [O III], H$\\alpha$, and [N II], in spite of the bright\nsky lines and strong atmospheric absorption in the near-IR.\n\n\\begin{figure*}\n\\epsscale{.8} \\plotone{f1.eps} \\caption{DEEP2 color-magnitude and\nmagnitude-stellar mass diagrams. The rest-frame $(U - B)$ vs.\n$M_B$ color-magnitude diagrams ({\\it top}) and the $M_B$ vs.\nstellar mass diagrams ({\\it bottom}) are for DEEP2 galaxies at\n$0.96 \\le z \\le 1.05$ ({\\it left}) and $1.36 \\le z \\le 1.50$ ({\\it\nright}). In each plot, DEEP2 galaxies from both the pilot sample\npresented in Paper I as well as objects with new NIRSPEC\nobservations are indicated with red squares. As shown here, all\nNIRSPEC targets were drawn from the ``blue cloud'' of the color\nbimodality. \\label{fig:colormag}.}\n\\end{figure*}\n\n\nThe absolute $B$ magnitude, $M_B$, and stellar mass estimates are\ngiven in Tables \\ref{tab:obs} and \\ref{tab:emi} for the new\nobjects and plotted as red squares in the lower panels of Figure\n\\ref{fig:colormag}, together with the data for the pilot sample.\nFor all the objects, we use the $M_B$ values from\n\\citet{willmer06} based on optical data alone and confirm their\ngood agreement with estimates based on fits to the $BRIK_S$ SEDs\nthat span through rest-frame $J$ ($I$) band for the objects at $z\n\\sim 1.0$ ($z \\sim 1.4$). Stellar masses for the objects in our\nsample are derived with $K_S$-band photometry, following the\nprocedure outlined in \\citet{bundy05}, which assumes a\n\\citet{chabrier03} stellar initial mass function (IMF). As\ndiscussed in detail in Paper I, the \\citet{bundy05} stellar mass\nmodelling technique agrees well with that used by\n\\citet{kauffmann03a} for SDSS galaxies, based on both spectral\nfeatures and broadband photometry. Stellar masses for galaxies in\nthe pilot sample have been updated to reflect both the most\ncurrent DEEP2 near-IR photometric catalog, and population\nsynthesis models limiting the stellar population age to be younger\nthan the age of the Universe at $z\\sim 1$. Thus, in a few cases,\nthe stellar masses differ slightly from those presented in Table\n(2) of Paper I. As shown in the lower panels of Figure\n(\\ref{fig:colormag}), the $z \\sim 1.4$ galaxies in our sample span\nthe full range of absolute $B$ luminosities in the DEEP2 survey,\nfrom $M_{B} \\sim$ -20 to -23, while the smaller set of $z \\sim\n1.0$ galaxies happens to cover the faint end of the luminosity\nfunction. Galaxies in both redshift intervals cover more than an\norder of magnitude in stellar mass and therefore should be able to\nprobe an interesting dynamic range. Since our goal was to study\nthe emission-line properties of galaxies, all objects in our\nsample lie in the blue component of the observed $(U-B)$ color\nbimodality in the DEEP2 survey, as shown in the upper panels of\nFigure \\ref{fig:colormag}.\n\n\n\n\\begin{deluxetable*}{lcccccccccr}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Emission Lines and Physical Quantities.\\label{tab:emi}}\n\\tablewidth{0pt}\n\\tablehead{\n & & & & & &\n \\multicolumn{2}{c}{12 + log(O\/H)} & & & \\\\\n \\cline{7-8} \\\\\n \\colhead{DEEP ID} & \\colhead{z$_{{\\rm H}\\alpha}$} & \\colhead{F$_{{\\rm H}\\beta}$\\tablenotemark{a}} &\n \\colhead{F$_{{\\rm [O III]}\\lambda5007}$\\tablenotemark{a}} & \\colhead{F$_{{\\rm H}\\alpha}$\\tablenotemark{a}} &\n \\colhead{F$_{{\\rm [N II]}\\lambda6584}$\\tablenotemark{a}} & \\colhead{N2\\tablenotemark{b}} &\n \\colhead{O3N2\\tablenotemark{c}} & \\colhead{$L_{H\\alpha}$\\tablenotemark{d}} &\n \\colhead{SFR$_{{\\rm H}\\alpha}$\\tablenotemark{e}} &\n \\colhead{log$(M_{\\ast}\/M_{\\odot})$\\tablenotemark{f}}\n }\n\\startdata\n42044579 & 1.0180 & 2.4$\\pm$0.4 & 5.4$\\pm$0.3 & 12.1$\\pm$0.3 & 2.8$\\pm$0.3 & 8.54$\\pm$0.18 & 8.41$\\pm$0.14 & 0.7 & 3 & 10.26$\\pm$0.15 \\\\\n22046630 & 1.0225 & 3.7$\\pm$0.5 & 8.7$\\pm$0.4 & 15.2$\\pm$0.3 & 1.8$\\pm$0.3 & 8.37$\\pm$0.18 & 8.32$\\pm$0.14 & 0.8 & 3 & 10.29$\\pm$0.16 \\\\\n22046748 & 1.0241 & 2.2$\\pm$0.4 & 6.4$\\pm$0.3 & 9.6$\\pm$0.2 & 2.3$\\pm$0.2 & 8.55$\\pm$0.18 & 8.39$\\pm$0.14 & 0.5 & 2 & 10.28$\\pm$0.10 \\\\\n42044575 & 1.0490 & 7.7$\\pm$0.7 & 22.9$\\pm$0.3 & 23.6$\\pm$0.3 & 3.6$\\pm$0.3 & 8.43$\\pm$0.18 & 8.32$\\pm$0.14 & 1.4 & 6 & 9.74$\\pm$0.06 \\\\\n42010638 & 1.3877 & 3.8$\\pm$0.9 & 17.3$\\pm$0.4 & 16.6$\\pm$0.5 & 2.8$\\pm$0.5 & 8.46$\\pm$0.19 & 8.27$\\pm$0.15 & 2.0 & 9 & 10.21$\\pm$0.06 \\\\\n42010637 & 1.3882 & 3.3$\\pm$0.9 & 3.7$\\pm$0.3 & 6.4$\\pm$0.4 & 2.9$\\pm$0.4 & 8.70$\\pm$0.18 & 8.60$\\pm$0.15 & 0.8 & 4 & 9.96$\\pm$0.12 \\\\\n42021412 & 1.3962 & 7.1$\\pm$0.8 & $>$ 2.0 & 12.7$\\pm$0.3 & 2.9$\\pm$0.3 & 8.54$\\pm$0.18 & $<$ 8.70 & 1.5 & 7 & 10.89$\\pm$0.13 \\\\\n42021652\\tablenotemark{g} & 1.3984 & 4.6$\\pm$0.5 & 9.7$\\pm$0.5 & 9.5$\\pm$0.3 & 1.2$\\pm$0.3 & 8.38$\\pm$0.19 & 8.33$\\pm$0.15 & 1.1 & 5 & ... \\\\\n\\enddata\n\\tablenotetext{a}{Emission-line flux and random error in units of $10^{-17}$ ergs s$^{-1}$ cm$^{-2}$.} \n\\tablenotetext{b}{Oxygen abundance deduced from the N2 relationship presented in \\citet{pp04}.} \n\\tablenotetext{c}{Oxygen abundance deduced from the O3N2 relationship presented in \\citet{pp04}.}\n\\tablenotetext{d}{H$\\alpha$ luminosity in units of $10^{42}$ ergs s$^{-1}$.} \n\\tablenotetext{e}{Star formation rate in units of $M_{\\odot}$ yr$^{-1}$, calculated from $L_{{\\rm H}\\alpha}$ using the calibration of \\citet{kennicutt98}, and divided by a factor of 1.8 to convert to a \\citet{chabrier03} IMF from the Salpeter IMF assumed by \\citet{kennicutt98}. Note that SFRs have not been corrected for dust extinction or aperture effects, which may amount to a factor of $2$ difference \\citep{erb06c}.}\n\\tablenotetext{f}{Stellar mass and uncertainty estimated using the methods described in Bundy et al. (2005), and assuming a \\citet{chabrier03} IMF.} \\tablenotetext{g}{This object has a double morphology. The separation between the two components is about 0.9$^{''}$, which corresponds to $\\sim$8 kpc at $z = 1.3984$. We measured line fluxes for the emission-line-dominated component, but do not have a robust estimate of the corresponding stellar mass. In the DEEP2 photometry the two components were counted as one source, and the resulting stellar-mass estimate has contribution from both components. We therefore do not include this stellar mass estimate in our sample.}\n\\end{deluxetable*}\n\n\nThe near-IR spectra were obtained on 2005 September 17 and 18 with\nthe NIRSPEC spectrograph \\citep{mclean98} on the Keck~II\ntelescope. Over the range of redshifts of the galaxies presented\nhere, two filter setups are required to measure the full set of\nH$\\beta$, [O III], H$\\alpha$, and [N II]. For objects at $z \\sim\n1.4$, the NIRSPEC 5 filter (similar to $H$ band) is used to\nobserve H$\\alpha$ and [N II], whereas the NIRSPEC 3 filter\n(similar to $J$ band) is used for H$\\beta$ and [O III]. For\nobjects at $z \\sim 1.0$, the NIRSPEC 3 filter is used to observe\nH$\\alpha$ and [N II], whereas the NIRSPEC 1 filter ($\\Delta\\lambda\n=$ 0.95-1.10 $\\mu$m) is used for H$\\beta$ and [O III]. All targets\nwere observed for 3$\\times$900 s in each filter with a\n0.76$^{''}$$\\times$42$^{''}$ long slit. The spectral resolution\ndetermined from sky lines is $\\sim10 {\\rm \\AA}$ for all four\nNIRSPEC filters used here. Photometric conditions and seeing were\nvariable throughout both nights, with seeing ranging from\n0.5$^{''}$ to 0.7$^{''}$ in the near-IR. In order to enhance the\nlong-slit observing efficiency, we targeted two galaxies\nsimultaneously by placing them both on the slit.\n\nWe observed a total of 10 DEEP2 galaxies, successfully measuring\nthe full set of H$\\beta$, [O III], H$\\alpha$, and [N II] for eight\nout of 10. For the remaining pair, we only detected H$\\alpha$ in\nthe $H$ band, but no H$\\beta$ nor [O III] in the $J$-band\nexposures, in which the background in between sky lines was\ncharacterized by a significantly higher level of continuum than\nusual. This anomalous background is likely due to an increased\ncontribution from clouds, which may have affected both $H$- and\n$J$-band observations of the pair. Since a clear measurement was\nnot obtained for these two objects, due to variable weather\nconditions, we exclude them from our study. The object, 42021652,\nhas a double morphology, with one component dominated by emission\nlines with a weak continuum, and another component dominated by\nstrong continuum, with only weak emission lines at roughly the\nsame redshift. The separation between the two components on the\nsky is $\\sim$0.9$^{''}$, which corresponds to $\\sim$8 kpc at $z =\n1.3984$, perhaps indicative of a merger event. This interpretation\nis supported by the small velocity difference of $\\Delta v\\sim\n125$~km s$^{-1}$ between the two components. We measure line\nfluxes for the component dominated by emission lines, since it\nprovides a more robust estimate of line ratios. Deblended optical\nand near-infrared magnitudes would be required to obtain robust\nstellar masses for the individual components. However, in the\nDEEP2 photometry the two components were counted as one source\nsince they are too close to be deblended and the stellar mass has\ncontribution from both of them. For now, we only include flux\nmeasurements of the emission-line component for the\ndiagnostic-line-ratio analysis but do not include this object in\nthe mass-metallicity studies. A summary of the observations\nincluding target coordinates, redshifts, and optical and near-IR\nphotometry is given in Table \\ref{tab:obs}.\n\n\n\n\n\\subsection{Data Reduction and Optimal Background Subtraction}\n\nData reduction was performed with a similar procedure to the one\ndescribed in Paper I and \\citet{erb03}, with the exception of an\nimproved background subtraction method applied to the\ntwo-dimensional galaxy spectral images \\citep[][private\ncommunication]{kelson03,becker06}.\nIn the custom NIRSPEC long-slit reduction package written by D. G.\nBecker (2006, private communication), optimal background\nsubtraction is performed on the unrectified science frames. First,\na transformation is calculated between CCD ($x$, $y$) coordinates\nand those of slit position and wavelength, using the\nwavelength-dependent traces of bright standard stars and the\nspatially dependent curves of bright sky lines. Then a\ntwo-dimensional model of the sky background is constructed as a\nfunction of slit position and wavelength, using a low-order\npolynomial in the slit-position dimension, and a $b$-spline\nfunction in the wavelength dimension. This two-dimensional model\nis iteratively fit in the differenced frame of adjacent science\nexposures and subtracted from the unrectified data. After\nbackground subtraction, cosmic rays were removed from each\nexposure, which was then rotated, cut out along the slit, and\nrectified. Finally, all background-subtracted, rectified exposures\nof a given science target were combined in two dimensions. This\nnew approach to reducing NIRSPEC spectra results in fewer\nartifacts around bright sky lines and cosmic rays, which are\ncommonly introduced when rectification is performed before sky\nsubtraction and cosmic-ray zapping.\nOne-dimensional spectra, along with error spectra, were then\nextracted and flux- calibrated using A-star observations,\naccording to the procedure described in Paper I and \\citet{erb03}.\n\n\n\n\\subsection{Measurements and Physical Quantities}\n\n\n\\begin{figure*}\n\\epsscale{1} \\plotone{f2.eps} \\caption{NIRSPEC spectra of DEEP2\ngalaxies in our new sample at $z\\sim 1.0$. H$\\beta$ and [O III]\nare observed in the NIRSPEC 1 filter, with H$\\alpha$ and [N II] in\nthe NIRSPEC 3 filter (similar to the $J$ band). The 1 $\\sigma$\nerror spectra are shown as dotted lines, offset vertically by $-5\n\\times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ for\nclarity.\\label{fig:spec1}} \\epsscale{1.}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f3.eps} \\caption{NIRSPEC spectra of DEEP2\ngalaxies in our new sample at $z\\sim 1.4$. H$\\beta$ and [O III]\nare observed in the NIRSPEC 3 filter, with H$\\alpha$ and [N II] in\nthe NIRSPEC 5 filter (similar to the $H$ band). The 1 $\\sigma$\nerror spectra are shown as dotted lines, offset vertically by $-5\n\\times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ for\nclarity.\\label{fig:spec2}} \\epsscale{1.}\n\\end{figure*}\n\nOne-dimensional, flux-calibrated NIRSPEC spectra along with the 1\n$\\sigma$ error spectra of galaxies in our new sample are shown in\nFigures \\ref{fig:spec1} and \\ref{fig:spec2}. Emission-line fluxes\nand uncertainties measured from the one-dimensional spectra are\ngiven in Table \\ref{tab:emi}. H$\\alpha$ and [N II] $\\lambda$6584\nemission-line fluxes were determined by first fitting a Gaussian\nprofile to the H$\\alpha$ feature to obtain the redshift and FWHM,\nand using these values to constrain the fit to the [N II] emission\nline. This method is based on the assumption that the H$\\alpha$\nand [N II] lines have exactly the same redshift and FWHM, with the\nH$\\alpha$ line having a higher signal-to-noise ratio (S\/N). For\nmost of the objects in our sample, the [N II] $\\lambda$6548 line\nwas too faint to measure. [O III] $\\lambda$5007 and H$\\beta$\nfluxes were determined with independent fits. In most cases,\nredshifts from H$\\alpha$, [O III] $\\lambda$5007, and H$\\beta$\nagree to within $\\Delta z = 0.0004$ ($\\Delta v = 50-60$ km\ns$^{-1}$ at $z=1.0-1.4$). For the object 42021412, [O III]\n$\\lambda$5007 lies on top of a bright sky line, and only a lower\nlimit is given.\n\n\nSFRs inferred from H$\\alpha$ luminosities using the calibration of\n\\citet{kennicutt98} are shown in Table (\\ref{tab:emi}). The\nresults have been converted from the Salpeter IMF used by\n\\citet{kennicutt98} to a \\citet{chabrier03} IMF by dividing the\nresults by a factor of $1.8$. In our whole sample of 20 galaxies,\nthe H$\\alpha$ fluxes range from $5.6 \\times 10^{-17}$ to $2.4\n\\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$. The mean H$\\alpha$ flux\nfor the sample at $z \\sim 1.0$ ($z \\sim 1.4$) is $1.3 \\times\n10^{-16}$ ($1.2 \\times 10^{-16}$), corresponding to a star\nformation rate of 3 (6) $M_{\\odot}$ yr$^{-1}$, uncorrected for\ndust extinction or aperture effects, which may amount to a factor\nof $2$ difference \\citep{erb06c}. Note that these characteristic\nH$\\alpha$ star formation rates, after being corrected for aperture\neffects, would be significantly higher than those of local\ngalaxies in the SDSS sample of \\citet{tremonti04} and the $\\langle\nz \\rangle = 0.4$ TKRS subsample of \\citet{kobulnicky04}, even\nbefore correction for dust extinction. We also note that the mean\nspecific SFR for both $z\\sim 1.0$ and $z \\sim 1.4$ samples is\n$\\log((SFR\/M_{\\star})\\mbox{ yr}^{-1})=-9.7$.\n\nAs discussed in Paper I, absolute line flux measurements suffer\nfrom several sources of systematic error, which can amount to at\nleast a $\\sim25\\%$ uncertainty \\citep{erb03}. This level of\nuncertainty is present even under photometric conditions, which\nmay not have applied through the full extent of our observations.\nFor the remainder of the discussion we therefore focus on the\nmeasured line {\\it ratios}, [N II] $\\lambda$6584\/H$\\alpha$ and [O\nIII] $\\lambda$5007\/H$\\beta$, which are not only unaffected by\nuncertainties in flux calibration and other systematics but also\nrelatively free from the effects of dust extinction, due to the\nclose wavelength spacing of the lines in each ratio. Hereafter, we\nuse ``[N II]\/H$\\alpha$\" to refer to the measured emission-line\nflux ratio between [N II] $\\lambda$6584 and H$\\alpha$, and ``[O\nIII]\/H$\\beta$\" for that between [O III] $\\lambda$5007 and\nH$\\beta$.\n\n\n\\section{The Oxygen Abundance}\n\nH II region metallicity is an important probe of galaxy formation\nand evolution, as it represents the integrated products of past\nstar formation, modulated by the inflow and outflow of gas. Oxygen\nabundance is often used as a proxy for metallicity since oxygen\nmakes up about half of the metal content of the interstellar\nmedium and exhibits strong emission lines from multiple ionization\nstates in the rest-frame optical that are easy to measure. For\ncomparison, we use the solar oxygen abundance expressed as 12 +\nlog(O\/H) = 8.66 \\citep{allende02,asplund04}.\n\nThe most robust way to estimate the oxygen abundance is the\nso-called direct $T_e$ method, based on the measurement of the\ntemperature-sensitive ratio of auroral and nebular emission lines.\nHowever, in distant galaxies the auroral lines are almost always\nundetectable \\citep[but see][]{hoyos05,kakazu07} since they become\nextremely weak at metallicities above $\\sim$0.5 solar. Even at\nlower metallicities, the auroral lines are typically beyond the\nreach of the low S\/N typical of the spectra of distant galaxies.\nFor distant star-forming galaxies, therefore, measuring strong\nemission-line ratios is the only viable way of obtaining the H II\nregion gas-phase oxygen abundance \\citep{kobulnicky99,pettini01}.\n\nGiven our NIRSPEC data set, and our desire to avoid the systematic\nuncertainties entailed in adding [O II] line fluxes obtained with\nthe DEIMOS spectrograph (without real-time flux-calibration) to [O\nIII] fluxes obtained with NIRSPEC, we focus on two strong-line\nratios as indicators for the oxygen abundance: N2$\\equiv$ log([N\nII]\/H$\\alpha$) and O3N2$\\equiv$ log\\{([O III]\/H$\\beta$)\/([N\nII]\/H$\\alpha$)\\}. These indicators have been calibrated by\n\\citet{pp04} using local H II regions, most of which have direct\n$T_e$ abundance determinations. The sensitivity of the indicators\nto oxygen abundance, as well as their limitations, have been\ndiscussed in \\citet{pp04} and Paper~I. Absolute estimates of\nmetal abundances are quite uncertain, as abundances determined\nwith different indicators or with different calibrations of the\nsame indicator may have substantial biases or discrepancies\n\\citep[e.g.][]{kennicutt03,kobulnicky04}. We therefore emphasize\nrelative abundances determined with the same method, using the\nsame calibration.\n\nThe N2 indicator, pointed out by several works\n\\citep{storchi94,raimann00,denicolo02}, is related to the oxygen\nabundance via\n\\begin{equation}\n{\\rm 12 + log(O\/H) = 8.90 + 0.57 \\times} {\\rm N2},\n\\end{equation}\nwhich is valid for ${\\rm7.50 < 12 + log(O\/H) < 8.75}$, with a 1\n$\\sigma$ scatter of $\\pm0.18$ dex \\citep{pp04}. It has been used\nby \\citet{erb06a} to estimate oxygen abundances for UV-selected $z\n\\sim 2$ galaxies. Note that the N2 indicator is not sensitive to\nincreasing oxygen abundance above roughly solar metallicity, as\nshown with photoionization models \\citep{kewley02}. Thus, for a\nsubset of 12 galaxies in our sample with measurements of the full\nset of H$\\beta$, [O III], H$\\alpha$, and [N II], we also use the\nO3N2 indicator introduced by \\citet{alloin79}, which is expected\nto be particularly useful at solar and super-solar metallicities\nwhere [N II] saturates but the strength of [O III] continues to\ndecrease with increasing metallicity. Using the same calibration\nsample, \\citet{pp04} show that O3N2 is related to the oxygen\nabundance via\n\\begin{equation}\n{\\rm 12 + log(O\/H) = 8.73 - 0.32 \\times} {\\rm O3N2},\n\\end{equation}\nwhich is valid for ${\\rm8.12 < 12 + log(O\/H) < 9.05}$, with a 1\n$\\sigma$ scatter of $\\pm0.14$ dex. Table \\ref{tab:emi} lists\noxygen abundances derived using these two indicators. The errors\non the oxygen abundances are dominated by the systematic\nuncertainties in the calibrations of the indicators.