EFFECT OF INCLINATION OF GALAXIES ON PHOTOMETRIC REDSHIFT

The photo-z error, z(photo)−z(spec), as a function of the inclination of the disk galaxies is shown in Figure 1. Three cases are considered: in Figure 1(a) the SDSS photo-z based on the template-fitting⁴ approach (Csabai et al. 2003, and references therein); in Figure 1(b) the SDSS photo-z by using the Artificial Neural Network approach (Oyaizu et al. 2008), in which the authors used an implementation similarly to that of Collister & Lahav (2004); and in Figure 1(c) the photo-z calculated in this work based on the Random Forest approach (Carliles et al. 2010, details are deferred to Section 4.3). The "CC1" photo-z of Oyaizu et al. (2008) are used, because they were obtained by employing only four SDSS colors u−g, g−r, r−i, and i−z in the training procedure, in this sense similar to Figure 1(c). While both the bias ($\left<\mathrm{\mbox{\it z}(photo)}-\mathrm{\mbox{\it z}(spec)} \right>$= −0.004 ± 0.001, 0.003 ± 0.001, 0.003 ± 0.0003) and the root mean square (rms = $\sqrt{\vphantom{A^A}\smash{\hbox{${ \left< [ \mathrm{\mbox{\it z}(photo)}-\mathrm{\mbox{\it z}(spec)} ]^{2}\right> }$}}}$ = 0.032 ± 0.002, 0.020 ± 0.001, 0.018 ± 0.0005)⁵ are, respectively, of the same order of magnitudes for all of the cases, the dependences on the inclination are noticeably different. In the template-fitting

Zoom In Zoom Out Reset image size

approach the bias in the photo-z increases from the face-on to edge-on galaxies, in such a way bias(edge-on)−bias(face-on) = 0.036. The photo-z bias also changes sign with inclination, showing that it is the ensemble bias (= −0.004 ± 0.001) being minimized instead of the bias for a particular group of galaxies. The statistics of the photo-z error in all of the inclination bins are given in Table 1. In contrast, the photo-z error does not show prominent dependence on the inclination of the disk galaxies in both of the machine learning approaches (Figures 1(b) and (c)). Because the inclination is not included explicitly in the training procedure in both of these approaches, this lack of inclination dependence is interpreted as the success of the methods in segregating the photometry of the disk galaxies by their inclination. The inclination of the galaxies therefore impacts their observed photometry, which in turn can be learned by a machine learning approach. We investigate how the inclination of galaxies impacts their observed photometry in the next section.

Next, we seek to understand why the photo-z error correlates with the inclination of the photometry of the disk galaxies. Our approach is to establish the variance in the galaxy sample and its relation to the parameter(s) of interest, or the inclination of the galaxies in the current context. The Principal Component Analysis (PCA), which is adopted here, was shown to be a powerful technique for this purpose (e.g., Madgwick et al. 2003; Yip et al. 2004). PCA identifies directions (or eigenvectors) in a multi-dimensional data space such that they represent the maximized sample variance. The lower the order of eigenvector, in this case the eigenspectrum (Connolly et al. 1995b), the more sample variance it describes. After relating the sample variance with the inclination, if possible, we can examine the involved eigenspectra to explain why in the edge-on galaxies the photo-z error is larger.

The first two eigenspectra calculated based on the photometry of the disk galaxies are shown in Figure 2. The first eigenspectrum resembles the mean spectrum of the galaxies. Perhaps more interestingly, the second eigenspectrum visually resembles the extinction curve obtained in Paper I, despite the fact that they are obtained by two completely different approaches (PCA versus composite spectra construction) and data sets (the photometry versus the spectroscopy of the galaxy sample). The discrepancy between the second eigenspectrum and the actual extinction curve is expected to be primarily due to variance in galaxy type within our disk galaxy sample.

Zoom In Zoom Out Reset image size

To confirm that the second mode represents the inclination effect on the photometry, we examine the distribution of the eigencoefficients of various orders as a function of the inclination (Figure 3). The eigencoefficients of a galaxy are the expansion coefficients of its photometry onto the eigenspectra. A clear separation is seen in the distribution of the second eigencoefficients (a₂) between the face-on and edge-on galaxies. This separation is not seen, or is not as prominent, in the other orders of eigencoefficients. In other words, the second eigencoefficient is an indicator for the inclination of the disk galaxies.

Zoom In Zoom Out Reset image size

Going back to Figure 2 to examine the eigenspectra, we see that the first eigenspectrum minus the second eigenspectrum results in a spectral energy distribution that is redder than the first eigenspectrum or the average galaxy spectrum. If the adopted theoretical model in the template-based photo-z does not take account of this reddening effect in the edge-on galaxies, the model would need to be shifted to a higher-than-true redshift in order to match the redder colors of the galaxies. This situation results in an over-estimation of the photo-z, or what is seen in Figure 1(a).