\n\n\n\n\\subsection{Composite Spectra}\n\n\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f4.eps} \\caption{Composite NIRSPEC spectra\nof the 18 $z\\sim 1.0-1.5$ galaxies with both stellar mass\nestimation as well as [N II] and H$\\alpha$ measurement in our\nsample, divided separately at $z \\sim 1.0$ and $z \\sim 1.4$. The 1\n$\\sigma$ error spectra are shown as dotted lines, offset\nvertically by $-3 \\times 10^{-18}$ ergs s$^{-1}$ cm$^{-2}$\n\\AA$^{-1}$ for clarity. The spectra are labeled with the mean\nstellar mass from each bin, and the H$\\alpha$, [N II] and [S II]\nlines are marked by dashed lines. Note that the spectra near the\ndensity-sensitive [S II] lines are very noisy, due to the large\ndispersion of the flux per count from near-IR standard star\ncalibration, caused by low efficiency near the filter\nedge.\\label{fig:compspec:n2}}\n\\end{figure*}\n\n\\begin{figure*}\n\\epsscale{.95} \\plotone{f5.eps} \\caption{Composite NIRSPEC spectra\nof the 12 $z\\sim 1-1.5$ galaxies with both stellar mass estimation\nas well as all-four-line measurement in our sample, divided\nseparately at $z \\sim 1.0$ and $z \\sim 1.4$. The 1 $\\sigma$ error\nspectra are shown as dotted lines, offset vertically by $-3 \\times\n10^{-18}$ ergs s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ for clarity. The\nspectra are labeled with the mean stellar mass from each bin, and\nthe H$\\beta$, [O III], H$\\alpha$, [N II] and [S II] lines are\nmarked by dashed lines.\\label{fig:compspec:o3n2}}\n\\end{figure*}\n\n\n\n\n\nRelative, average abundances determined from composite spectra can\nbe more accurately determined than those from individual spectra.\nAs discussed by \\citet{erb06a}, making a composite spectrum not\nonly reduces the uncertainties associated with the strong-line\ncalibration by a factor $N^{1\/2}$, where $N$ is the number of\nobjects included in the composite spectrum, but also enhances the\nspectrum S\/N since sky lines generally lie at different\nwavelengths for spectra at different redshifts. In addition, one\nof our goals is to determine the average properties of subgroups\nof galaxies in our sample. For the subset of 18 galaxies in our\nsample with H$\\alpha$ and [N II] measurements, as well as stellar\nmass estimates, we divide the sample into four bins by stellar\nmass, with two bins at $z \\sim 1.0$ and two bins at $z \\sim 1.4$.\nFor the subset of 12 galaxies with measurements of not only\nH$\\alpha$ and [N II], but also H$\\beta$ and [O III], we also\ndivide the sample into four bins by stellar mass with two bins\neach at $z \\sim 1.0$ and at $z \\sim 1.4$.\n\nTo make the composite spectra, we first shift the individual\none-dimensional flux-calibrated spectra into the rest frame and\nthen combine them by generating the median spectrum, which\npreserves the relative fluxes of the emission features\n\\citep{vanden01}. We use N2 and N2+O3 composite spectra to refer\nto the composites with H$\\alpha$ and [N II], and those with all\nfour lines, respectively. The N2 and N2+O3 composite spectra,\nlabelled with mean stellar mass in each bin, are shown in Figures\n\\ref{fig:compspec:n2} and \\ref{fig:compspec:o3n2}. The\ncorresponding emission-line flux ratios along with uncertainties\nmeasured from the composite spectra, as well as the inferred\noxygen abundances, are listed in Tables \\ref{tab:comp:n2} and\n\\ref{tab:comp:o3n2}. The listed errors in 12 + log(O\/H) include\nthe uncertainties from the propagation of emission-line flux\nmeasurements, as well as the systematic scatter from the\nstrong-line calibration.\nAs shown in paper I, the systematic discrepancies between the N2-\nand O3N2-based abundances are mainly due to the fact that, on\naverage, DEEP2 galaxies are offset from the excitation sequence\nformed by local H II regions and star-forming galaxies. We discuss\nthis issue in detail in \\S \\ref{sec:offset}.\n\n\n\n\n\\section{The Mass-Metallicity Relation}\\label{sec:mz_relation}\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f6.eps} \\caption{Mass-metallicity relation\nobserved at $z \\sim 1.0$ and $z \\sim 1.4$. In the left panel,\nmetallicities are inferred from the N2 indicator, while the right\npanel shows metallicities estimated from the O3N2 indicator. In\nboth plots, open (filled) diamonds and squares are used for N2\n(N2+O3) composite spectra at $z \\sim 1.0$ and $z \\sim 1.4$,\nrespectively (see section 3.1). Errors show the uncertainties\npropagated from the strong-line ratio measurement, while the\nsystematic uncertainties from the strong-line method calibration\n\\citep{pp04} are shown in the lower right corner of each plot as\ndotted (solid) error bars for N2 (N2+O3) composite spectra. The\nsystematic uncertainties from the calibration are reduced by a\nfactor $N^{1\/2}$, where $N$ is the number of individual spectra\nincluded in each composite. For stellar masses, the horizontal\nbars give the mass range in each bin. Data points for individual\nobjects are shown as filled triangles and circles for $z \\sim 1.0$\nand $z \\sim 1.4$, respectively. Associated error bars are listed\nin Table 2 in both this paper and in Paper~I. For comparison,\nmetallicities as a function of stellar mass are also shown, for\nboth local SDSS galaxies ({\\it grey contours and dots}) and the\n\\citet{erb06a} $z \\sim 2$ sample ({\\it grey open circles}, {\\it\nleft panel}). Note that here and throughout, we use contours and\ndots to show SDSS objects. On each such applicable plot, SDSS data\npoints were mapped onto 10 evenly spaced levels according to\nnumber density, where objects on the lowest level are denoted by\ndots while other levels are presented by contours. Solar\nmetallicity is marked with a horizontal dashed line. It can be\nseen that the N2 indicator saturates near the solar abundance.\n\\label{fig:massz}}\n\\end{figure*}\n\n\nThe redshift evolution of the luminosity-metallicity and\nmass-metallicity relations provides important constraints on\nmodels of galaxy evolution. A correlation between gas-phase\nmetallicity and stellar mass can be explained by either the\ntendency of lower mass galaxies to have larger gas fractions and\nlower star formation efficiencies\n\\citep{mcgaugh97,bell00,kobulnicky03}, or the preferential loss of\nmetals from galaxies with shallow potential wells by\ngalactic-scale winds \\citep{larson74}. In the local universe,\nstrong correlations between rest-frame optical luminosity and the\ndegree of chemical enrichment have been observed in both\nstar-forming and early-type galaxies\n\\citep{garnett87,brodie91,tremonti04}. The correlation has also\nbeen observed in intermediate- and high- redshift samples\n\\citep{kobulnicky03,lilly03,kobulnicky04,erb06a}, although caution\nmust be taken when comparing samples with metallicities determined\nfrom different methods. Physically, the correlation between\nstellar mass and metallicity is more fundamental than that between\nluminosity and metallicity \\citep[Paper I;][]{tremonti04,erb06a}.\nWe therefore focus on the mass-metallicity relation in the\nfollowing discussion.\n\n\n\n\n\n\n\nThe left panel of Figure \\ref{fig:massz} shows the average\nmetallicity of the galaxies in each mass bin determined from the\nN2 composite spectra plotted against their average stellar mass at\n$z \\sim 1.0$ ({\\it open diamonds}) and at $z \\sim 1.4$ ({\\it open\nsquares}). Although our sample is still small, we do see evidence\nfor mass-metallicity relations at both $z \\sim 1.0$ and $z \\sim\n1.4$. These trends are also present when we examine the\nmetallicities and stellar masses for individual objects, which are\nplotted in the figure as well. For comparison, the local SDSS\ngalaxies discussed by \\citet{tremonti04} are denoted by contours\nand dots\\footnote{Note that here and throughout, we use a\ncombination of contours and dots to indicate SDSS objects. On each\nsuch applicable plot, SDSS data points were mapped onto 10 evenly\nspaced levels according to surface density, where objects on the\nlowest level are denoted by dots while other levels are presented\nby contours.} and the $z \\sim 2$ \\citet{erb06a} sample as open\ncircles. Metallicities for SDSS galaxies were calculated using the\nsame strong-line indicator that was applied to the DEEP2 galaxies\nand not the Bayesian O\/H estimate from \\citet{tremonti04}.\n\nAt fixed stellar mass, the metallicities of our $z \\sim 1.0-1.5$\nsample as a whole are lower than those of local galaxies yet\nhigher than those of the $z \\sim 2$ sample. However, there is\nevidence for a reverse trend between the subsets of our sample at\n$z \\sim 1.0$ and at $z \\sim 1.4$. In Paper I, this difference was\nattributed to the fact that the $z \\sim 1.0$ sample was on average\nfainter and less massive than the $z\\sim 1.4$ sample. With a\nlarger sample, however, we find that the higher mass $z \\sim 1.0$\nbin does have lower metallicity than the lower mass $z \\sim 1.4$\nbin. Differences in outflow or inflow rate of unenriched gas at $z\n\\sim 1.0$ and at $z \\sim 1.4$ could give rise to this trend.\nHowever, the interval in cosmic time between $z\\sim 1.4$ and $1.0$\nis small enough that typical gas inflow rates at fixed stellar\nmass, and the corresponding star formation and outflow rates, will\nnot significantly evolve. Therefore, this explanation is not a\nlikely cause of the reverse trend. A different average degree of\ndust reddening at $z \\sim 1.0$ and at $z \\sim 1.4$ is also not a\nlikely cause, since the N2 indicator is based on emission lines\nwith very close spacing in wavelength. On the other hand, if the\n$z \\sim 1.0$ galaxies have systematically different physical\nconditions or less significant contributions from AGN activity\nrelative to the $z \\sim 1.4$ objects, metallicities estimated with\nthe same calibration would be systematically biased between the\ntwo samples in such a way to produce the observed trend. As\ndiscussed in \\S \\ref{sec:offset}, we propose that the most likely\ncause for the reverse trend is this difference in H II region\nphysical conditions. Since the systematic uncertainty from\nstrong-line calibration is large, and our sample is still too\nsmall to draw any solid conclusion, it will become feasible to\nclarify this issue only when a statistically large enough sample\nis assembled, and both the high- and low- mass ends are spanned at\n$z \\sim 1.0$ as well as at $z \\sim 1.4$.\n\n\nWe also plot the mass-metallicity relation from the N2+O3\ncomposite spectra in Figure \\ref{fig:massz}, where the left panel\nshows metallicities determined from the N2 indicator, while the\nright panel shows those determined from the O3N2 indicator, at\nboth $z \\sim 1.0$ ({\\it filled diamonds}) and $z \\sim 1.4$ ({\\it\nfilled squares}). The O3N2-based abundances are systematically\nlower than those based on N2. As suggested in Paper I and\ndiscussed in detail in \\S \\ref{sec:offset}, these systematic\ndiscrepancies between N2- and O3N2-based abundances are due to the\nfact that DEEP2 galaxies depart from the local H~II region\nexcitation sequence. In addition, the reverse trend between $z\n\\sim 1.0$ and $1.4$ in metallicity estimated from O3N2 is much\nless significant than the one in metallicity estimated from N2. We\nreturn to this issue as well in \\S \\ref{sec:offset}. Despite these\ndiscrepancies, the overall correlation between average stellar\nmass and metallicity observed among the N2 composite spectra is\nstill present for the N2+O3 spectra. This is evidence that the\ncorrelation is insensitive to the spectrum of any particular\nobject, as it is robust to analyses using different binning\nschemes. At the lower mass end ($M_{\\star} \\sim 8\\times10^{9}$\n$M_{\\sun}$), the average metallicity of $z \\sim 1.0-1.5$ galaxies\nbased on N2 is at least 0.22 dex lower than the local typical\nvalue. Since the N2 indicator saturates near solar abundance, as\ndiscussed in \\citet{erb06a}, it is difficult to determine the true\nmetallicity offset between two samples at different redshifts\nusing this indicator. We can also determine the offset from the\nO3N2-based abundances, particularly near the solar abundance.\nBased on O3N2 abundances, the metallicity offset between our DEEP2\nobjects and the local SDSS sample is at least 0.21 dex at the\nhigh-mass end ($M_{\\star} \\sim 5\\times10^{10}$ $M_{\\sun}$).\nHowever, as we discuss in \\S \\ref{sec:offset}, these estimates\nbased on strong-line indicators may still be subject to systematic\nuncertainties from using the calibration of H II regions with\nsignificantly different physical properties.\n\n\n\n\\begin{deluxetable}{lccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Oxygen Abundances from N2 composite spectra.\\label{tab:comp:n2}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~Bin~~~~~} & \n\\colhead{N\\tablenotemark{a}} & \n\\colhead{$\\langle {\\rm z}_{{\\rm H}\\alpha} \\rangle$\\tablenotemark{b}} & \n\\colhead{log$(M_{\\ast}\/M_{\\odot})$\\tablenotemark{c}} & \n\\colhead{N2\\tablenotemark{d}} &\n\\colhead{12 + log(O\/H)\\tablenotemark{e}}\n}\n\\startdata\n1\\dotfill & 3 & 1.0375 & $9.88^{+0.10}_{-0.13}$ & -0.96$\\pm$0.03 & 8.35$\\pm$0.11 \\\\\n2\\dotfill & 4 & 1.0207 & $10.37^{+0.05}_{-0.06}$ & -0.74$\\pm$0.02 & 8.48$\\pm$0.09 \\\\\n3\\dotfill & 5 & 1.3930 & $10.10^{+0.06}_{-0.06}$ & -0.63$\\pm$0.02 & 8.54$\\pm$0.09 \\\\\n4\\dotfill & 6 & 1.3903 & $10.83^{+0.05}_{-0.05}$ & -0.49$\\pm$0.02 & 8.62$\\pm$0.08 \\\\\n\\enddata\n\\tablenotetext{a}{Number of objects contained in each bin.}\n\\tablenotetext{b}{Mean redshift for each bin.}\n\\tablenotetext{c}{Mean stellar mass and uncertainty from error propagation.}\n\\tablenotetext{d}{N2 $\\equiv$ log([N II]$\\lambda6584$\/H$\\alpha$).}\n\\tablenotetext{e}{Oxygen abundance deduced from the N2 relationship presented in \\citet{pp04}.}\n\\end{deluxetable}\n\n\n\\begin{deluxetable*}{lccccccc}\n\\centering\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Oxygen Abundances from N2+O3 composite spectra.\\label{tab:comp:o3n2}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~~~~~~Bin~~~~~~~~~~} & \n\\colhead{~~~N~~~} & \n\\colhead{~~~$\\langle {\\rm z}_{{\\rm H}\\alpha} \\rangle$~~~} &\n\\colhead{~~~log$(M_{\\ast}\/M_{\\odot})$~~~} & \n\\colhead{~~~N2~~~} & \n\\colhead{~~~O3N2\\tablenotemark{a}~~~} &\n\\colhead{~~~[12 + log(O\/H)]$_{N2}$\\tablenotemark{b}~~~} & \n\\colhead{~~~[12 + log(O\/H)]$_{O3N2}$\\tablenotemark{c}~~~}\n}\n\\startdata\n1\\dotfill & 3 & 1.0372 & $9.98^{+0.09}_{-0.11}$ & -0.89$\\pm$0.03 & 1.37$\\pm$0.03 & 8.39$\\pm$0.10 & 8.29$\\pm$0.08 \\\\\n2\\dotfill & 3 & 1.0216 & $10.40^{+0.06}_{-0.06}$ & -0.79$\\pm$0.02 & 1.17$\\pm$0.06 & 8.45$\\pm$0.10 & 8.35$\\pm$0.08 \\\\\n3\\dotfill & 3 & 1.3923 & $10.06^{+0.06}_{-0.06}$ & -0.61$\\pm$0.04 & 1.09$\\pm$0.08 & 8.55$\\pm$0.11 & 8.38$\\pm$0.08 \\\\\n4\\dotfill & 3 & 1.3858 & $10.63^{+0.07}_{-0.09}$ & -0.51$\\pm$0.02 & $>$ 0.62 & 8.61$\\pm$0.10 & $<$ 8.53 \\\\\n\\enddata\n\\tablenotetext{a}{O3N2 $\\equiv$ log$\\{$([O III]$\\lambda$5007\/H$\\beta$)\/([N II] $\\lambda$6584\/H$\\alpha$)$\\}$.}\n\\tablenotetext{b}{Oxygen abundance deduced from the N2 relationship presented in \\citet{pp04}.} \n\\tablenotetext{c}{Oxygen abundance deduced from the O3N2 relationship presented in \\citet{pp04}.}\n\\end{deluxetable*}\n\n\n\n\\section{The Offset in Diagnostic Line Ratios of High-Redshift Galaxies}\\label{sec:offset}\n\n\\subsection{Emission-Line Diagnostics}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=80mm]{f7a.eps}\n \\includegraphics[width=80mm]{f7b.eps}\n \\caption{H II region log[N II]\/H$\\alpha$ vs. log[O III]\/H$\\beta$ diagram.\n On the left, the $z\\sim$1.0-1.5 DEEP2 galaxies are shown as indicated on\n the plot. The right panel shows the average diagnostic line ratios from\n the composite spectra of DEEP2 objects. Note the bin with the highest\n [N II]\/H$\\alpha$ value only has a lower limit on [O III]\/H$\\beta$ because\n it includes the object 42021412. SDSS emission-line galaxies that satisfy\n the criteria described in \\S \\ref{subsec:sdsspreselect} are shown\n as grey contours and dots. The dashed line is an empirical demarcation\n from Ka03 based on the SDSS galaxies, whereas the dotted\n line is the theoretical limit for star-forming galaxies from Ke01.\n Nearby star-forming galaxies and H II regions form a well-defined excitation\n sequence of photoionization by massive stars, below and to the left of these\n curves. The $z \\sim 1.0-1.5$ DEEP2 objects on average are offset from this\n excitation sequence, with objects at $z\\sim$1.4 more offset than those at $z\\sim$1.0.}\n \\label{fig:subfig:bpt}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=160mm]{f8.eps}\n \\caption{H II region diagnostic diagram: log[N II]\/H$\\alpha$ vs.\n log[O III]\/H$\\beta$. SDSS Main, Offset-AGN, Offset-ambiguous,\n and Offset-SF samples are shown as grey contours with dots, blue triangles,\n green crosses, and red circles, respectively (see \\S \\ref{subsec:sdsspreselect}\n for more information on sample selection rules). The $z\\sim1.0-1.5$\n DEEP2 objects are also shown for a comparison using the same legends\n as in the left panel of Figure \\ref{fig:subfig:bpt}, and the dotted\n and dashed lines have the same meanings as those in Figure \\ref{fig:subfig:bpt}.}\n \\label{fig:subfig:bptn2}\n\\end{figure*}\n\n\nThere is evidence that physical conditions in the H II regions of\nhigh-redshift galaxies hosting intense star formation are\ndifferent from those of local H II regions (Paper I). The most\ncommon method for probing H~II region physical conditions, and\ndiscriminating between star-forming galaxies and AGNs, is based on\nthe positions of objects in the \\citet*{bpt81} empirical\ndiagnostic diagrams (hereafter BPT diagrams). These plots feature\nthe optical line ratios [N II]\/H$\\alpha$, [O I]\/H$\\alpha$, [S\nII]\/H$\\alpha$, and [O III]\/H$\\beta$, and have been updated by many\nauthors, including \\citet{osterbrock85}, \\citet{veilleux87},\n\\citet[][hereafter Ke01]{kewley01a}, and \\citet[][hereafter\nKa03]{kauffmann03b}. A considerable fraction of high-redshift\nstar-forming galaxies at both z $\\sim$ 1 (paper I) and z $\\geq$ 2\n\\citep{erb06a} do not follow the local excitation sequence formed\nby nearby H II regions and star-forming galaxies on the\nemission-line diagnostic diagram of [N II]\/H$\\alpha$ versus [O\nIII]\/H$\\beta$; on average, they lie offset upward and to the\nright. These differences must be taken into account when applying\nempirically calibrated abundance indicators to galaxy samples at\ndifferent redshifts. Several possible causes for the offset have\nbeen discussed in Paper I, including differences in the ionizing\nspectrum, ionization parameter, electron density, and the effects\nof shocks and AGNs.\n\n\nIn Figure \\ref{fig:subfig:bpt}, [O III]\/H$\\beta$ and [N\nII]\/H$\\alpha$ ratios are plotted in the left panel for the 13\ngalaxies in our sample with the full set of emission lines, and\nthe average [O III]\/H$\\beta$ and [N II]\/H$\\alpha$ ratios from the\nN2+O3 composite spectra are also shown in the right panel for bins\nat $z \\sim 1.0$ and $1.4$, respectively. A subset of emission-line\nobjects from SDSS are also shown as grey contours and dots for\ncomparison. The dotted curve is from Ke01, representing a\ntheoretical upper limit on the location of star-forming galaxies\nin the diagnostic diagram. The dashed curve is from Ka03 and\nserves as an empirical discriminator between star-forming galaxies\nand AGNs. On average, the H$\\alpha$ flux of galaxies below this\ncurve should have $<1\\%$ contribution from AGNs\n\\citep{brinchmann04}.\nThe effect observed in Paper I is still present in our larger\nsample, in the sense that the $z \\sim 1.0-1.4$ sample is, on\naverage, significantly offset from the excitation sequence formed\nby star-forming galaxies from SDSS. Furthermore, the average\noffset for the $z\\sim 1.4$ objects is larger than for those at\n$z\\sim 1.0$. A similar, if not stronger, effect is observed in\nstar-forming galaxies at $z \\sim 2$ \\citep{erb06a}. As discussed\nin Paper I, unaccounted-for stellar Balmer absorption is not the\nexplanation for the offsets in emission-line ratios, since the\ncorrections would shift the DEEP2 galaxies by no more than 0.1 dex\ndownward and by an insignificant amount in [N II]\/H$\\alpha$.\n\nIsolating the causes of the offset of the high-redshift samples\nfrom the local excitation sequence on the diagnostic diagram will\nprovide important insight into the physical conditions in distant\nstar-forming galaxies. These conditions also comprise an essential\nsystematic factor determining the emergent strong emission-line\nratios and associated calibration of chemical abundances.\nUnfortunately, we do not currently have a large high-redshift\nsample with available additional spectral features, stellar\npopulation parameters, and morphological information, which is\nneeded for a direct study of how the diagnostic line ratios vary\nas functions of galaxy properties. However, as seen in Figure\n\\ref{fig:subfig:bpt}, while only a tiny fraction of the SDSS\ngalaxies reside in the region of BPT parameter space inhabited by\nour most extremely offset high-redshift galaxies, in between the\nstar-forming excitation sequence and the AGN branch, the sheer\nsize of the SDSS sample still results in a set of $\\sim 100$ such\nlocal objects. These objects can serve as possible local\ncounterparts for our DEEP2 objects, with the added benefit of high\nS\/N photometric, spectroscopic, and morphological information from\nSDSS. We will make use of this detailed information to understand\nthe cause of the local galaxies' offset in the BPT diagram, and,\nby extension, the likely cause of the offset among the\nhigh-redshift galaxies.\n\nIn the following sections, we compare in detail these anomalous\nSDSS objects against more typical SDSS star-forming galaxies.\nAccordingly, we analyze possible causes for their offset in terms\nof different physical conditions in the ionized regions, which\ninclude H~II region electron density, hardness of the ionizing\nspectrum, ionization parameter, the effects of shock excitation,\nand contributions from an AGN. We further try to unravel possible\nconnections between physical conditions of these anomalous SDSS\nobjects and their host galaxy properties and use them to interpret\nour observations of high-redshift star-forming galaxies.\n\n\n\\subsection{Local Counterparts: SDSS Main and Offset Samples}\\label{subsec:sdsspreselect}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=160mm]{f9.eps}\n \\caption{H II region diagnostic diagrams. Descriptions for SDSS objects\nare the same as those in Figure \\ref{fig:subfig:bptn2}. On the [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$ diagrams, the dotted (blue\nsolid) lines are empirical curves separating star-forming galaxies\n(Seyferts) and AGN (LINERs) from \\citet{kewley06}. The log[N\nII]\/[O II] vs. log[O III]\/H$\\beta$ diagram is useful in separating\nionization parameter and abundance but becomes insensitive to\nionization parameter above solar metallicity \\citep{dopita00}. The\n[N II]\/[O II] flux ratio has been corrected for dust extinction,\ngiven the wide spacing in wavelength between [N II] and [O II].}\n \\label{fig:subfig:agnbpt}\n\\end{figure*}\n\n\nWithin this work we use the SDSS Data Release 4\n\\citep[DR4;][]{adelman06} spectroscopic galaxy sample, which\ncontains $u$-, $g$-, $r$-, $i$-, and $z$-band photometry and\nspectroscopy of 567,486 objects. As described in\n\\citet{tremonti04}, emission-line fluxes of these galaxies were\nmeasured from the stellar-continuum subtracted spectra with the\nlatest high spectral resolution population synthesis models by\n\\citet{bc03}.\n\nBefore being further divided into groups with different\nemission-line diagnostic ratios, our sample was selected from the\n567,486 galaxy DR4 sample according to the following criteria:\n\n(1) Redshifts in the range $0.005 < z < 0.25$. (2) Signal-to-noise\nratio (S\/N) $>$ 3 in the strong emission-lines [O II]\n$\\lambda\\lambda$3726, 3729, H$\\beta$, [O III]\n$\\lambda\\lambda$4959, 5007, H$\\alpha$, [N II] $\\lambda$6584, and\n[S II] $\\lambda\\lambda$6717, 6731; and S\/N $>$ 1 in the weak\nemission-line [O I] $\\lambda$6300. In practice, $\\sim 99$\\% of the\nobjects also have [O I] $\\lambda$6300 S\/N$>$3. (3) The fiber\naperture covers at least 20\\% of the total $g$-band photons. (4)\nStellar population parameter estimates are available in the\ncatalog derived using methods described by \\citet{kauffmann03a}.\n\nThe first criterion on redshift range is the same as that adopted\nby \\citet{tremonti04}, which enables a fair comparison with the\nmass-metallicity relation from their work. The second criterion\nselects galaxies with well-measured emission-line fluxes, required\nby our study based on emission-line diagnostic ratios. The S\/N\nvalues were obtained using errors of the emission-line fluxes,\nwhich were first taken from the emission-line flux catalog on the\nMPA DR4 Web site\\footnote{See\nhttp:\/\/www.mpa-garching.mpg.de\/SDSS\/} and then scaled by the\nrecommended factors (see the Web site for more details about the\nemission-line flux error). The third criterion is required to\navoid significant aperture effects on the flux ratios\n\\citep{kewley05,tremonti04}. Lower aperture fraction can cause\nsignificant discrepancies between aperture and global parameter\nestimates. The fourth criterion enables a comparison of stellar\npopulation properties with diagnostic emission-line ratios.\nRoughly 30\\% of the initial set of DR4 galaxies were ruled out\nbecause they do not have stellar population parameter estimates;\nan additional $\\sim$20\\% was cut out according to the redshift and\naperture-coverage constraints. Rules on the S\/N in both strong and\nweak emission lines removed another $\\sim$45\\% of the objects. In\nall, these selection criteria leave us with $\\sim$31,000\nemission-line objects, shown as grey contours and dots in Figure\n\\ref{fig:subfig:bpt}, which is about 5\\% of the 567,486 galaxy DR4\nsample.\n\n\n\nWith the exception of one of the 13 DEEP2 objects, which lies on\nthe part of the [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram\npopulated by many H II\/AGN composites, the high-redshift galaxies\non average fall between the excitation sequence of typical SDSS\nstar-forming galaxies and the AGN branch. Although many of the\nDEEP2 objects have a less extreme offset relative to typical SDSS\nstar-forming galaxies than those of the sample of \\citet{erb06a},\nand the most extreme $z\\sim 1.4$ objects in Paper~I, there are\nstill several that are significantly offset from the main\nsequence. We proceed by studying properties of the SDSS galaxies\nwith similar [N II]\/H$\\alpha$ and [O III]\/H$\\beta$ values to those\nof the significantly offset DEEP2 objects, and comparing these\nunusual objects with typical SDSS star-forming galaxies along the\nexcitation sequence. In this sense we further divide the SDSS\nsample into the ``Main'', and ``Offset'' subgroups, as shown in\nFigure \\ref{fig:subfig:bptn2}.\n\nObjects in the Main sample are selected to lie below the Ka03\nempirical curve, whereas Offset objects are located in between the\nKa03 and the Ke01 curves. The Offset sample is also constrained to\nhave log([N II]\/H$\\alpha$) $\\leq$ -0.44 and log([O III]\/H$\\beta$)\n$\\geq$ 0.26, according to the two emission-line ratios of the\nDEEP2 object with the second largest [N II]\/H$\\alpha$ value. We do\nnot use the galaxy with the largest [N II]\/H$\\alpha$ value in our\nDEEP2 sample for the selection criteria of the two line ratios,\nbecause the number density in the composite region increases very\nrapidly as the AGN branch is approached, yet most of our DEEP2\ngalaxies clearly do not fall in that regime. Also, it is worth\npointing out that, while they are significantly displaced from the\nlocal excitation sequence, more than half of the DEEP2 objects\nactually lie below the Ka03 curve. Indeed, the SDSS Offset sample\nis constructed according to our DEEP2 objects that are offset the\nmost, in order to create a strong contrast with the Main sample.\nIt is also representative of the $z\\sim 2$ galaxies observed by\n\\citet{erb06a}. As we describe in \\S 5.6, the conclusions drawn\nfrom this extreme sample are corroborated by the work of\n\\citet{brinchmann07}, where objects with less extreme offsets are\nconsidered and therefore should be valid for typical objects in\nour DEEP2 sample. Finally, the Offset sample only covers a certain\nrange of stellar masses, so we further select the Main control\nsample according to the same stellar-mass range in order to ensure\na fair comparison. This leaves us with $\\sim$21,000 objects for\nthe Main sample and 101 objects for the Offset sample.\n\nFigure \\ref{fig:subfig:agnbpt} displays how objects in the Offset\nsample are distributed in the additional BPT diagrams featuring [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$. These plots indicate that\nOffset objects selected solely on the basis of their [O\nIII]\/H$\\beta$ and [N II]\/H$\\alpha$ ratios span a diverse range of\nproperties in other physical parameter spaces. High [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$ both occur when there is a hard\nionizing radiation field, including significant contribution from\nX-ray photons. In this case, there is an extended, partially\nionized zone, where H I and H II coexist, and [O I] and [S II] are\ndominant forms of O and S. The extended zone of partially ionized\nH does not exist in H II regions photoionized by OB stars\n\\citep{evans85,veilleux87}. High [O I]\/H$\\alpha$ and [S\nII]\/H$\\alpha$ are also produced in gas that has been heated by\nfast, radiative shocks, which also produce partially ionized\nshock-precursor regions \\citep{dopita95,dopita96}. Material in\nsupernova remnants provides an example of shocked gas. While it is\nsensitive to shocks, [S II] is more susceptible to collisional\nde-excitation than [O I], given its critical density\n(2$\\times10^3$ cm$^{-3}$), as opposed to that of [O I]\n(2$\\times10^6$ cm$^{-3}$), and therefore might be suppressed in\nregions of high electron densities \\citep{dopita97,kewley01b}.\nAnother point worth mentioning is that [O I]\/H$\\alpha$ should\nreveal larger differences in regions ionized by hard spectrum as\nopposed to that of stars, than [S II]\/H$\\alpha$ does. This is\nbecause the ionization potential of [O I] matches that of H I, so\n[O I] enhancement should reflect increased presence of partially\nionized zone. However, [S II] exists in both completely and\npartially ionized zones. So the contrast between AGN- and\nstellar-ionized regions for [S II]\/H$\\alpha$ is not as great as\nthat for [O I]\/H$\\alpha$.\n\nIn summary, the various locations of the Offset objects on the [O\nI]\/H$\\alpha$ and [S II]\/H$\\alpha$ diagrams indicate different\nlevels of contribution from AGN and shock excitations to the\nemerging spectra. We therefore further divide the Offset sample on\nthe [O I]\/H$\\alpha$ and [S II]\/H$\\alpha$ diagrams, according to\nthe theoretical scheme for classifying starburst galaxies and AGNs\n\\citep{kewley01a,kewley06}. It is worth noting here that all\nobjects in the Offset sample have S\/N$>$3 in both [S II] and [O I]\nemission lines, so this division should not be compromised by\nspurious measurements. While there may still be a low-level\n($<10$\\%) AGN contribution to the spectra of objects classified as\nstarbursts by this scheme, it serves as a rough guide to the range\nof properties in the Offset sample. This classification leaves us\nwith (1) ``Offset-AGN'' (43), where objects lie above both the\nKe01 curves of the two diagrams; (2) ``Offset-ambiguous'' (33),\nwhere objects lie above one Ke01 curve and under the other one;\nand (3) ``Offset-SF'' (25), where objects lie below both the Ke01\ncurves. The division of Offset-AGN and Offset-SF is only based on\nhardness of the ionizing spectrum and the contribution of shock\nexcitation. We show in Section \\ref{sec:sub:comparison} that these\ntwo offset samples have different host galaxy properties.\n\n\n\\subsection{H II Region Emission-Line Diagnostic Ratio, Physical Conditions, and Galaxy Properties}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=160mm]{f10a.eps}\n \\includegraphics[width=160mm]{f10b.eps}\n \\includegraphics[width=160mm]{f10c.eps}\n \\includegraphics[width=160mm]{f10d.eps}\n \\caption{Distribution of row (1): redshift and metallicities\n based on various strong-line indicators; row (2): ionization-parameter\n indicator log($O_{32}$), electron density indicator [S II] $\\lambda6717$\/[S II] $\\lambda6731$,\n D$_n(4000)$ and H$\\delta_{A}$; row (3): stellar mass,\n fiber-corrected specific star formation rate, $r$-band Petrosian\n concentration index $C = R_{90}\/R_{50}$, and color-magnitude diagram; row\n (4): surface mass density log($\\mu_{\\star}$), SFR surface density\n log($\\dot{\\Sigma}_{\\star}$), $R_{50}(z)$, which is the radius enclosing\n 50\\% of the Petrosian $z$-band luminosity of a galaxy, and H$\\alpha$\n emission equivalent widths, for the SDSS Main ({\\it black dotted line}),\n Offset-AGN ({\\it blue dashed line}), and Offset-SF samples ({\\it red solid line}; see \\S \\ref{subsec:sdsspreselect}\n for more information). For clarity, the distribution functions are\n normalized to three different levels, with the maxima being 1.0,\n 0.7, and 0.5 for the SDSS Main, Offset-AGN, and Offset-SF samples, respectively.}\n \\label{fig:subfig:dis}\n\\end{figure*}\n\n\nIn order to determine the origin of the observed difference in\nemission-line ratios between the Main and Offset samples, we\nconsider a large set of galaxy properties. Measurements of\nemission lines as well as photometric, physical, and environmental\nproperties of stellar population are available for our SDSS\nsamples. In this section, we describe the galaxy properties of\ninterest, while in the following section, a comparison of the Main\nand Offset sample distributions in these properties is presented.\n\nFirst, using emission-line ratios, we examine the main factors\ncontrolling the emission-line spectrum in an H~II region, i.e.,\nthe gas phase metallicity, the shape or hardness of the ionizing\nradiation spectrum, and the geometrical distribution of gas with\nrespect to the ionizing sources, which can be represented by the\nmean ionization parameter and electron density \\citep{dopita00}.\nSeveral H~II region emission-line ratios can be used as\ndiagnostics of these factors.\n\nApart from the N2 and O3N2 indicators, we also use the ratio N2O2\n$\\equiv$ log([N II] $\\lambda$6584\/[O II] $\\lambda\\lambda$3726,\n3729), suggested by \\citet{vanzee98}, as an abundance diagnostic\nfor SDSS objects, because N2O2 is virtually independent of\nionization parameter and also strongly sensitive to metallicity.\nThis indicator is monotonic between 0.1 and over 3.0 times solar\nmetallicity \\citep{dopita00,kewley02}. Concerns about reddening\ncorrection and reliable calibration over such a large wavelength\nbaseline have hampered the use of this ratio. As shown in\n\\citet{kewley02}, however, the use of classical reddening curves\nand standard calibration are quite sufficient to allow this [N\nII]\/[O II] diagnostic to be used as a reliable abundance\nindicator. The \\citet{bresolin07} calibration of N2O2 is used to\ninfer oxygen abundance. We also use the $R_{23} \\equiv$ log\\{([O\nII] $\\lambda$3727 + [O III] $\\lambda\\lambda$4959, 5007)\/H$\\beta$\\}\nparameter, introduced by \\citet{pagel79}, as an abundance\nindicator, to make comparison with several works\n\\citep{lilly03,savaglio05}, although this indicator has some\nwell-documented drawbacks \\citep[e.g.][]{kobulnicky99,kewley02}.\n\n$O_{32} \\equiv$ log([O III] $\\lambda$5007\/[O II]\n$\\lambda\\lambda$3726, 3729) is used as an indicator of the\nionization parameter \\citep{mcgaugh91,dopita00}. However, $O_{32}$\nis sensitive to both ionization parameter and abundance; both low\nionization parameter and high abundances produce low values of\n$O_{32}$ \\citep{dopita00,kewley02}. Thus, when $O_{32}$ is used as\na diagnostic of the relative ionization parameters of different\nsamples, the metallicities of these samples must also be taken\ninto account for a fair comparison. A diagnostic plot, for\nexample, N2O2 versus $O_{32}$, can be used to separate ionization\nparameter and abundance. The ratio [S III]\/[S II] provides another\ndiagnostic of the ionization parameter, which is independent of\nmetallicity except at very high values of the ratio\n\\citep{kewley02}. Unfortunately, the SDSS spectra do not cover [S\nIII] $\\lambda\\lambda$9069, 9532 from SDSS, so it is not included\nhere.\n\nFinally, [S II] $\\lambda6717$\/[S II] $\\lambda6731$ is adopted as\nan electron-density indicator. We do not use the density-sensitive\nratio, [O II] $\\lambda3726$\/[O II] $\\lambda3729$, since this\ndoublet is typically blended in SDSS spectra.\n\nWe note that emission-line ratios, such as $O_{32}$, N2O2, and\n$R_{23}$ have large separations in wavelength and thus need to be\ncorrected for dust reddening. The Balmer decrement method with the\nreddening curve of \\citet{calzetti00} is used to correct these\nquantities for dust extinction, assuming an intrinsic ratio of\n$H\\alpha\/H\\beta=2.78$, appropriate for $T_e=10,000$ K.\n\n\nNext, we turn to the question of the stellar populations and\nstructural and environmental properties of the Main and the Offset\nsample host galaxies. To examine stellar ages, we adopt the narrow\ndefinition of the 4000 ${\\rm \\AA}$ break denoted as D$_n(4000)$\n\\citep{balogh99}, which is small for young stellar populations and\nlarge for old, metal-rich galaxies \\citep{kauffmann03a}. We use\nthe H$\\delta_{A}$ index as an indicator of recent starburst\nactivities \\citep[e.g.][]{kauffmann03a,worthey97}. Strong\nH$\\delta$ absorption lines arise in galaxies that experienced a\nburst of star formation that ended $\\sim$0.1-1 Gyr ago. For galaxy\nmorphology, we use the standard concentration parameter defined as\nthe ratio $C = R_{90}(r)\/R_{50}(r)$, where $R_{90}(r)$ and\n$R_{50}(r)$ are the radii enclosing 90\\% and 50\\% of the Petrosian\n$r$-band luminosity of the galaxy. For galaxy size, we study\n$R_{50}(z)$, the radius enclosing 50\\% of the Petrosian $z$-band\nluminosity of a galaxy. We also examine the surface mass density\n$\\mu_{\\star}$, defined as $M_{\\star}\/[2\\pi R^2_{50}(z)]$\n\\citep{kauffmann03c}, the SFR surface density\n$\\dot{\\Sigma}_{\\star}\\equiv {\\rm SFR}\/[2\\pi R^2_{50}(r)]$, where\nwe use a radius defined in the $r$ band rather than the $z$ band,\nas it is more appropriate for H$\\alpha$ luminosities. Additional\nparameters include the fiber-corrected specific star formation\nrate SFR\/M$_{\\star}$, H$\\alpha$ equivalent width (EW), and the\ncolor-magnitude diagram $(g-i)^{0.1}$ versus $M_r$, where\n$(g-i)^{0.1}$ denotes the $(g-i)$ color $k$-corrected to $z =\n0.1$, and $M_r$ stands for the $k$-corrected $r$-band absolute\nmagnitude \\citep{kauffmann03a}. In order to search for any\nenvironmental dependence of the Offset sample properties, we count\nthe number of spectroscopically observed galaxies that are located\nwithin 2 Mpc in projected radius and $\\pm$500 km s$^{-1}$ in\nvelocity difference from each object in our SDSS samples. Here we\nfollow the procedures described in \\citet{kauffmann04},\nconstructing a volume-limited tracer sample using the SDSS DR4\ndata.\n\n\n\\subsection{Comparison with Typical SDSS Star-forming Galaxies}\\label{sec:sub:comparison}\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f11.eps} \\caption{Ionization-parameter\nindicator $O_{32}$, as a function of various parameters.\nDescriptions are the same as those in Figure\n\\ref{fig:subfig:agnbpt}.\\label{o32_dep}}\n\\end{figure*}\n\n\nIn this section, we compare the properties of Offset and Main\nsample objects, in terms of their H~II region physical conditions,\nstellar populations, structural parameters, and environments. In\nparticular, we find striking differences in H~II region ionization\nparameter, electron density, galaxy size, and star formation rate\nsurface density. Figure \\ref{fig:subfig:dis} shows relative\ndistributions of diagnostic-line ratios and galaxy properties of\nthe SDSS Main, Offset-AGN, and Offset-SF samples. For clarity, the\ndistribution functions are normalized to three different levels,\nwith the maxima being 1.0, 0.7 and 0.5 for the SDSS Main,\nOffset-AGN, and Offset-SF samples, respectively.\n\nFirst, we consider H~II region physical conditions. Compared to\nthe Main sample, both Offset samples on average have larger\nionization parameters. The Offset-SF and Offset-AGN samples have\n$O_{32} = 0.0$ and -0.2, respectively, as opposed to -0.7 for the\nMain sample. Here and throughout, we refer to the median values of\nthe distributions. Since $O_{32}$ also depends on metallicity, we\nneed to make a fair comparison for the ionization parameter at a\nfixed metallicity. Figure \\ref{o32_dep} shows $O_{32}$ as a\nfunction of several emission-line diagnostic ratios and galaxy\nproperties. In the lower four panels, we can see that, regardless\nof which metallicity indicator is used, the Offset-SF sample has\nbigger $O_{32}$ values than the Main sample at a fixed strong-line\nindicator value. Therefore, the Offset-SF sample has larger\naverage ionization parameter than Main sample objects having\nsimilar metallicities, independent of the metallicity indicator.\nCompared with the Main sample, all indicators except $R_{23}$\nsuggest that the Offset-AGN has higher ionization parameters at\nfixed metallicities.\n\nThe Offset-SF sample also has significantly larger electron\ndensities, on average, than the Main sample ([S II]\n$\\lambda6717$\/[S II] $\\lambda6731$ of $\\sim$1.23 compared to\n$\\sim$1.40, corresponding to electron densities of $\\sim$208\ncm$^{-3}$ compared to $\\sim$47 cm$^{-3}$), whereas the Offset-AGN\nsample has only slightly higher electron densities than those of\nthe Main sample ([S II] $\\lambda6717$\/[S II] $\\lambda6731$ of\n$\\sim$1.37, corresponding to electron densities of $\\sim$67\ncm$^{-3}$). As discussed in Paper I, for fixed ionization\nparameter, metallicity, and input ionizing spectrum, the\nphotoionization models presented in \\citet{kewley01a} display a\ndependence on electron density, in the sense that model grids with\nhigher electron density have an upper envelope in the space of [O\nIII]\/H$\\beta$ versus [N II]\/H$\\alpha$ that is offset upward and to\nthe right, relative to model grids with lower electron density.\nThis theoretical shift in the BPT diagram due to increased\nelectron density is qualitatively reflected in the properties of\nOffset-SF objects.\n\nNext, we consider photometric and spectroscopic stellar population\nproperties. The Offset-SF sample is similar to the Main sample in\nterms of median color [$(g-i)^{0.1}$ $\\sim$ 0.64 compared to\n$\\sim$0.66], stellar age [the same value of D$_n$(4000) = 1.24],\nand fiber-corrected specific star formation rates (the same value\nof $\\log((SFR\/M_{\\star})\\mbox{ yr}^{-1}) = -9.7$), whereas the\nOffset-AGN sample has redder colors [$(g-i)^{0.1}$ of $\\sim$0.86],\nolder stellar ages [D$_n$(4000) = 1.50], and lower specific star\nformation rates ($\\log((SFR\/M_{\\star})\\mbox{ yr}^{-1}) = -10.0$).\nAlso, the Offset-SF sample has a larger H$\\alpha$ EW than the Main\nsample (62 ${\\rm \\AA}$ compared to 40 ${\\rm \\AA}$), whereas the\nOffset-AGN sample has a much smaller H$\\alpha$ EW (15${\\rm \\AA}$).\nThe Offset-SF sample has a smaller burst fraction in stellar mass\nthan the Main sample (H$\\delta_{A}$ of 4.3 compared to 5.5), while\nthe discrepancy between the Offset-AGN and Main samples is even\nlarger (H$\\delta_{A}$ of 3.3 compared to 5.5).\n\nIn terms of galaxy structure, the Offset-SF sample has larger\nconcentration ($C$ = 2.58 compared to 2.36), smaller half-light\nradii [$R_{50}(z)$ of 1.2 kpc compared to 1.9 kpc], higher surface\nstellar mass density [log($\\mu_{\\star}$) of 8.78 M$_{\\odot}$\nkpc$^{-2}$ compared to 8.61 M$_{\\odot}$ kpc$^{-2}$], and higher\nSFR surface density [log($\\dot{\\Sigma}_{\\star}$) of -0.86\nM$_{\\odot}$ yr$^{-1}$ kpc$^{-2}$ compared to -1.12 M$_{\\odot}$\nyr$^{-1}$ kpc$^{-2}$] than the Main sample, whereas the Offset-AGN\nhas larger concentration ($C$ of 2.66), moderately smaller sizes\n[$R_{50}(z)$ of 1.5 kpc], moderately higher surface stellar mass\ndensity [log($\\mu_{\\star}$) of 8.68 M$_{\\odot}$ kpc$^{-2}$], and\nsimilar SFR surface density [log($\\dot{\\Sigma}_{\\star}$) of -1.12\nM$_{\\odot}$ yr$^{-1}$ kpc$^{-2}$].\n\nIn terms of environment, we find that 90\\%-95\\% of objects in the\nMain, Offset-SF, and Offset-AGN samples have zero or one neighbor,\nthe lowest density bin in \\citet{kauffmann04}. Furthermore, the\ndistributions of environments for the Offset samples are similar\nto that of the Main sample. Therefore, it appears that the vast\nmajority of galaxies considered here reside in the lowest density\nenvironment defined by \\citet{kauffmann04}. This result is not\nsurprising, as our SDSS samples all contain emission-line\ngalaxies, which are more likely to be found in low-density\nenvironments \\citep{kauffmann04}. Currently we have too few Offset\nobjects to draw any solid conclusion about the question on galaxy\nenvironmental dependence.\n\n\n\n\n\\citet{groves06} suggest that the offset on the BPT diagram seen\nin some high-redshift galaxies is mostly caused by contribution\nfrom AGNs, as they found properties of their candidate\nlow-metallicity H II\/AGN composites similar to those of the host\ngalaxies of AGNs \\citep{kauffmann03b,heckman04}; these objects\nhave on average significantly higher stellar masses, older stellar\npopulations, and redder colors than the sample of pure\nstar-forming galaxies used for comparison. \\citet{groves06} arrive\nat this conclusion after constructing their offset sample in a\nregion bounded by the Ka03 and the Ke01 curves in the [N\nII]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram, and the Seyfert\nbranch [log([O III]\/H$\\beta$) $\\geq$ 3log([N II]\/H$\\alpha$)]. When\nwe reproduced their offset sample, we found the median values of\nlog([N II]\/H$\\alpha$) $\\sim$ -0.286 and log([O III]\/H$\\beta$)\n$\\sim$ -0.024, whereas the corresponding values for our DEEP2\nobjects are -0.638 and 0.344. As the density of objects increases\nrapidly towards increasing [N II]\/H$\\alpha$, the offset sample\ndefinition of \\citet{groves06} is weighted heavily towards H\nII\/AGN composites and not necessarily representative of the\naverage [N II]\/H$\\alpha$ or [O III]\/H$\\beta$ of high-redshift\ngalaxies. Furthermore, it is not clear that their ``typical''\ncomparison sample, which was defined as all galaxies within\n$\\pm$0.05 dex of [-0.55, 0.10] in the [N II]\/H$\\alpha$ versus [O\nIII]\/H$\\beta$ diagram, represents a fair one, as they have not\ncontrolled for any galaxy property. Given the strong correlations\namong galaxy properties, it is crucial to compare galaxies that\nare similar in at least some basic parameters.\nHere we want to emphasize that in order to make a controlled\ncomparison, the Main sample was selected to have similar range and\nmedian of stellar masses as that of the Offset sample. The Main\nsample also has a similar median [N II]\/H$\\alpha$ value. With this\ncontrolled comparison of typical and Offset SDSS objects, we find\nthat the unusual objects on the BPT diagram are populated not only\nby likely AGN hosts (Offset-AGN) but also by the Offset-SF sample,\nmost of which do not resemble typical AGN-host galaxies.\nFurthermore, there is no segregation of the two offset samples in\nthe [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram.\n\nAs suggested in \\citet{groves06}, another possible test of the\npresence of an AGN would require the detection of either He II\n$\\lambda$4686 or [Ne V] $\\lambda$3426 lines. The [Ne V]\n$\\lambda$3426 line is not redshifted into the SDSS band for the\nmajority of our objects: we have only four such objects in\nOffset-AGN, of which the spectra near [Ne V] $\\lambda$3426 are too\nnoisy to provide any meaningful constraints. The He II\n$\\lambda$4686 line is expected to be very weak: a 20\\% AGN\ncontribution to H$\\beta$ implies He II\/H$\\beta$ = 0.05. Note that\nbesides the AGN photoionization, possible mechanisms for producing\nHe II emission also include hot stellar ionizing continua and\nshock excitation \\citep{garnett91}. Another important goal is to\ninfer metallicities of the Offset-SF objects using a method that\nis independent of strong-line indicators, since these objects have\nabnormal strong-line ratios. A more calibration-independent way to\naddress metallicity is through the classic $T_e$ method, which\nrelies on measuring weak auroral lines. Both the test of the\npresence of AGNs, and the determination of metallicity with the\ndirect $T_e$ method, require the measurement of very weak lines.\nThese measurements become feasible through the use of composite\nspectra.\n\n\n\n\n\n\n\n\n\\subsection{SDSS Composite Spectra}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=150mm]{f12a.eps}\n \\includegraphics[width=150mm]{f12b.eps}\n \\includegraphics[width=150mm]{f12c.eps}\n \\caption{SDSS composite spectra. For clarity, only the portion\n of the spectrum containing the weak emission lines of interest\n is presented here. The original composite and stellar-continuum-subtracted\n spectra for samples (A) Main, (B) Offset-AGN, and (C)\n Offset-SF\n are shown as black solid curves. Stellar-continuum fits using\n \\citet{bc03} population synthesis models are plotted as red dotted\n curves. The [O III] $\\lambda$4363 and He II $\\lambda$4686 lines are marked by dotted lines.}\n \\label{fig:sdsscompspec}\n\\end{figure*}\n\n\nIn order to measure He II $\\lambda$4686 as another test of\npossible AGN contribution to the ionizing spectrum, and [O III]\n$\\lambda$4363 to infer the oxygen abundance using the $T_e$\nmethod, we construct composite spectra for both Offset-SF and\nOffset-AGN objects. For comparison, we also make composite spectra\nfor the Main sample. We adopt the same method of making composite\nspectra used for our DEEP2 objects, employing median scaling to\npreserve the relative fluxes of the emission features. The\nvariance spectrum is combined in the same way and the error for\nthe corresponding composite spectrum is the square root of the\ncomposite variance spectrum divided by the number of objects.\n\nThe stellar continuum was modelled over the rest-frame wavelength\nrange of 3800 - 7500 ${\\rm \\AA}$ and then subtracted from the\ncomposite spectra, with a method similar to the one described in\n\\citet{tremonti04} and \\citet{brinchmann04}. This technique\n\\citep[][private communication]{charlot07} relies on the\n\\citet{bc03} stellar population synthesis model, but with two\nmajor improvements. First, the number of the metallicity grids was\nexpanded from three to four; second, and most importantly, the\nstellar continuum fitting was carried out using both the STELIB\n\\citep{leborgne03} and the MILES \\citep{sanchez06} libraries of\nobserved stellar spectra. Models using the MILES library fit the\ndata better than those with the STELIB one, in terms of the\nagreement around Balmer lines. It is especially essential to\nobtain a reasonable stellar continuum fit near H$\\gamma$, which is\nvery close in wavelength to the weak [O III] $\\lambda$4363 auroral\nline. Models using the STELIB library tend to over-estimate\nstellar continua, especially around Balmer lines. We therefore\nadopt the stellar continuum models based on the MILES library.\n\nThe resultant continuum-subtracted spectra, along with the\noriginal composite spectra and the best-fit model of the stellar\ncontinua, are shown in Figure (\\ref{fig:sdsscompspec}). We only\npresent portions of the entire composite spectra to enable\nscrutiny of the relevant weak lines ([O III] $\\lambda$4363 and He\nII $\\lambda$4686) and our capability of making a reasonably good\nstellar continuum fit. The corresponding error spectra are\nestimated by combining the measurement error for the composite\nspectra and the systematic uncertainties from the stellar\ncontinuum fitting. The systematic error from the stellar continuum\nfitting is estimated using the rms of a portion of the\ncontinuum-subtracted spectrum, which is relatively free of\nemission-line features. This method of estimating continuum fit\nerror is motivated by the fact that the rms would be zero if the\nfitting was ideally good and if there were no emission-line\nfeatures. At all of the wavelengths considered, the systematic\nerror is significantly larger than the measurement error for the\ncomposite spectra and therefore dominates the total error budget.\nEmission-line fluxes and flux ratios with the total uncertainties\nare given in Table \\ref{tab:sdsscomp}.\n\nThere are a few caveats that must be mentioned in the analysis of\ncomposite spectra described here. First, we are averaging over\ntens of thousands of galaxies for the SDSS Main sample and dozens\nof galaxies for the Offset-SF and Offset-AGN samples; second, for\neach single galaxy, we are dealing with integrated spectra\ncontaining contributions from multiple H II regions, which may\nshow a range of metallicities, ionization parameters, ionizing\nspectra, and electron densities. Also, even though we select the\nsamples to have fiber apertures covering more than 20\\% of the\ntotal $g$-band photons, because of the presence of radial\ngradients in galaxy properties, uncertainties still remain since\nthe spectrum is weighted towards the nucleus. Despite these\ncaveats, however, we want to emphasize that the analysis of\ncomposite integrated fibre-based spectra is still meaningful in\nterms of determining average properties and the relative\ndifferences among samples.\n\n\n\n\n\\begin{deluxetable*}{lccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Emission-line flux measurements of SDSS composite spectra.\\label{tab:sdsscomp}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~~~~~Sample~~~~~~~~~} & \n\\colhead{F$_{{\\rm [O II]}\\lambda\\lambda3726, 3729}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm [O III]}\\lambda4363}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm He II}\\lambda4686}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm H}\\beta}$\\tablenotemark{a}} &\n\\colhead{F$_{{\\rm [O III]}\\lambda4959}$\\tablenotemark{a}} & \n\\colhead{F$_{{\\rm [O III]}\\lambda5007}$\\tablenotemark{a}} &\n\\colhead{F$_{{\\rm [O I]}\\lambda6300}$\\tablenotemark{a}}\n}\n\\startdata\nMain\\dotfill & 327.3$\\pm$0.3 & 0.6$\\pm$0.2 & 1.4$\\pm$0.2 & 153.5$\\pm$0.2 & 45.2$\\pm$0.1 & 137.0$\\pm$0.3 & 19.1$\\pm$0.2 \\\\\nOffset-AGN\\dotfill & 271.9$\\pm$0.8 & 2.8$\\pm$0.5 & 4.9$\\pm$0.5 & 90.4$\\pm$0.5 & 86.4$\\pm$0.2 & 261.7$\\pm$0.6 & 24.3$\\pm$0.6 \\\\\nOffset-SF\\dotfill & 629.1$\\pm$1.0 & 7.1$\\pm$0.6 & 12.5$\\pm$0.7 & 321.0$\\pm$0.7 & 266.6$\\pm$0.3 & 807.4$\\pm$0.8 & 38.8$\\pm$0.7 \\\\\n\\hline\n~~~~~~~~~Sample~~~~~~~~~ & F$_{{\\rm H}\\alpha}$\\tablenotemark{a} & F$_{{\\rm [S II]}\\lambda6717}$\\tablenotemark{a} & F$_{{\\rm [S II]}\\lambda6731}$\\tablenotemark{a} & $\\frac{{\\rm F_{[O III] \\lambda \\lambda 4959, 5007}}}{{\\rm F_{[O III] \\lambda 4363}}}$\\tablenotemark{b} & $\\frac{{\\rm F_{He II \\lambda 4686}}}{{\\rm F_{H\\beta}}}$ & $\\frac{{\\rm F_{[O I] \\lambda 6300}}}{{\\rm F_{H\\alpha}}}$ & $\\frac{{\\rm F_{[S II]\\lambda\\lambda6717+6731}}}{{\\rm F_{H\\alpha}}}$ \\\\\n\\hline\n & & & & & & & \\\\\nMain\\dotfill & 633.6$\\pm$0.3 & 117.6$\\pm$0.3 & 83.9$\\pm$0.3 & 249$\\pm$82 & 0.009$\\pm$0.001 & 0.0301$\\pm$0.0004 & 0.3180$\\pm$0.0006 \\\\\nOffset-AGN\\dotfill & 337.3$\\pm$0.7 & 80.6$\\pm$0.6 & 58.5$\\pm$0.6 & 110$\\pm$19 & 0.054$\\pm$0.006 & 0.0720$\\pm$0.0018 & 0.4124$\\pm$0.0028 \\\\\nOffset-SF\\dotfill & 1272.8$\\pm$0.9 & 170.6$\\pm$0.8 & 137.3$\\pm$0.8 & 129$\\pm$11 & 0.039$\\pm$0.002 & 0.0305$\\pm$0.0006 & 0.2419$\\pm$0.0009 \\\\\n\\enddata\n\\tablenotetext{a}{Emission-line flux and 1 $\\sigma$ error in units of $10^{-17}$ ergs s$^{-1}$ cm$^{-2}$ from SDSS composite spectra.}\n\\tablenotetext{b}{Dust-reddening corrected using Balmer decrement method.}\n\\end{deluxetable*}\n\n\n\n\\subsubsection{AGN Contribution}\n\nThe composite spectra we construct can be used to address the\nlevel at which AGN ionization contributes to the anomalous\nemission-line ratios, discovered in both nearby SDSS Offset\nobjects and high-redshift DEEP2 galaxies. Based on their [S\nII]\/H$\\alpha$ and [O I]\/H$\\alpha$ ratios, we expect the Offset-AGN\nobjects to have a non-negligible contribution to their ionization\nfrom AGNs and\/or shock heating. Based on the composite spectrum,\nwe can also probe the weaker He II line, which is another\nindicator of AGN activity.\n\nFrom the He II\/H$\\beta$ ratio (0.054$\\pm$0.006), we can see that\nthe Offset-AGN sample may have up to $\\sim$20\\% AGN contribution\nto the Balmer emission lines \\citep{groves06}. The Offset-SF\nsample, on the other hand, shows a lower level of possible AGN\ncontamination (He II\/H$\\beta$ of 0.039$\\pm$0.002). This\ndistinction is also suggested by the different host-galaxy\nproperties of the two samples. The Offset-AGN objects with similar\nstellar masses to the Main sample have redder colors, older\nstellar populations, lower starburst stellar mass fraction,\nsmaller H$\\alpha$ EWs, and smaller specific star formation rates,\nwhich are similar to host galaxies of AGN \\citep{kauffmann03b},\nwhereas the Offset-SF sample is similar to the Main sample in\nterms of these galaxy properties. Furthermore, by construction,\nthe Offset-SF objects have low [S II]\/H$\\alpha$ and [O\nI]\/H$\\alpha$ in the range favored for typical star-forming\ngalaxies.\n\nWe note, however, that the Offset-SF sample still has a higher\nvalue of He II\/H$\\beta$ than the Main sample (0.009$\\pm$0.001).\nThere might be some mixing with the Offset-AGN objects because of\nuncertainties in emission-line diagnostic ratios. Indeed, three\nobjects in the Offset-SF sample have cross-identifications with\n$ROSAT$ sources, and two of them have optical spectra indicating\nbroad Balmer lines. However, $\\sim$90\\% of the Offset-SF sample\nshow no evidence of AGNs in terms of $ROSAT$-source\ncross-identifications and broad Balmer lines. In addition, we have\nexamined the He II\/H$\\beta$ ratios from individual galaxy spectra\nin the Offset-SF sample, finding that five objects (including the\nthree with $ROSAT$ cross-identifications) of the 25 have He\nII\/H$\\beta$ $\\sim$ 0.05-0.10, while the remaining 20 objects all\nhave He II\/H$\\beta$ $\\lesssim$ 0.01, similar to the typical He\nII\/H$\\beta$ of the Main sample. Even the five objects with He\nII\/H$\\beta$ $\\sim$ 0.05-0.10 could have other possible ionizing\nsources including hot stars and shocks \\citep{garnett91}. It is\nworth mentioning that nondetection in $ROSAT$ or broad Balmer\nlines cannot fully exclude the possibility of AGN contamination,\nas the majority of obscured AGNs are completely absorbed in the\nsoft X-ray range and do not show broad Balmer lines. However, if\nthe dominant cause of the Offset-SF on the BPT diagram for our\nOffset-SF sample was mildly obscured AGN contamination, namely, if\ntheir optical spectra were contaminated by AGN ionization at a\nlower level than the Offset-AGN sample, then we would expect them\nto exhibit intermediate emission-line diagnostic ratios and\nhost-galaxy properties, relative to the Main and the Offset-AGN\nsamples. This is not the case. Instead, the Offset-SF objects have\neven higher ionization parameters and electron densities than the\nOffset-AGN sample, relative to the Main sample. In addition, as\ndiscussed in \\S \\ref{sec:sub:comparison}, Offset-SF objects have\nhost-galaxy properties different from those of typical weak AGNs\nin SDSS \\citep{kauffmann03b}. Therefore, it appears that AGN\nexcitation does not provide the dominant cause of the anomalous\nemission-line ratios for most of the Offset-SF sample.\n\n\\subsubsection{Electron Temperature and Oxygen Abundance}\\label{sec:sub:te}\n\n\\begin{figure*}\n\\epsscale{1.} \\plotone{f13.eps} \\caption{Mass-metallicity relation\nfor the SDSS Main ({\\it grey contours}) and Offset-SF ({\\it red\nopen circles}) samples, based on various strong-line indicators.\nCalibrations are from \\citet[][for $R_{23}$]{zaritsky94},\n\\citet[][for N2 and O3N2]{pp04}, and \\citet[][for\nN2O2]{bresolin07}, respectively. \\label{fig:sdss:massz}}\n\\end{figure*}\n\n\n\n\n\\begin{deluxetable*}{lcccccc}\n\\centering\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Physical Quantities from SDSS composite spectra.\\label{tab:sdssabundance}}\n\\tablewidth{0pt}\n\\tablehead\n{\n\\colhead{~~~~~~~~~~Sample~~~~~~~~~~} & \n\\colhead{~~~~~N$_e$\\tablenotemark{a}~~~~~} &\n\\colhead{~~~~~T$_e$[O III]\\tablenotemark{b}~~~~~} &\n\\colhead{~~~~~T$_e$[O II]\\tablenotemark{b}~~~~~} & \n\\colhead{~~~~~O$^+$\/H$^+$\\tablenotemark{c}~~~~~} &\n\\colhead{~~~~~O$^{++}$\/H$^+$\\tablenotemark{c}~~~~~} & \n\\colhead{~~~~~12 + log(O\/H)\\tablenotemark{d}~~~~~}\n}\n\\startdata\nMain\\dotfill & 28$\\pm$ 5 & 0.950$^{+0.124}_{-0.072}$ & 0.965$^{+0.087}_{-0.050}$ & 13.2$^{+3.7}_{-4.1}$ & 3.7$\\pm$1.3 & 8.23$^{+0.11}_{-0.17}$ \\\\\nOffset-SF\\dotfill & 190$\\pm$12 & 1.170$^{+0.039}_{-0.033}$ & 1.119$^{+0.027}_{-0.023}$ & 6.5$\\pm$0.6 & 5.2$\\pm$0.5 & 8.07$\\pm$0.04 \\\\\n\\enddata\n\\tablenotetext{a}{Electron density and 1 $\\sigma$ error in units of cm$^{-3}$ derived from $({\\rm [S \\,\\,II]} \\lambda 6717\/{\\rm [S \\,\\,II]} \\lambda 6731$).}\n\\tablenotetext{b}{Electron temperature and 1 $\\sigma$ error in units of $10^4$ K.}\n\\tablenotetext{c}{Ionic abundance and 1 $\\sigma$ error from collisionally excited lines in units of $10^{-5}$.}\n\\tablenotetext{d}{Total oxygen abundance and 1 $\\sigma$ error assuming O\/H = (O$^{+}$ + O$^{++}$)\/H$^{+}$.}\n\\end{deluxetable*}\n\n\n\n\n\nBy making composite spectra, and hence enhancing the S\/N of weak\nauroral lines, we can determine metallicities with a method\nindependent of strong-line indicators. We have already discussed\nthe fact that the Offset-SF objects have larger ionization\nparameters, as indicated by their higher $O_{32}$ values. However,\nit is still unclear whether the Offset-SF sample has comparable\nmetallicity with the Main one, on average, as suggested by N2 and\nN2O2, or if it has $\\sim$0.3 dex lower metallicity, as suggested\nby O3N2 and $R_{23}$. Settling this question is important for\nmultiple reasons.\n\nFirst, gas-phase metallicity and electron temperature, themselves,\nserve as key parameters of H~II regions. Second, understanding the\nrelative metallicities of the Main and the Offset-SF samples on\naverage can help us determine the difference in average ionization\nparameter quantitatively. If the Offset-SF sample has comparable\nmetallicities with the Main sample, their ionization-parameter\ndifference would be very large; on the other hand, if the\nOffset-SF sample has much lower metallicities than the Main one\ndoes, their ionization-parameter difference would be much smaller,\nsince both lower metallicity and higher ionization parameter can\ncause large $O_{32}$. Finally, if the Offset-SF objects actually\nhave much lower metallicities than the Main objects, then they\nwill not follow the local mass-metallicity relation\n\\citep{tremonti04}, given their comparable stellar masses.\n\n\nIn order to determine oxygen abundances using the direct method,\nestimates of both the electron density and temperature are\nrequired. Based on the ratio [S II] $\\lambda$6717\/[S II]\n$\\lambda$6731, we establish that on average all of our objects are\nin the low-density regime. Then using the five-level atom program\n{\\tt nebular} implemented in IRAF\/STSDAS \\citep{shaw95}, we derive\nthe electron temperature $T$[O III] from the ratio of auroral line\n[O III] $\\lambda$4363 to nebular lines [O III] $\\lambda\n\\lambda$4959, 5007 \\citep{agn2} and correct for dust extinction\nusing the Balmer decrement method. As for $T$[O II], we adopt a\nsimple scaling relation between the temperatures in different\nionization zones of an H II region predicted by the\nphotoionization models of Garnett (1992):\n\\begin{equation}\nT{\\rm [O \\,\\,II]} = 0.70T{\\rm [O\\,\\, III]} + 3000 \\,\\,{\\rm K},\n\\end{equation}\nwhich is applicable in a wide range of $T$[O III] (2000-18,000 K).\nIonic abundances O$^{++}$\/H$^{+}$ and O$^{+}$\/H$^{+}$ are then\ndetermined using programs in the {\\tt nebular} package, and,\nfinally, we obtain the total gas phase oxygen abundance assuming\nO\/H = (O$^{+}$ + O$^{++}$)\/H$^{+}$. The results are summarized in\nTable \\ref{tab:sdssabundance}.\n\n\nFirst, the Offset-SF sample has $\\sim$2200 K higher electron\ntemperatures than the Main one does on average. These objects are\nunusual in the sense that they have higher electron densities and\nelectron temperatures, hence ambient interstellar pressures\nassuming pressure equilibrium between H II regions and ambient\ngas, and larger ionization parameters. It is not clear what is at\nthe root of the special H II region physical conditions, in terms\nof galaxy properties, or larger scale environments. However, in\nthe simple theoretical estimates for local starburst galaxies, the\nISM pressure scales roughly linearly with the SFR surface density\n\\citep[e.g.,][]{thompson05}. Indeed, the Offset-SF sample has\nseveral prominently related properties, including higher\nconcentration ($r$-band Petrosian concentration index\n$R_{90}\/R_{50}$ of $\\sim$2.6 compared to $\\sim$2.4), smaller\nhalf-light radii [$R_{50}(z)$ of 1.2 kpc compared to 1.9 kpc], and\nmost notably, higher SFR surface density\n[log($\\dot{\\Sigma}_{\\star}$) of -0.86 M$_{\\odot}$ yr$^{-1}$\nkpc$^{-2}$ compared to -1.12 M$_{\\odot}$ yr$^{-1}$ kpc$^{-2}$)],\nas shown in Figure \\ref{fig:subfig:dis}, compared to typical\nstar-forming galaxies. This higher SFR surface density may\ntherefore account for the higher interstellar pressure seen in the\nH II regions of Offset-SF objects.\n\n\n\nIt is also possible to relate electron temperature, electron\ndensity, and ionization parameter in a single H II region, through\nthe classical picture of \\citet{stromgren39}. With higher ambient\ninterstellar pressure due to higher electron densities and\ntemperatures, H II regions are surrounded by denser molecular gas\ndust and therefore have smaller radii under the stall condition.\nIn an idealized case of a fully ionized spherical H II region with\npure hydrogen, the radius $R$ of the ionized region is given by $R\n= [3Q\/(4\\pi N_e^2 \\alpha_B)]^{1\/3}$, where $Q$ is the rate of\nemission of hydrogen ionizing photons, and $\\alpha_B$ is the case\nB recombination coefficient \\citep{agn2}. The ionization parameter\n$U$ defined by $[F\/4\\pi c R^2 N_{{\\rm H II}}]$, where $F$ is the\nflux of ionizing photons, $c$ is the speed of light, and $N_{{\\rm\nH II}}$ is the particle density in H II region, is then\nproportional to $N_e^{1\/3}$ for a given central hot star.\nTherefore, in this simple picture, H II regions with higher\nelectron densities and temperatures will have higher ionization\nparameters. Accordingly, the higher electron densities and\ntemperatures inferred from the integrated spectra of the Offset-SF\nobjects may result in the observed higher ionization parameters as\nwell.\n\nSecond, the Offset-SF sample has $\\sim $0.16 dex lower\nmetallicities than the Main sample. Note that the absolute\nabundance values inferred using the $T_e$ method are significantly\nlower than those based on any strong-line indicators. This\ndifference may stem from temperature fluctuations in H~II regions\nand the strong $T_e$ dependence of line emissivities, from which\nthe net effect is that the electron temperature derived from\nauroral lines tends to overestimate the real temperature and hence\nsystematically underestimate the abundance \\citep{garnett92}.\nTherefore, we only focus on relative differences based on the same\nmethod between the Offset-SF and Main samples. Given that the\nOffset-SF and Main samples are characterized by similar average\nstellar masses, this difference in average metallicity suggests\nthat Offset-SF objects do not follow the local mass-metallicity\nrelation, in addition to being more compact and sustaining higher\ninterstellar pressures than the Main sample.\n\nTo investigate this question further, we examine several versions\nof the mass-metallicity relation for the Offset-SF and the Main\nobjects, based on different strong-line metallicity indicators.\nFigure \\ref{fig:sdss:massz} includes these mass-metallicity\ncorrelations for both the Main and the Offset-SF samples,\ndemonstrating that one would draw different conclusions about the\nrelationship between the Offset-SF and the Main samples, depending\non which strong-line indicator was used. Between the Main and\nOffset-SF samples, the relative average metallicity differences at\nfixed mass based on $R_{23}$, N2, O3N2, and N2O2 are $\\sim$0.28,\n0.00, 0.18, and 0.00 dex, respectively. This ambiguity could be\nunderstood as the result of adopting the same calibration for\nsamples with different physical conditions, of which the most\nimportant are ionization parameter and electron density.\n\nThe discrepancy between conclusions drawn from the strong-line\nindicators and the $T_e$ method can be understood using the\nphotoionization models of \\citet{kewley02}. For example, according\nto $R_{23}$, the average metallicity difference between the\nOffset-SF and Main samples is $\\sim$0.28 dex, whereas this\nmetallicity difference based on the $T_e$ method is $\\sim$0.16\ndex. This $\\sim$0.12 dex discrepancy can be explained by examining\nthe theoretical grids in the $R_{23}$ versus 12 + log(O\/H) space,\ngiven by photoionization models with different values of electron\ndensity and ionization parameter. If the incorrect assumption is\nadopted that both the Main and Offset-SF samples have electron\ndensities $N_e \\sim 10$ cm$^{-3}$ and ionization parameters\n$q=3\\times10^{7}\\mbox{cm s}^{-1}$ -- values appropriate for the\nMain but not Offset-SF sample -- we obtain that the Offset-SF\nsample is $\\sim 0.25$~dex lower in oxygen abundance than the Main\nsample. On the other hand, if an electron density $N_e \\sim 350$\ncm$^{-3}$ and ionization parameter $q=5\\times10^{7}\\mbox{cm\ns}^{-1}$ are assumed for the Offset-SF sample -- more appropriate\nbased on their inferred H~II region physical conditions -- the\ndifference in estimated Main and Offset-SF oxygen abundance is\nonly $\\sim$0.10 dex. This difference is $\\sim$0.15 dex smaller\nthan the one inferred with the incorrect assumption that the\nOffset-SF and Main samples have the same electron densities and\nionization parameters. The actual difference in electron density\nbetween the two samples is less extreme ($N_e \\sim 190$ cm$^{-3}$\nof the Offset-SF compared to $N_e \\sim 28$ cm$^{-3}$ of the Main),\nso this discrepancy may be smaller, as observed ($\\sim$0.12 dex).\nAnalogous effects occur for the N2 and O3N2 indicators.\n\n\n\n\nIn summary, the metallicities of the Offset-SF objects based on\nstrong-line indicators can be either overestimated or\nunderestimated, depending on the relative shifts of the\ntheoretical grids caused by varying physical conditions, and an\nincorrect assumption of the values of ionization parameter and\nelectron density. For the SDSS Offset-SF sample, the metallicities\nbased on strong-line indicators can either be systematically too\nlarge by $\\sim$0.16 dex (N2, N2O2), or too low by $\\sim$0.12 dex\n($R_{23}$). The bias of O3N2 is little (too large by $\\sim$0.02\ndex) for our SDSS Offset-SF objects, although it may not always be\nnegligible given other ranges of physical conditions in terms of\nmetallicity, ionization parameter, and electron density. The\nreason for the Offset-SF objects' anomalous positions on the BPT\ndiagram is the same: H~II regions with distinct ionization\nparameters and electron densities fall on different surfaces in\nthe diagnostic-line parameter space.\n\n\n\nOne caveat that should be mentioned here is the fact that we are\ninferring average biases over the stellar mass and metallicity\nranges spanned by the Offset-SF sample. Another caveat is that\nresults of the SDSS data are all based on integrated fiber spectra\nand hence limited in terms of spatial information. We plan to\nfurther study the spatial dependence of H~II region emission lines\nin these offset objects using long-slit spectrographs in the\nfuture.\n\n\n\n\n\\subsection{Implications for DEEP2 Objects}\n\nWe have analyzed emission-line diagnostic ratios, physical\nconditions and galaxy properties of SDSS objects with similar [N\nII]\/H$\\alpha$ and [O III]\/H$\\beta$ values to those of DEEP2\ngalaxies with the most extreme offset in our sample, and $z \\sim\n2$ star-forming galaxies \\citep{erb06a}. There are two major\ncauses for their offset: one is different H II region physical\nconditions characterized by higher electron density and\ntemperature, and hence larger ionization parameter, compared to\ntypical SDSS star-forming galaxies. These physical conditions of\nthe SDSS Offset-SF objects are also connected to their host galaxy\nproperties, particularly the higher SFR surface density. The other\npossible cause for the offset is contribution from AGNs and\/or\nshock excitation. We cannot rule out either of these two\npossibilities, or a combination of both, to explain the offset of\nDEEP2 objects on the diagnostic diagram.\n\nAs for the question of H II region physical conditions, it is\npossible to determine electron densities for our DEEP2 objects\nusing the [O II] doublet contained in the DEIMOS spectra. Note\nthat we will not directly compare electron densities of the $z\n\\sim 1.0-1.5$ objects to those of SDSS local galaxies, because\nthey were estimated using different density-sensitive doublets,\nand therefore the comparison might suffer from the associated\nsystematic uncertainties. However, when we divide all of the\nobjects in our DEEP2 sample that have reliable [O II]-doublet\nmeasurements into two groups, according to the empirical Ka03\ncurve in the [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram, we\nfind that for the group below the Kauffmann curve, the median\nvalue of electron density inferred from [O II] $\\lambda$3726\/[O\nII] $\\lambda$3729 is $\\sim$23 cm$^{-3}$, whereas for the group\nabove the Kauffmann curve, the median value is $\\sim$159\ncm$^{-3}$. This difference in electron density as a function of\nposition on the BPT diagram is qualitatively consistent with what\nwe have found for SDSS objects (Table (\\ref{tab:sdssabundance});\nelectron density N$_e$ [cm$^{-3}$] of 190$\\pm$12 for the Offset-SF\ncompared to 28$\\pm$5 for the Main), lending independent support\nfor our method of using Offset-SF objects as local analogs for the\nH~II regions in our high-redshift sample. In principle, we could\nexamine the ionization parameter indicator, $O_{32}$, for our\nDEEP2 objects, using [O II] from DEIMOS and [O III] from NIRSPEC\nspectra. However, at this point, the comparison does not seem\nuseful, as it would suffer from large systematic uncertainties,\ndue to the manner in which the optical and near-IR spectra were\ncollected and flux-calibrated.\n\nA related question is whether objects that are more offset in the\n[N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram have higher\naverage SFR surface density than objects with less offset within\nour DEEP2 $z \\sim 1-1.5$ sample. The spatial extent of H$\\alpha$\nemission contains information about galaxy sizes. However, we can\nonly measure the extent along the slit, which does not necessarily\nreflect the actual galaxy size. In addition, the size estimated\nfrom the spatial extent of H$\\alpha$ emission is subject to\nuncertainties due to the variation in the seeing FWHM, which is\ncomparable to the H$\\alpha$ size itself. As we do not have\nreliable estimates for the sizes of our DEEP2 objects, the above\nquestion can be finally answered only when the galaxy\nmorphological information is robustly gathered. Furthermore, the\nSFRs of our DEEP2 objects have not been corrected for dust\nextinction or aperture effects, which may amount to factor of 2\ndifferences \\citep{erb06c}. Limited by the associated systematic\nuncertainties, we are not able to compare the SFR surface\ndensities of our DEEP2 objects directly with those of the SDSS\nsamples. However, after correcting the SFR-associated systematic\nuncertainties for their $z \\sim 2$ sample, \\citet{erb06b} found a\nmean log($\\dot{\\Sigma}_{\\star}$) $\\approx$ 0.46, significantly\nhigher than that of SDSS local star-forming galaxies. In addition,\nobservational evidence exists that the galaxy size at fixed\nmass\/luminosity decreases with increasing redshift out to z $\\sim$\n3 \\citep{trujillo06,dahlen07}. At the same time, galaxies with\nstar formation rates significantly higher than those found among\ntypical SDSS emission-line galaxies are more commonly found at\nhigher redshifts \\citep[e.g.,][]{dahlen07,reddy07}. Therefore,\ngalaxies with high SFR surface densities are more prevalent in the\nhigh-redshift universe. For such objects, a noticeable difference\nin H~II region physical conditions is expected.\n\n\nAs for the question of AGN contribution, the DEEP2 object 42010637\nclearly falls on the part where many objects have contribution\nfrom AGNs in their ionizing spectra. However, it does not have\nmulti-wavelength information that can either confirm or rule out\nthe AGN excitation. In addition, all of the $z\\sim 2$ objects with\n[O III]\/H$\\beta$ and [N II]\/H$\\alpha$ measurements are offset by\nan amount similar to our most extremely offset $z\\sim 1.0-1.5$\nobjects but show no evidence of AGN contamination in at least\ntheir rest-frame UV spectra \\citep{erb06a}. The lack of such\nevidence in the rest-frame UV rules out the presence of at least a\nfairly unobscured AGN. Note that \\citet{daddi07} find that roughly\n20-30\\% of star-forming galaxies with $M\\sim 10^{10}$-$10^{11}\nM_{\\odot}$ at $z\\sim 2$ display a mid-IR excess, as evidence for\nhosting an obscured AGN. These authors show that this fraction\nincreases with stellar mass, reaching $\\sim$50-60\\% for an extreme\nmass range of $M > 4\\times10^{10} M_{\\odot}$. Our DEEP2 $z\\sim\n1.0-1.5$ sample covers a range of $M \\sim 5\\times10^9$- $10^{11}\nM_{\\odot}$, not all of which are in the ``massive'' range of\n\\citet{daddi07}, and only two of 20 are in the extreme range. If\nthe mid-IR excess, hence AGN contamination, found by\n\\citet{daddi07} is also applicable to our DEEP2 $z\\sim 1.0-1.5$\nobjects, the fraction of objects containing potential AGN\ncontamination would be at most $\\sim 20$\\%-30\\%. However, nine (or\n10, one with a lower limit) of the 13 objects in our sample with\nall four line measurements show evidence for being offset in the\n[O III]\/H$\\beta$ versus [N II]\/H$\\alpha$ BPT diagram (i.e., more\nthan 20-30\\%). More generally, the associated space densities of\nboth the blue DEEP2 galaxies \\citep{coil07} and UV-selected $z\n\\sim 2$ objects for which we have NIRSPEC spectra\n\\citep{adelberger05,reddy07} are significantly higher than that of\nthe obscured AGN population featured in \\citet{daddi07}.\nTherefore, as one of the possible causes for the offset on the BPT\ndiagram, AGN contamination cannot account for nor is consistent\nwith all the rest-frame optical emission-line measurements\npresented here.\n\nWe will address the AGN-contamination issue in the future by\nlooking at more DEEP2 objects for which multi-wavelength\ninformation is available and obtaining spatially resolved spectra\nwith an integral field unit assisted by adaptive optics to isolate\nthe contribution from the nucleus. Also, the measurements of\nflux-calibrated emission lines including [O II] $\\lambda$3727, [O\nIII] $\\lambda$4363, H$\\beta$, [O III] $\\lambda$5007, [O I]\n$\\lambda$6300, [N II] $\\lambda$6584, H$\\alpha$ and [S II]\n$\\lambda\\lambda$6717, 6731, as well as host-galaxy morphological\ninformation are required, in order to finally settle the causes of\nthe offset in the [N II]\/H$\\alpha$ versus [O III]\/H$\\beta$ diagram\nfor high-redshift star-forming galaxies. [O III] $\\lambda$4363 may\nbe difficult to measure for all but the most metal-poor objects or\nin deep $z \\sim 1$ composite spectra, and intensity limits on [O\nI] $\\lambda$6300 may also be difficult to obtain. Yet measurements\nof the stronger emission lines for a statistical sample of objects\nat $z \\geq 1$ will be feasible with the next generation of\nground-based multi object near-IR spectrographs.\n\n\nIn an independent study, \\citet{brinchmann07} have also analyzed\nthe possible causes for the high-redshift galaxies' offset in the\nBPT diagram, using theoretical models of nebular emission from\nstar-forming galaxies \\citep{charlot01} and the SDSS DR4 data.\nThey have found a relationship in SDSS galaxies between their\nlocation in the BPT diagram and their excess specific SFRs and\nlarger H$\\alpha$ EWs relative to galaxies of similar mass. We note\nthat they have only examined SDSS star-forming galaxies below the\nKa03 curve, whereas our SDSS Offset samples have been selected to\nbe above this curve, in order to probe exactly the regime where\nthe most offset objects in our DEEP2 $z \\sim 1-1.5$ sample and the\n\\citet{erb06a} $z - 2$ star-forming galaxies reside.\n\\citet{brinchmann07} have inferred that an elevated ionization\nparameter $U$ is at the root of the excess specific SFRs of the\nmore offset objects within their pure star-forming galaxy sample\nand further speculated that higher electron densities and escape\nfractions of hydrogen ionizing photons might be the factors\nresponsible for the systematically higher values of $U$ in the H\nII regions of high-redshift galaxies. Using a different technique\nand sample of galaxies, we have reached a similar conclusion about\nthe higher ionization parameter and larger H$\\alpha$ EWs in our\nSDSS Offset-SF samples. In addition, we have also uncovered that\nthe higher electron density and temperature, hence higher\ninterstellar ambient pressure, is at the root of the higher\nionization parameter. We have further shown that these unusual H\nII region physical conditions are well connected to the higher SFR\nsurface density of host galaxies. The trend of higher electron\ndensity with increasing BPT diagram offset found within our DEEP2\n$z \\sim 1-1.5$ sample, and the observational evidence that\ngalaxies with high SFR surface densities are more prevalent at\nhigh redshifts, lend further support to our conclusions drawn\nbased on the SDSS local emission-line galaxies.\n\n\nThese differences discovered in H II region physical conditions,\nwhich may commonly apply to $z \\sim 1-1.5$ star-forming galaxies,\nmust be taken into account when strong-line abundance indicators\nare used to study the evolution of galaxy metallicity with\nredshift. The resulting systematic bias in inferred oxygen\nabundance can be estimated quantitatively either via detailed\nphotoionization models, given the difference in H II region\nphysical conditions inferred from the relevant density- and\nionization parameter-sensitive line ratios, or through empirical\ncomparisons, as we have illustrated for the SDSS Main and\nOffset-SF samples.\n\nWe have shown in \\S \\ref{sec:sub:te} that, for the SDSS Offset-SF\nobjects, the metallicities based on various strong-line indicators\ncan either be systematically too large by $\\sim$0.16 dex (N2,\nN2O2) or too low by $\\sim$0.12 dex ($R_{23}$). For N2 in\nparticular, the inferred metallicities of the SDSS Offset-SF\nobjects can be systematically too large by $\\sim$0.16 dex, which\nis already comparable to the inherent scatter in the N2\ncalibration \\citep[1 $\\sigma$ dispersion of 0.18 dex;][]{pp04}.\nThis systematic bias may be even larger if the actual difference\nin H II region physical conditions is more extreme. In addition,\nwe note that this systematic uncertainty stemming from the\ndifference in H II region physical conditions has different\neffects from that of the inherent scatter in the calibration, when\nthe strong-line relation calibrated with local H II regions is\napplied to high-redshift star-forming galaxies. As there is\nevidence that the average ionization parameter and electron\ndensity in high-redshift star-forming galaxies are systematically\nhigher than the local typical values, there will be a systematic\n``bias,'' instead of a scatter as a result. For example, the\nN2-based metallicities for the DEEP2 $z\\sim 1.4$ sample may be\nsystematically overestimated by as much as $\\sim$ 0.16 dex.\nFurthermore, as discussed in \\S 4, the fact, that our DEEP2 $z\\sim\n1.4$ sample is more offset from the local excitation sequence than\nthe $z\\sim 1.0$ sample, may result in N2-based metallicities that\nare more significantly overestimated as well. This effect may\ncause the apparent reverse trend of average O\/H with redshift\nwithin the DEEP2 sample, such that the $z\\sim 1.4$ sample is\ndescribed by apparently higher metallicities at fixed stellar mass\nthan the one at $z\\sim 1.0$, despite the general trend of\nincreasing O\/H towards lower redshift. The reverse trend within\nthe DEEP2 sample based on O3N2 is less significant than that based\non N2, which is consistent with the fact that the O3N2-based\nmetallicity bias due to the offset in the BPT diagram is much less\nthan that of N2.\n\nIn addition, we can quantify the potential bias for the sample of\nstar-forming galaxies at $z\\sim2$ presented by \\citet{erb06a}. If\nthe four $z\\sim2$ objects with the full set of H$\\beta$, [O III]\n$\\lambda 5007$, H$\\alpha$, and [N II] $\\lambda 6584$ measurements\nare representative of the larger population of star-forming\ngalaxies in \\citet{erb06a} in terms of diagnostic line ratios,\nsuch that the $z\\sim 2$ star-forming galaxies are even more\nsignificantly offset than the DEEP2 $z\\sim 1.4$ sample on average,\nthen their metallicities based on N2 would also be overestimated\nby $\\sim$0.16 dex. Therefore, the true metallicity offset between\nthe \\citet{erb06a} $z\\sim2$ sample and local SDSS objects may be\nas much as $\\sim$0.16 dex larger than it appears now in Figure\n(\\ref{fig:massz}a). Accounting for the systematic differences in\nconverting strong emission-line ratios to oxygen abundances is\ntherefore a crucial component of comparing galaxy metallicities at\ndifferent redshifts.\n\n\\section{Summary}\n\nWe have compiled a sample of 20 star-forming galaxies at $1.0 < z\n< 1.5$ drawn from the blue cloud of the color bimodality observed\nin the DEEP2 survey, to study the correlation between stellar mass\nand metallicity, across a dynamical range of 2 orders of magnitude\nin stellar mass, as well as H~II region physical conditions at\nthis redshift range. In order to gain some insights on the causes\nof the offset in the BPT diagram observed in high-redshift\nstar-forming galaxies, we have examined the H II region diagnostic\nline ratios and host galaxy properties of the small fraction of\nSDSS galaxies that have similar diagnostic ratios to those of the\nDEEP2 sample. Our main results are summarized as follows:\n\\begin{enumerate}\n\n\\item[1.] There is a correlation between stellar mass and\ngas-phase oxygen abundance in DEEP2 star-forming galaxies at $z\n\\sim 1.0$ and at $z \\sim 1.4$. We have found that the zero point\nof the $M-Z$ relationship evolves with redshift, in the sense that\ngalaxies at fixed stellar mass become more metal-rich at lower\nredshift, by comparing the $1.0 < z < 1.5$ sample with UV-selected\n$z \\sim 2$ and SDSS local star-forming galaxies. At the low-mass\nend ($M_{\\star} \\sim 8\\times10^9 M_{\\odot}$), the relation at $1.0\n< z < 1.5$ is offset by $\\sim$0.2 (0.35) dex from the local\nmass-metallicity relation according to the N2 (O3N2) indicator.\nThe N2-based offset could be larger by as much as $\\sim$0.16 dex,\nwhen the systematic bias due to difference in H II region physical\nconditions between $1.0 < z < 1.5$ and the local universe is taken\ninto account. At the high-mass end ($M_{\\star} \\sim 5\\times10^{10}\nM_{\\odot}$), the metallicity offset between the DEEP2 $1.0 < z <\n1.5$ sample and the local SDSS sample is at least $\\sim 0.2$ dex,\naccording to the O3N2 indicator.\n\n\n\n\\item[2.] As observed previously for a very small sample of\nhigh-redshift galaxies, on average our new DEEP2 sample at $1.0 <\nz < 1.5$ is offset from the excitation sequence formed by nearby\nH~II regions and SDSS emission-line galaxies. By examining the\nsmall fraction of SDSS galaxies that have similar diagnostic\nratios to those of the DEEP2 sample, we have found two likely\ncauses for the anomalous emission-line ratios. One is the\ncontribution from AGN and\/or shock excitation at the level of\n$\\sim 20$\\%. The other is the difference in H~II region physical\nconditions, characterized by significantly larger ionization\nparameters, as a result of higher electron densities and\ntemperatures, and hence higher interstellar ambient pressure, than\nthe typical values of local star-forming galaxies with similar\nstellar mass. These unusual physical conditions are possibly\nconnected to the host-galaxy properties, most importantly smaller\nsizes and higher star-formation rate surface densities. Our\nconclusion drawn from analyzing the SDSS data has been further\nverified by the fact that the DEEP2 objects more offset from the\nlocal excitation sequence in the BPT diagram also have higher\nelectron densities than those closer to the local sequence. We\ncannot rule out either the contribution from AGN and\/or shock\nexcitation, or the difference in H~II region physical conditions,\nfor the unusual emission-line diagnostic ratios of high-redshift\nstar-forming galaxies.\n\n\\item[3.] We have quantified the effects of different H II region\nphysical conditions on the strong-line metallicity calibrations.\nThe direct electron temperature method was used to estimate the\n``true'' metallicity difference between offset SDSS objects with\nanomalous line ratios and more typical objects of similar stellar\nmass. Strong-line indicators were also used to estimate this\ndifference. A comparison of these results reveals potential biases\nin the strong-line indicators. According to our test, the\nmetallicities based on strong-line indicators can either be\nsystematically too large by $\\sim$0.16 dex (N2, N2O2), or too low\nby $\\sim$0.12 dex ($R_{23}$), for objects with similar H~II region\nphysical conditions to those observed in high-redshift galaxies.\nThe bias of O3N2 is much less significant (too large by $\\sim$0.02\ndex) for offset SDSS objects with anomalous line ratios, although\nit may not always be negligible given other ranges of physical\nconditions in terms of metallicity, ionization parameter, and\nelectron density.\n\\end{enumerate}\n\n\n\nThe difference in H~II region physical conditions, which may\ncommonly apply to $z\\sim 1.0-1.5$ star-forming galaxies, must be\ntaken into account when strong-line abundance indicators are used\nto study the evolution of galaxy metallicity with redshift. There\nare at least two methods to remove the systematic bias from the\neffect of significantly different H~II region physical conditions\non the strong-line abundance calibrations. One is to gather the\nabundance information with direct $T_e$ method for a sample of\nhigh-redshift H~II regions as the calibration sample, which may be\nhard to achieve, as auroral lines are difficult to measure except\nfor very metal-poor objects or in deep, composite spectra. The\nother, which is currently feasible yet relies on photoionization\nmodels, is to quantify the biases in strong-line indicators when\ncertain physical conditions are present and then compensate the\nbiases when inferring abundances from strong-line indicators for\nhigh-redshift galaxies.\n\nIn this study we have presented evidence that high-redshift\nstar-forming galaxies possess distinct H~II region physical\nproperties, as characterized by on average larger ionization\nparameters, higher electron densities, and temperatures, which are\npossibly connected to their relatively smaller sizes and higher\nSFR surface densities. These conditions may be quite common during\nthe epoch at $z \\geq 1$ when at least 50\\% of the local stellar\nmass density was formed \\citep{bundy06,drory05}. Therefore, they\nshould be characterized in more detail for a full understanding of\nthe star formation history of the universe as well as the buildup\nof heavy elements in galaxies. The next generation of ground-based\nnear-IR multi-object spectrographs will play a key role in\nassembling rest-frame optical emission-line measurements for large\nsamples of high-redshift galaxies, enabling the detailed study of\nstar-forming galaxies in the early universe.\n\n\\acknowledgments We are indebted to the DEEP2 team, whose\nsignificant efforts in establishing such a tremendous\nspectroscopic sample at $z\\sim1$ made this project possible. We\nalso thank Kevin Bundy for his assistance with estimating stellar\nmasses, and Bruce Draine and Jenny Greene for helpful discussions.\nA. E. S. acknowledges support from the David and Lucile Packard\nFoundation and the Alfred P. Sloan Foundation. A. L. C. is\nsupported by NASA through Hubble Fellowship grant HF-01182.01-A\nawarded by the Space Telescope Science Institute, which is\noperated by the Association of Universities for Research in\nAstronomy, Inc., for NASA, under contract NAS 5-26555. C. P. M. is\nsupported in part by NSF grant AST 04-07351. Funding for the DEEP2\nsurvey has been provided by NSF grants AST95-09298, AST-0071048,\nAST-0071198, AST-0507428, and AST-0507483 as well as NASA LTSA\ngrant NNG04GC89G. Funding for the Sloan Digital Sky Survey (SDSS)\nhas been provided by the Alfred P. Sloan Foundation, the\nParticipating Institutions, the National Aeronautics and Space\nAdministration, the National Science Foundation, the U.S.\nDepartment of Energy, the Japanese Monbukagakusho, and the Max\nPlanck Society. The SDSS Web site is http:\/\/www.sdss.org\/. We wish\nto extend special thanks to those of Hawaiian ancestry on whose\nsacred mountain we are privileged to be guests. Without their\ngenerous hospitality, most of the observations presented herein\nwould not have been possible.\n\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn recent years, ubiquitous computer vision applications are pervading our lives. Smartphones, self-driving terrestrial and aerial vehicles, Visual Sensor Networks (VSNs) are capable of acquiring visual data and performing complex analysis tasks. In particular, VSNs are expected to play a major role in the advent of the \\emph{Internet-of-Things} paradigm. Such computer vision tasks usually exploit a concise yet effective representation of the acquired visual content, rather than being based on the pixel-level content. In this context, local features represent an effective solution that is being successfully exploited for a number of tasks such as content-based retrieval, object tracking, image registration, etc. Local feature extraction algorithms usually consist of two distinct components. First, a keypoint detector aims at identifying salient regions (e.g. corners, blobs) within a given image. Second, a descriptor assigns each identified keypoint a descriptor, in the form of a set of values, based on the local characteristics of the image patch surrounding such keypoint. Such information is further processed in order to extract a semantic representation of the acquired content, e.g., by identifying and tracking objects, recognizing faces, monitoring the environment and recognizing events. \n\nAs regards visual feature extraction algorithms, SIFT~\\cite{DBLP:journals:ijcv:Lowe04} is widely considered as the state-of-the-art for a large number of tasks. It consists in a keypoint detector based on the Difference-of-Gaussians (DoG) algorithm, and in a scale- and rotation-invariant real-valued descriptor, based on local intensity gradients. Besides, SURF~\\cite{DBLP:conf:eccv:BayTG06} is partially inspired by SIFT and aims at achieving a similar level of accuracy at a lower computational cost. More recently, several low-complexity algorithms have been proposed, with the objective of alleviating the computational burden required by both traditional keypoint detectors and descriptors. For example, FAST~\\cite{rosten_2005_tracking} and AGAST~\\cite{mair2010_agast} are computationally efficient detectors capable of identifying stable corners. As for descriptors, binary-valued features are emerging as an efficient alternative to traditional real-valued features. BRIEF~\\cite{DBLP:conf\/eccv\/CalonderLSF10}, BRISK~\\cite{DBLP:conf\/iccv\/LeuteneggerCS11}, FREAK~\\cite{DBLP:conf\/cvpr\/AlahiOV12} and BAMBOO~\\cite{baroffio:2014:ICIP:BAMBOO} are instances of such category. For each identified keypoint, they compute a descriptor vector in the form of a sequence of binary values, each of which is obtained by comparing the (smoothed) intensities of a pair of pixels sampled around the keypoint. \nIn some cases, ad-hoc software-based implementations are available for specific hardware architectures~\\cite{baroffio:2014:ICIP:BRISKOLA}. \n\nLocal feature detection in video sequences has been addressed in the past literature, with the goal of identifying keypoints that are stable across time. For example, Shi and Tomasi~\\cite{Shi94goodfeatures} propose a widely adopted detector suitable for tracking applications. Zhang et. al propose a complex video-retrieval system based on color, shape and texture features extracted from the key-frames of a video~\\cite{Zhang1997643}. More recently, Zha et al. propose a method to extract spatio-temporal features from video content~\\cite{2009:Zha:ACCV}. Besides being a key to tasks such as object tracking, event identification and video calibration, temporally stable features improve the efficiency of coding architectures tailored to features extracted from video content~\\cite{BaroffioRCTT:TIP, BaroffioRCTT:ICIP2013, Makar:2014:TIP}.\nMore recently, Girod et al.~\\cite{journals\/ijsc\/MakarTCCG13} propose a feature detection and coding algorithm inspired by traditional motion estimation methods. Such algorithm selects a set of features corresponding to canonical image patches whose content is stable across frames, leading to a significant reduction of the transmission bitrate thanks to ad-hoc coding primitives. Although such algorithm represents a good solution for applications that require the efficient transmission of local features for further processing, it might not be the best in terms of computational complexity. Considering low-power devices, computationally intensive operations might significantly reduce the detection frame rate, possibly impairing performance of time-critical tasks or introducing undue delay. \nIn this paper, we introduce a fast detection algorithm based on BRISK~\\cite{DBLP:conf\/iccv\/LeuteneggerCS11} and tailored to the context of video sequences, aimed at reducing the computational complexity and thus enabling high frame rates, without significantly affecting performance in terms of accuracy.\n\nThe rest of this paper is organized as follows. Section~\\ref{sec:BRISK} introduces the main concepts behind BRISK. Section~\\ref{sec:algorithm} illustrates the proposed fast detection architecture. Section~\\ref{sec:experiments} defines the experimental setup and presents results. Finally, conclusions are drawn in Section~\\ref{sec:conclusions}.\n\n\n\n\n\\section{Binary Robust Invariant Scalable Keypoints (BRISK)}\\label{sec:BRISK}\n\nLeutenegger et al.~\\cite{DBLP:conf\/iccv\/LeuteneggerCS11} propose the Binary Robust Invariant Scalable Keypoints (BRISK) algorithm as a computationally efficient alternative to traditional local feature detectors and descriptors. The algorithm consists in two main steps: i) a keypoint detector, that identifies salient points in a scale-space and ii) a keypoint descriptor, that assigns each keypoint a rotation- and scale- invariant binary descriptor. Each element of such descriptor is obtained by comparing the intensities of a given pair of pixels sampled within the neighborhood of the keypoint at hand. \n\nThe BRISK detector is a scale-invariant version of the lightweight FAST~\\cite{rosten_2005_tracking} corner detector, based on the Accelerated Segment Test (AST). Such a test classifies a candidate point $p$ (with intensity $I_p$) as a keypoint if $n$ contiguous pixels in the Bresenham circle of radius 3 around $p$ are all brighter than $I_p + t$, or all darker than $I_p - t$, with $t$ a predefined threshold. Thus, the highest the threshold, the lowest the number of keypoints which are detected and vice-versa.\n\nScale-invariance is achieved in BRISK by building a scale-space pyramid consisting of a pre-determined number of octaves and intra-octaves, obtained by progressively downsampling the original image.\nThe FAST detector is applied separately to each layer of the scale-space pyramid, in order to identify potential regions of interest having different sizes. Then, non-maxima suppression is applied in a 3x3 scale-space neighborhood, retaining only features corresponding to local maxima. Finally, a three-step interpolation process is applied in order to refine the correct position of the keypoint with sub-pixel and sub-scale precision.\n\n\n\n\n\n\n\\section{Fast video feature extraction}\\label{sec:algorithm}\nLet $\\mathcal{I}_n$ denote the $n$-th frame of a video sequence of size $N_x \\times N_y$, \nwhich is processed to extract a set of local features $\\mathcal{D}_n$. First, a keypoint detector is applied to identify a set of interest points. Then, a descriptor is applied on the (rotated) patches surrounding each keypoint. Hence, each element of $d_{n,i} \\in \\mathcal{D}_n$ is a visual feature, which consists of two components: i) a 4-dimensional vector $\\mathbf{p}_{n,i} = [x, y, \\sigma, \\theta]^T$, indicating the position $(x,y)$, the scale $\\sigma$ of the detected keypoint, and the orientation angle $\\theta$ of the image patch; ii) a $P$-dimensional \\emph{binary} vector $\\mathbf{\\descr}_{n,i} \\in \\{0,1\\}^P$, which represents the descriptor associated to the keypoint $\\mathbf{p}_{n,i}$.\n\nTraditionally, local feature extraction algorithms have been designed to efficiently extract and describe salient points within a single frame. Considering video sequences, a straightforward approach consists in applying a feature extraction algorithm separately to each frame of the video sequence at hand. \nHowever, such a method is inefficient from a computational point of view, as the temporal redundancy between contiguous frame is not taken into consideration. The main idea behind our approach is to apply a keypoint detection algorithm only on some regions of each frame. To this end, for each frame $\\mathcal{I}_n$, a binary \\emph{Detection Mask} $\\mathcal{M}_n \\in \\{0, 1\\}^{N_x \\times N_y}$ having the same size of the input image is computed, exploiting the information extracted from previous frames. Such mask defines the regions of the frame where a keypoint detector has to be applied. That is, considering an image pixel $\\mathcal{I}_n(x,y)$, a keypoint detector is applied to such a pixel if the corresponding mask element $\\mathcal{M}_n(x,y)$ is equal to 1. Furthermore, we assume that if a region of the $n$-th frame is not subject to keypoint detection , the keypoints that are present in such an area in the previous frame, i.e. $\\mathcal{I}_{n-1}$, are still valid. Hence, such keypoints are propagated to the current set of features. That is,\n\n\\begin{equation}\n\t\\mathcal{D}_n = \\{d_{n,i} : \\mathcal{M}_n(\\mathbf{p}_{n,i}) = 1 \\;\\cup\\; d_{n-1, j} : \\mathcal{M}_n(\\mathbf{p}_{n-1, j}) = 0\\}\n\\end{equation} \n\nNote that the algorithm used to compute the \\emph{Detection Mask} needs to be computationally efficient, so that the savings achievable by skipping detection in some parts of the frame are not offset by this extra cost. In the following, two efficient algorithms for obtaining a \\emph{Detection Mask} are proposed: \\emph{Intensity Difference Detection Mask} and \\emph{Keypoint Binning Detection Mask}.\n\n\\vspace{-3 mm}\n\n\\subsection{Intensity Difference Detection Mask}\\label{sec:IDDM}\n\nThe key tenet is to apply the detector only to \nthose regions that change significantly across the frames of the video. In order to identify such regions and build the \\emph{Detection Mask}, we exploit the scale-space pyramid built by the BRISK detector, thus incurring in no extra cost. \nConsidering frame $\\mathcal{I}_n$ and $\\mathcal{O}$ detection octaves, pyramid layers $\\mathcal{L}_{n,o}, o = 1, \\dots, \\mathcal{O}$ are obtained by progressively smoothing and half-sampling the original image, as illustrated in Section~\\ref{sec:BRISK}. Then, considering two contiguous frames $\\mathcal{I}_{n-1}$ and $\\mathcal{I}_{n}$ and octave $o$, a subsampled version of the \\emph{Detection Mask} is obtained as follows:\n\n\n\\begin{equation}\n\t\\mathcal{M}'_{n,o}(k, l) =\n\t\t\\begin{cases} 1 &\\mbox{if } \\lvert \\mathcal{L}_{n, o}(k,l) - \\mathcal{L}_{n-1, o}(k,l) \\rvert \\le \\mathcal{T}_{I} \\\\ \n\t\t\t0 & \\mbox{if } \\lvert \\mathcal{L}_{n, o}(k,l) - \\mathcal{L}_{n-1, o}(k,l) \\rvert > \\mathcal{T}_{I},\n\t\t\\end{cases}\n\\end{equation}\n\nwhere $\\mathcal{T}_{I}$ is an arbitrarily chosen threshold and $(k,l)$ the coordinates of the pixels in\nthe intermediate representation $\\mathcal{M}'_{n,o}$. \nFinally, the intermediate representation $\\mathcal{M}'_{n,o}$ resulting from the previous operation needs to be upsampled in order to obtain the final mask $\\mathcal{M}_n \\in \\{0, 1\\}^{N_x \\times N_y}$. Masks can then be applied to detection in different fashions: i) exploiting the mask obtained resorting to each scale-space layer $o = 1, \\dots, \\mathcal{O}$ in order to detect keypoint at the corresponding layer $o$; ii) use a single detection mask for all the scale-space layers. \n\n\\vspace{-3 mm}\n\n\\subsection{Keypoint Binning Detection Mask}\\label{sec:KBDM}\n\nConsidering two contiguous frames of a video sequence, the amount of features identified in a given area are often correlated~\\cite{Eriksson:2014:ICASSP}. To exploit such information, \nthe detector is applied to a region of the input image only if the number of features extracted in the co-located region in the previous frame is greater than a threshold. Specifically, in order to obtain a \\emph{Detection Mask} for the $n-$th frame, a spatial binning process is applied to the features extracted from frame $\\mathcal{I}_{n-1}$. To this end, we define a grid consisting of $\\mathcal{N}_{r} \\times \\mathcal{N}_{c}$ spatial bins $\\mathcal{B}_{i, j}, i = 0, \\dots, \\mathcal{N}_{r}, j = 0, \\dots, \\mathcal{N}_{c}$. Thus, each bin refers to a rectangular area of $S_x \\times S_y$ pixels, where $S_x = \\nicefrac[]{N_x}{\\mathcal{N}_{c}}$ and $S_y = \\nicefrac[]{N_y}{\\mathcal{N}_{r}}$. Then, a two-dimensional spatial histogram of keypoints is created by assigning each feature to the corresponding bin as follows:\n\n\\begin{equation}\n\t\\mathcal{M}''_{n}(k, l) = \\lvert d_{n-1, i} \\in \\mathcal{D}_{n-1} \\rvert : \\lfloor \\nicefrac{x_{n-1, i}}{S_x}\\rfloor = k , \\lfloor \\nicefrac{y_{n-1, i}}{S_y}\\rfloor = l,\n\n\\end{equation}\n\nwhere $(x_{n-1, i}, y_{n-1, i})$ represents the location of feature $d_{n-1, i}$ and $|\\cdot|$ the number of elements in a set.\nThen, a binary subsampled version of the \\emph{Detection Mask} is obtained by thresholding such histogram, employing a tunable threshold $\\mathcal{T}_{H}$:\n\n\\begin{equation}\n\t\\mathcal{M}'_{n}(k, l) =\n\t\t\\begin{cases} 1 &\\mbox{if } \\mathcal{M}''_{n}(k, l) \\ge \\mathcal{T}_{H} \\\\ \n\t\t\t0 & \\mbox{if } \\mathcal{M}''_{n}(k, l) < \\mathcal{T}_{H},\n\t\t\\end{cases}\n\\end{equation}\n\nFinally, the \\emph{Detection Mask} $\\mathcal{M}_n$ having size $N_x \\times N_y$ pixels is obtained by upsampling the intermediate representation $\\mathcal{M}'_{n}$. Such a detection mask is applied to all scale-space octaves.\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\\label{sec:experiments}\n\\textbf{Dataset: }\nWe evaluated the proposed algorithms with respect to three different test scenarios. First, we exploited the \\emph{Stanford MAR dataset}~\\cite{Makar:2014:TIP}, containing the four VGA size, 200 frames long video sequences \\emph{Alicia Keys}, \\emph{Fogelberg}, \\emph{Anne Murray} and \\emph{Reba}. Each sequence contains a CD cover recorded with a hand-held mobile phone, under different imaging conditions such as illumination, zoom, perspective, rotation, glare, etc. Furthermore, for each sequence, the dataset contains the ground truth information, in the form of a still image of the corresponding CD cover, having a size of $500 \\times 500$ pixels. \n\nAs a second test, we evaluated the approaches resorting to the \\emph{Rome Landmark Dataset}. Such dataset includes a set of 10 query video sequences, each capturing a different landmark in the city of Rome with a camera embedded in a mobile device~\\cite{romelandmark}. The frame rate of such sequences is equal to 24fps, whereas the resolution ranges from 480x360 pixels (4:3) to 640x360 pixels (16:9). The first 50 frames of each video were used as query. On average, each query video corresponds to 9 relevant images representing the same physical object under different conditions and with heterogeneous qualities and resolutions. Then, distractor images randomly sampled from the \\emph{MIRFLICKR-1M} dataset~\\cite{huiskes08}, so that the final database contains 10k images.\n\nFinally, we tested our method on the \\emph{Stanford MAR multiple object} video set~\\cite{Makar:2014:TIP}. Such a set is made up of 4 video sequences, each consisting of 200 frames at 640x480 resolution. Each video is recorded with a handheld camera and portrays three different objects, one at a time.\n\n\\textbf{Methods: \nWe tested the two detection methods presented in Section~\\ref{sec:algorithm}, that is, \\emph{Intensity Difference Detecion Mask} and \\emph{Keypoint Binning Detection Mask}. In both cases, we employed the original BRISK implementation from the authors\\footnote{http:\/\/www.asl.ethz.ch\/people\/lestefan\/personal\/BRISK}, setting the number of octaves to 4 and the detection threshold to 55 and 70 for the \\emph{Stanford MAR dataset} and the \\emph{Rome landmark dataset}, respectively.\nAs regards \\emph{Intensity Difference Detection Mask}, we built the mask testing several different configurations. \nWe tested our algorithm with the 4 layers corresponding to each scale-space octaves. \nSince the performance was similar when using different layers, we resorted to the top-layer, i.e., the one with the lowest spatial resolution and processing cost. \nBoth \\emph{Intensity Difference Detection Mask} and \\emph{Keypoint Binning Detection Mask} require a threshold to be set in order to obtain the final detection mask. We tested several different configurations, each representing a tradeoff between computational efficiency and task accuracy. \n\nWe compared our algorithms with a \\emph{Temporally Coherent Detector} based on non-canonical patch matching~\\cite{Makar:2014:TIP}, which also exploits temporal redundancy in the detected keypoints.\nSuch algorithm aims at propagating stable keypoints across frames, exploiting a pixel-level representation of local features. In details, a traditional keypoint detector is applied to the first frame of a Group Of Pictures of size $\\Delta$. Given an identified keypoint, a non-canonical square image patch is extracted from the neighborhood of such a point. Then, considering the following frame, we searched for a matching patch in a window surrounding such a keypoint. Two patches are assumed to be a match if the Sum of Absolute Differences (SAD) between their pixels is below a given threshold $\\mathcal{T}_{BM}$. Finally, keypoints for which a match is found are propagated to the next frame, and their position is determined by the aforementioned block matching procedure. \nIn our tests, according to the prescriptions of~\\cite{Makar:2014:TIP}, we employed patches of $16 \\times 16$ pixels and we set $\\Delta = 10$ and $\\mathcal{T}_{BM} = 1800$. Furthermore, to make the procedure faster, we implemented a coarse-to-fine matching algorithm, where the first step consists in a spiral search algorithm with a precision of $2$ in a search window of $24 \\times 24$ pixels, whereas the second step in a spiral search algorithm with quarter-pixel precision in a search window of $1.75 \\times 1.75$ pixels. Finally, to further speed-up the process, we set an early termination SAD threshold $\\mathcal{T}_{ET} = 1000$. \nThis detector was originally proposed with the goal of maximizing coding efficiency, when patches around the detected keypoints need to be compressed and transmitted. To this end, this method can also adopt more sophisticated matching strategies, e.g., based on affine warping. However, in this paper we consider an implementation based on block matching to minimize the computational complexity. \n\n\n\\textbf{Evaluation methods and measures: }In the case of the \\emph{Stanford MAR dataset}, for a given video sequence, we extracted a set of features for each frame.\nThen, the set of features extracted from a frame is matched with the ones extracted from the ground truth frame. A radius match algorithm is used, where the matching threshold is set to $\\mathcal{T}_{M} = 0.18*512 \\simeq 102$. Finally, geometric coherence of matches is enforced resorting to the RANSAC algorithm. Finally, the number of \\emph{Matches-Post-Ransac} (MPR) is employed as the accuracy measure. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=0.43\\textwidth]{figures\/AliciaKeys_55_40_MPSvsTime_BLOCK_MATCHING.pdf}\n\t\\caption{Accuracy, measured as the number of matches post-RANSAC (MPR), and computational time for each frame of the \\emph{Alicia Keys} test sequence.}\n\t\\label{fig:MPR_over_time_Alicia}\n\t\\vspace{-3 mm}\n\\end{figure}\n\nIn the case of the \\emph{Rome Landmark dataset}, the accuracy of the task was evaluated according to the \\emph{Mean Average Precision} (MAP). Given an input query sequence $q$, for each frame $\\mathcal{I}_{q,n}$ it is possible to define the \\emph{Average Precision} as \n\\begin{equation}\n\tAP_{q,n} = \\frac{\\sum_{k=1}^Z P_{q,n}(k)r_{q,n}(k)}{R_{q,n}},\n\\end{equation}\nwhere $P_{q,n}(k)$ is the precision (i.e., the fraction of relevant documents retrieved) considering the top-$k$ results in the ranked list of database images; $r_{q,n}(k)$ is an indicator function, which is equal to 1 if the item at rank $k$ is relevant for the query, and zero otherwise; $R_{q,n}$ is the total number of relevant document for frame $\\mathcal{I}_{q,n}$ of the query sequence $q$ and $Z$ is the total number of documents in the list.\nThe overall accuracy for the query sequence $q$ is evaluated according to\n\n\\begin{equation}\n\tAP_q = \\frac{\\sum_{n = 1}^N AP_{q,n}}{N}, \n\\end{equation}\n\nwhere $N$ is the total number of frames of the query video $q$. \n\nFinally, the \\emph{Mean Average Precision} is obtained as\n\\begin{equation}\\label{eq:MAP}\nMAP = \\frac{\\sum_{q = 1}^Q AP_q}{Q}, \n\\end{equation}\nthat is, the mean of the $MAP_q$ measure over all the query sequences.\n\nIn the case of the \\emph{Stanford MAR multiple object} video set, the accuracy is measured according to a combined detection and tracking precision metric. In particular, for each frame, the goal is to correctly detect the portrayed database object and to identify its position within the frame. Each frame of the video sequences is matched against all the database object. Radius match and geometric verification steps are performed as in the case of~\\emph{Stanford MAR dataset} scenario. The matching object is the one with the highest number of matches-post-RANSAC. The bounding box for the identified object is obtained by projecting the database object corners according to the homography computed with the RANSAC algorithm at the previous step. Each frame is deemed as correct if the correct object is detected, and if the estimated position is consistent with the ground-truth information. As to the latter, the estimated object position is deemed correct if the displacement between the estimated centroid and the ground truth one is lower than a threshold. We set the value of such a threshold to 10 pixels. \n\nWe evaluated the complexity of the feature extraction methods by means of the required CPU time. We performed our tests on a laptop equipped with a 2.5GHz Intel Core i5 processor and 10 GB of RAM. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0.5cm 0cm 0cm 0cm, clip=true,width=0.40\\textwidth]{figures\/StanfordMAR_MPSvsTime2_new.pdf}\n\t\\caption{Energy-Accuracy curves for \\emph{Stanford MAR} dataset.}\n\t\\label{fig:energyaccuracy} \n\t\\vspace{-3 mm}\n\\end{figure}\n\n\\textbf{Results: } As an illustrative example, Figure~\\ref{fig:MPR_over_time_Alicia} shows the results obtained for the \\emph{Alicia Keys} sequence. The charts also report the results obtained when detection is performed independently on a frame-by-frame basis (full detection) to serve as a reference. \nWe observe that the method using the \\emph{Intensity Difference Detection Mask} (threshold 20) achieves an accuracy level similar to that of full detection ($MPR = 55$ vs. $56$), at a reduced computational time (20.5 ms vs. 24.5 ms). As for \\emph{Temporally Coherent Detector}, it leads to a significant loss in terms of accuracy ($MPR = 41$), while being quite computationally intensive (72 ms on average). While accuracy could be further improved by resorting to matching based on affine warping, this would further increase its complexity. This confirms the fact that this detector was originally designed with the goal of maximizing coding efficiency rather than computational cost. Since this is confirmed also on other test sequences, we do not report additional results for this detector. \n\nIt is interesting to observe the energy-accuracy trade-off that can be achieved by varying the threshold used by the algorithms based on detection masks. To this end, Figure~\\ref{fig:energyaccuracy} compares the performance of \\emph{Intensity Difference Detection Mask} and \\emph{Keypoint Binning Detection Mask} with that of full detection, averaging the results on the \\emph{Stanford MAR dataset}. The two methods based on a detection mask performs on a par, reducing the required computational time by 30\\% while losing as few as 4 matches. \n\nFurthermore, we tested our approach based on a \\emph{Detection Mask} on the \\emph{Rome Landmark Dataset}. \nFigure~\\ref{fig:RomeLandmarkMAP} compares the results of \\emph{Intensity Difference Detection Mask} with that of full detection, showing that computational time can be reduced by about 35\\% without affecting task accuracy. Furthermore, the feature extraction process can be speeded un by 3 times at the cost of 0.03\\% lower \\emph{Mean Average Precision}.\n\nFinally, the results of our fast detection algorithms on the \\emph{Stanford MAR multiple object} video set are reported in Figure~\\ref{fig:StanfordMultiple}. The computational time can be be reduced up to 40\\% without significantly impairing object detection and tracking performance. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=0.40\\textwidth]{figures\/RomeLandmarkMAPvsTime.pdf}\n\t\\caption{Energy-Accuracy curve for the \\emph{Rome Landmark} dataset, when using the \\emph{Intensity Difference Detection Mask} in order to reduce the detection area and with different values for the thresholding parameter. The computational time for each frame can be reduced from 28ms to 18ms, without significantly affecting the accuracy of the task.}\n\t\\label{fig:RomeLandmarkMAP} \n\t\\vspace{-1 mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=0.40\\textwidth]{figures\/StanfordMultiple_1_thr10.pdf}\n\t\\caption{Energy-Accuracy curves for the \\emph{Stanford MAR} multiple object sequences.}\n\t\\label{fig:StanfordMultiple} \n\t\\vspace{-4 mm}\n\\end{figure}\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this paper we presented a method for fast keypoint detection in video sequences based on \\emph{Detection Masks}. Results show that the proposed approach allows for a reduction in terms of computational complexity of up to 35\\% without significantly impair task performance.\nIn our future investigation we plan to further improve the \\emph{Detection Mask} building process, by introducing more sophisticated yet computationally efficient solutions. \n\n\n\n\\balance\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMany real-world networks can be formulated as graphs for modeling different relationships among nodes, such as social networks, chemical molecule structures and citation networks. Recently, there have been various attempts to extend Convolutional Neural Network (CNNs) and pooling methods to graph-structured data. These methods are named as Graph Neural Networks (GNNs) and have been successfully applied to different graph related tasks containing graph classification, node classification and link prediction \\cite{zhang2018deep,zhou2018graph}.\nGraph classification aims to extract accurate information from graph-structured data for classification. However, most existing GNNs based graph classification methods overlook that it's complicated and time consuming to collect or label the graph data. Learning with few labeled graph data is still a challenge for the practical applications of graph classification. \n\\begin{figure*}[!t]\\small\n\n\\centering\n\\subfigure[\\scriptsize{Graph spectral measures method \\cite{Chauhan2020FEW-SHOT}.}]{\n\\label{fig:gmsdemo}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pic\/gsm.PNG}\n}\n\\subfigure[\\scriptsize{Our method.}]{\n\\label{fig:ourdemo}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pic\/assumption1.PNG}\n}\n\\caption{Comparison of different methods. (a) The method from the global structure of dataset. The most representative graph of each class is viewed as class-prototype graph. The super-classes are clustered from training classes. (b) Our method from the view of local structure. $\\Theta$ is meta-learner's parameters and $\\theta_{1},\\theta_{2}$ are task specific parameters derived from meta-learner's fast adaptation. The components in dotted boxes have similar triangle structure. We assume the similarity can be discovered by a well initialized meta-learner within a few adaptation steps.}\n\\label{fig:first_image}\n\\end{figure*} \n\nFew-shot learning, aiming to label the query data when given only a few labeled support data, is a natural way to alleviate the problem. There are many papers discussing few-shot learning with meta-learning \\cite{FinnAL17}, data augmentation \\cite{NIPS2018_7504} or regularization \\cite{Yin2020Meta-Learning}, but most of them don't consider the graph data. \nFurthermore, there have been several methods for few-shot node classification \\cite{SatorrasICLR2018graph,Kim_2019_CVPR,meta_gnn} and few-shot link prediction \\cite{NIPS2017_7266,recommendation,MeLU,adCold-Start}, but they only focus on node-level embedding. Recently, Chauhan et al. \\cite{Chauhan2020FEW-SHOT} proposed few-shot graph classification based on graph spectral measures and got satisfactory performance. \nFrom the global structure of dataset, they \\textit{bridge} the test classes and training classes by assuming that the test classes belong to the same set of super-classes clustered from training classes. However, the above methods based on graph spectral measures might have some limitations for the following reasons: (1) the label spaces of training classes and test classes usually do not overlap in few-shot settings; (2) the bridging methods above may diminish the model to capture the local structure of test data. \n\n\nFrom the perspective of the graph's local structure, we observe that the graphs of training classes and test classes have similar sub-structures. For example, different social networks usually have similar groups; different protein molecules usually have similar spike proteins. We assume these similarities can be discovered by a well initialized meta-learner within a few adaptation steps. \nTherefore, we consider fast adaptation by a meta-learner from learned graph classification tasks to new tasks. Figure \\ref{fig:first_image} illustrates the existing method and our assumption. \n\nCurrently, GNNs have reliable ability to capture local structures over graphs by convolutional operations and pooling operations, but lack of fast adaptation mechanism when dealing with never seen graph classes. Inspired by Model Agnostic Meta-Learning (MAML, \\cite{FinnAL17}), which has attracted great attention because of its fast adaptation mechanism, we leverage GNNs as graph embedding backbone and meta-learning as a training paradigm to rapidly capture task-specific knowledge in graph classification tasks and transfer them to new tasks.\n\nHowever, directly applying MAML for fast adaptation is suboptimal due to the following reasons: (1) MAML requires painstaking hyperparameter searches to stabilize training and achieve high generalization \\cite{antoniou2018how}; (2) unlike images, graphs have arbitrary node size and sub-structure, which brings uncertainty for adaptation. There have some variants of MAML trying to overcome these problems by incorporating an online hyperparameter adaptation \\cite{behl2019alpha}, reducing optimization difficulty \\cite{nichol2018firstorder} or increasing context parameters for adaptation \\cite{zintgraf2019fast}, but they don't consider the structure of graph data. In this paper, we design a novel component named as adaptive step controller to learn optimal adaptation step for meta-learner to improve its learning robustness and generalization. The controller evaluates the meta-learner and decides when to stop adaptation by two kinds of inputs: (1) graphs' embedding quality, which is viewed as a meta-feature and indicated with Average Node Information (ANI, the average amount of node information in a batch of graphs); (2) meta-learner's training state, which is indicated with training loss of classification. \n\nWe formulate our framework as \\textbf{A}daptive \\textbf{S}tep MAML (AS-MAML). To the best of our knowledge, we are the first to consider the few-shot graph classification problem from the view of graph's local structure and propose a fast adaptation mechanism on graphs via meta-learning. Our contributions are summarized as follows:\n\\begin{itemize}\n\\item We propose a novel GNNs based graph meta-learner, which captures the features efficiently of sub-structures on unseen graphs by fast adaptation mechanism.\n\\item We design a novel controller for meta-learner. Driven by Reinforcement Learning (RL, \\cite{rl}), the controller provide optimal adaptation step for the meta-learner via graph's embedding quality and training loss. Our ablation experiments show its effectiveness to improve learning robustness and generalization.\n\\item We perform quantitative analysis and provide a generalization guarantee of key algorithms via a graph-dependent upper bound. \n\n\\item We evaluate our framework's performance against different baselines on four graph datasets and achieve state-of-the-art performance in almost all the tasks. We also evaluate the transferability of popular graph embedding modules on our few-shot graph classification tasks.\n\\end{itemize}\n\\section{Related Works}\n\\subsection{Graph Classification}\n\nIn graph classification tasks, each full graph is assigned a class label. There exist several branches for graph classification. The first is graph kernel methods which design kernels for the sub-structures exploration and exploitation of graph data. The typical kernels include Shortest-path Kernel \\cite{borgwardt2005shortest}, Graphlet Kernel \\cite{shervashidze2009efficient} and Weisfeiler-Lehman Kernel \\cite{shervashidze2011weisfeiler}.\n\nAs the main branch in recent years, GNNs have been successfully applied to graph classification. GNNs focus on node representations, which are iteratively computed by message passing from the features of their neighbor nodes using a differentiable aggregation operation. GCN \\cite{kipf2016semi} proposed Graph Convolutional Neural Network (termed as GCN) and got satisfying results based on directly feature aggregation from neighborhood nodes. GAT \\cite{GAT} imported attention mechanism for graph convolutional operations. GraphSAGE \\cite{SAGE} proposed an inductive framework which leverages node features to generate node embeddings efficiently for unseen nodes. In our framework, we use these classical methods to update nodes of graphs, while other methods like Graph Isomorphism Network (GIN) \\cite{xu2018how} are also applicable. \n\nIn the meantime, inspired by pooling in CNNs, a bunch of researchers concentrates on efficient pooling methods for accurate graph summary and computation efficiency. Beyond pooling layers in CNNs, graph pooling layers can enable GNNs to reason and get global representation from adjacent nodes. More and more evidence shows that graph pooling promotes the graph classification performance \\cite{lee2019self,diehl2019edge,unet}. SAGPool \\cite{lee2019self} implemented self-attention pooling on graphs considering both node features and graph topology. EdgePool \\cite{diehl2019edge} implemented a localized and sparse pooling transform backed by the notion of edge contraction. Graph U-nets \\cite{unet} implemented novel graph pooling and unpooling operations. Based on these operations, they developed a new model containing graph encoder and graph decoder and got satisfactory performance on graph classification tasks.\n\n\\subsection{Few-Shot Learning and Meta-Learning}\nFew-shot classification aims to learn a model under the circumstances of low sample resources and is usually powered by meta-learning in recent years. Meta-learning was also known as learning to learn, with a meta-learner observing various task learning processes and summarizing meta-knowledge to accelerate the learning efficiency of new tasks. Baxter et al. \\cite{BaxterBiasLearning} proposed a model to learn inductive bias from the perspective of bias learning, and they analytically showed that the number of examples required of each task decreases as the number of task rises.\n\nRecent meta-learning related works can be classified into three categories: optimization (or gradients) based methods, metric learning based methods and memory based methods. Optimization based methods aim to train a model to learn optimization \\cite{RaviICLR2017,LiICML2018}, learn a good initialization \\cite{FinnAL17} for rapid adaptation, or train parameter generator for task-specific classifier \\cite{RusuICLR2019}. Metric learning based methods aim to learn a feature space shared with new tasks \\cite{VinyalsBLKW16,SnellSZ17}. Moreover, memory based methods learn new tasks by reminiscence mechanism in virtue of physical memory \\cite{SantoroBBWL16}.\n\nFurthermore, almost all the previous few-shot learning methods are devised for image data, where images are prone to be represented in Euclidean space. Because we all have the idea that CNNs based models can perform efficient transfer in Euclidean space by feature reuse \\cite{Raghu2020Rapid}, in virtue of that different images usually share common edge features and corner features. Graph data such as social networks, which are appropriate to be formed into non-Euclidean space instead of Euclidean space. Few-shot learning in non-Euclidean space is addressed in our work.\n\\subsection{GNNs and its Generalization on Graph data}\nWe have seen several works of few-shot node classification promoting performance via GNNs \\cite{SatorrasICLR2018graph, Kim_2019_CVPR, liu2019GPN, yang2020dpgn,Yao2020Automated,NIPS2019_8389,yao2019graph,Gidaris_2019_CVPR,Liu2019PrototypePN}, but they just leverage the message passing mechanism of GNNs to enhance the performance on node classification, without involving the generalization of GNNs themselves and compatibility with graph classification task. For graph classification, Knyazev et al. \\cite{NIPS2019_8673} focus on the ability of attention GNNs to generalize to larger, more complex or noisy graphs. Lee et al. \\cite{LeeTransfer} imported domain transfer method by transferring the intrinsic geometric information learned in the source domain to the target. Hu et al. \\cite{Hu*2020Strategies} systematically studied the effectiveness of pre-training strategies on multiple graph datasets. Based on graph spectral measures, Chauhan et al. \\cite{Chauhan2020FEW-SHOT} proposed few-shot graph classification using the notion of super-graph by two steps: (1) they define the \\(p\\)-th Wasserstein distance to measure the spectral distance among graphs and select the most representative graph as prototype graph for each class; (2) by clustering the prototype graphs based on spectral distance, they clustered the prototype graph again into a super-graph consisting of super-classes. Therefore, they assume that the test classes belong to the same set of super-classes clustered from the training classes. We loosen the assumption and emphasize fast adaptation to boost few-shot graph classification. \n\n\\section{Problem Setup}\n\\begin{figure*}[!ht]\n \\centering\n \\includegraphics[width=0.70\\textwidth]{pic\/frameworkYZ.pdf}\n \\caption{Diagram of the AS-MAML framework's learning process in a single episode on the 2-way-1-shot graph classification task. The yellow arrows show meta-learner's $T$ step adaptations on support graphs. The blue dash arrows show $T$ step evaluations (Accuracies) on the query graphs. The orange dash arrows show the backpropagation (BP) according to $T$-th loss on query graphs. The step controller receives ANIs and classification losses on support graphs of each step. After that, the controller outputs the adaptation step $T$. Finally, the controller receives accuracies on query graphs as rewards and updates its own parameters. }\n \\label{fig:fm}\n\\end{figure*}\n\nWe form the few-shot problem as N-way-K-shot graph classification. Firstly, given graph data $\\mathcal{G}=\\left\\{({G}_{1},\\mathbf{y}_{1}), ({G}_{2},\\mathbf{y}_{2}), \\cdots, ({G}_{n},\\mathbf{y}_{n})\\right\\}$, where ${G}_{i}= \\left(\\mathcal{V}_{i}, \\mathcal{E}_{i}, \\mathbf{X}_{i}\\right)$. We use ${n}_{i}$ to denote the number of node set $ \\mathcal{V}_{i}$. So each graph ${G}_{i}$ has an adjacent matrix $\\mathbf{A}_{i} \\in \\mathbb{R}^{n_{i} \\times n_{i}}$ and a node attribute matrix $\\mathbf{X}_{i} \\in \\mathbb{R}^{n_{i} \\times d}$, where $d$ is the dimension of node attribute. Secondly, according to label $\\mathbf{y}$, we split $\\mathcal{G}$ into $\\{(\\mathcal{G}^{train},\\mathbf{y}^{train})\\}$ and $\\{(\\mathcal{G}^{test},\\mathbf{y}^{test})\\}$ as training set and test set respectively. Notice that $\\mathbf{y}^{train}$ and $\\mathbf{y}^{test}$ must have no common classes. We use episodic training method, which means at the training stage we sample a task $\\mathcal{T}$ each time, and each task contains support data $D^{train}_{sup}={ \\{ (G_{i}^{train},\\mathbf{y}_{i}^{train}) \\} }_{i=1}^{s}$ and query data $D^{train}_{que}={ \\{ (G_{i}^{train},\\mathbf{y}_{i}^{train}) \\} }_{i=1}^{q}$,\nwhere $s$ and $q$ are the number of support data and query data respectively. Given labeled support data, our goal is to predict the labels of query data. Please note that in a single task, support data and query data share the same class space. If $s={N \\times K}$, which means that support data contain N classes and K labeled samples per class, we name the problem as N-way-K-shot graph classification. At test stage when performing classification tasks on unseen classes, we firstly fine tune the meta-learner on the support data of test classes $D^{test}_{sup}={ \\{ (G_{i}^{test},\\mathbf{y}_{i}^{test}) \\} }_{i=1}^{s}$ , then we report classification performance on $D^{test}_{que}={ \\{ (G_{i}^{test},\\mathbf{y}_{i}^{test}) \\} }_{i=1}^{q}$.\n\\section{Proposed Framework}\n\nOverall, our few-shot graph classification framework consists of GNNs based meta-learner and a step controller to decide the adaptation steps of meta-learner. We use MAML to implement a fast adaptation mechanism for meta-learner because of its model agnostic property. Du et al. \\cite{recommendation} proposed an RL based step controller to guide meta-learner for link prediction. We argue that classification loss is suboptimal to be viewed as rewards for overcoming overfitting. Therefore, we adopt a novel step controller to accelerate training and overcome overfitting. Our step controller is also driven by RL but learns the optimal adaptation step by using ANIs and losses as inputs and classification accuracy as rewards. Figure \\ref{fig:fm} demonstrates the training process of our framework.\n\n\\subsection{Graph Embedding Backbone }\nWe explain our proposed framework with typical graph convolutional modules and pooling modules as embedding backbone, due to that novel graph convolutional modules or pooling modules are out of concern for this paper. The first step to represent a graph is to embed the nodes it contains. We investigate several embedding methods such as GCN, GAT, GraphSAGE and GIN. Here we focus on GraphSAGE as following reasons: (1) GraphSAGE has more flexible aggregators in few-shot learning scenarios; (2) Errica et at. \\cite{Errica2020A} set GraphSAGE as a strong baseline when compared to GIN for graph classification task. Hence we use mean aggregator of GraphSAGE as follows: \n\\begin{equation}\n\\mathbf{h}_{v}^{l} = \\sigma\\left(\\mathbf{W} \\cdot \\operatorname{mean}\\left(\\left\\{\\mathbf{h}_{v}^{l-1}\\right\\} \\cup\\left\\{\\mathbf{h}_{u}^{l-1}, \\forall u \\in \\mathcal{N}(v)\\right\\}\\right)\\right.,\n\\end{equation}\nwhere $\\mathbf{h}_{v}^{l}$ is the $l$-th layer representation of node $v$, $\\sigma$ is the sigmoid function, $\\mathbf{W}$ is the parameters and $\\mathcal{N}(v)$ contains the neighborhoods of $v$. Please note that this mean aggregator just belongs to the group of typical aggregators we use in experiments. We will provide concrete analysis for other aggregators in Section \\ref{section:bound} and Section \\ref{section:exp}.\n\nAfter that, we discuss existing pooling operations. Under the circumstances of few-shot learning, the meta-learner urgently needs a flexible pooling strategy with learning capability to strengthen its generalization. Here, we focus on self-attention pooling (SAGPool) \\cite{lee2019self} as our pooling layer thanks to its flexible attention parameters. The main step of SAGPool is to calculate the attention score matrix of graph ${G}_{i}$ as follows: \n\\begin{equation}\n\\mathbf{S}_{i}=\\sigma\\left(\\mathbf{\\tilde{D}}_{i}^{-\\frac{1}{2}} \\mathbf{\\tilde{A}}_{i} \\mathbf{\\tilde{D}}_{i}^{-\\frac{1}{2}} \\mathbf{X}_{i} \\mathbf{\\Theta}_{a t t}\\right),\n\\end{equation}\nwhere the $\\mathbf{S}_{i} \\in \\mathbb{R}^{n_{i} \\times 1}$ indicates the self-attention score, $n_{i}$ is node number of the graph. $\\sigma$ is the activation function (e.g., tanh), $\\mathbf{\\tilde{A}}_{i} \\in \\mathbb{R}^{n_{i} \\times n_{i}}$ is the adjacency matrix with self-connections, $\\mathbf{\\tilde{D}}_{i} \\in \\mathbb{R}^{n_{i} \\times n_{i}}$ is the diagonal degree matrix of $\\mathbf{\\tilde{A}}_{i} $, $\\mathbf{X}_{i} \\in \\mathbb{R}^{n_{i} \\times d}$ is $n$ input features with dimension $d$, and $\\mathbf{\\Theta}_{a t t} \\in \\mathbb{R}^{d \\times 1}$ is the learnable parameters of pooling layer. Based on the attention score, we select top $c