We derive the following formulae for correcting rest-frame magnitudes of the whole disk galaxies in the SDSS u, g, r, i, z bands

$$\begin{eqnarray} M_u(1) & = & M_u(b/a) - 1.14\cdot \log^{2}_{10}(b/a),\\[4pt] M_g(1) & = & M_g(b/a) - 0.76\cdot \log^{2}_{10}(b/a),\\[4pt] M_r(1) & = & M_r(b/a) - 0.39\cdot \log^{2}_{10}(b/a),\\[4pt] M_i(1) & = & M_i(b/a) - 0.17\cdot \log^{2}_{10}(b/a), \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} M_z(1) & = & M_z(b/a) - 0.00\cdot \log^{2}_{10}(b/a). \end{eqnarray} \tag{ 5 }$$

The underlying calculation is similar to that in Paper I; as such we fit to the relative extinction versus b/a data the following relation:

$$\begin{equation} M(b/a) - M(1) = \eta \, \log^{2}_{10}(b/a), \end{equation} \tag{ 6 }$$

where M(b/a) is the K-corrected absolute magnitude of a galaxy at a given inclination. This functional form is taken to be the same for all of the SDSS bands, and the proportional constant η is fitted for each band. The left-hand side of Equation (6) is the relative extinction because M(b/a) − M(1) = A^intrinsic(b/a) − A^intrinsic(1) (see the Appendix for details). The actual relative extinction versus b/a values are given in Table 2. The magnitudes of the whole galaxies are considered in Equations (1)–(5), instead of the central 3''-diameter areas of the galaxies that were considered in Paper I.⁶ In particular, the model magnitudes ("modelMag") from the SDSS are used because they give unbiased colors of galaxies, a result of the flux being measured through equivalent apertures in all bands (Stoughton et al. 2002). The 1σ uncertainties for the best-fit η are, respectively, 0.05, 0.03, 0.02, 0.01 in u, g, r, and i. The corresponding reduced chi-squares are 3.04, 2.79, 1.10, and 1.18. The points for the relative extinction z(b/a) − z(1) versus inclination are scattered around zero and do not suggest any non-trivial functional form. We therefore do not attempt to fit the above relation in the z band and assign zero to the proportional constant (Equation (5)).

For the purpose of photo-z estimation, we will show in Section 4.3 that the color corrections derived from Equations (1)–(5) perform well, in the sense that the resultant photo-z are unbiased with inclination. On the other hand, the larger chi-squares in the u and g bands suggest that the chosen relation may not be ideal. We therefore encourage the interpolation to the actual data listed in Table 2 when higher-accuracy corrections are required. We choose Equation (6) for the purpose of a direct comparison with literature (e.g., Unterborn & Ryden 2008; Yip et al. 2010), in which the powers of log₁₀(b/a) have been considered. Only even integers are allowed in the power index because log₁₀(b/a) is negative for all b/a values except unity. A power index of 4 is confirmed to provide a bad fit to our data, and a power index of 0 contradicts the data because it gives no b/a dependence. We plan to find other functional forms that may be unconventional but better describe the data.

The rest-frame M_u(b/a) − M_g(b/a) versus M_g(b/a) − M_r(b/a) color–color diagram of our disk galaxies is shown in Figure 4, before and after the above inclination-dependent magnitude corrections. The average and the 1σ sample scatter of the colors of the edge-on galaxies are, before the corrections: 1.37 ± 0.25 (for color M_u(b/a) − M_g(b/a)), 0.57 ± 0.14 (M_g(b/a) − M_r(b/a)), and after the corrections: 1.14 ± 0.26 (M_u(b/a) − M_g(b/a)), 0.35 ± 0.14 (M_g(b/a) − M_r(b/a)). The before-and-after color offset is ≈0.2 for both the M_u(b/a) − M_g(b/a) and M_g(b/a) − M_r(b/a) colors. Obviously, the colors of the face-on galaxies remain unchanged: 1.11 ± 0.12 (M_u(b/a) − M_g(b/a)) and 0.39 ± 0.08 (M_g(b/a) − M_r(b/a)). For both colors, the offsets in the systematic locations between the edge-on galaxies and the face-on ones are greatly reduced after the corrections.

Zoom In Zoom Out Reset image size

The second-order power dependence of the relative extinction of the whole galaxies on log₁₀(b/a) agrees with that obtained by Unterborn & Ryden (2008). For the center of the disk galaxies (within 0.5 half-light radius), however, the extinction–inclination relation is steeper than a log²₁₀(b/a) dependence (Paper I). The difference likely reflects a higher extinction in the center relative to the edge of the galaxies, or an extinction radial gradient. We will investigate this finding in a separate paper.

The determination of many properties of galaxies, including the photo-z, involves fitting to the observational data a theoretical stellar population model. The model is defined by the related physical parameters, such as the stellar age and metallicity, at the correct amplitudes. In this kind of analysis, instead of correcting the observational data against the inclination effect as discussed previously, one can correct the theoretical model itself. The latter approach is at the expense of (or/and has the merit of) introducing the inclination of a galaxy as an extra parameter, which is to be determined simultaneously with the other properties during the minimization. Given a theoretical spectrum from a stellar population model, f_λ(b/a = 1), its inclined flux densities can be calculated as follows:

$$\begin{equation} f_{\lambda }(b/a) = f_{\lambda }(1) \cdot s_{\lambda }(b/a), \end{equation} \tag{ 7 }$$

where

$$\begin{equation} s_{\lambda }(b/a) = 10^{- 0.4 \, \eta _{\lambda } \, \log ^{4}_{10}(b/a)} \end{equation} \tag{ 8 }$$

is derived from the extinction curve of the disk galaxies and its variation with inclination as given in Paper I. Here,

$$\begin{equation} \eta _{\lambda } = \sum _{j = 0}^{3} \frac{a_{j} \, \tilde{\nu }^{j}}{\log ^{4}_{10}\left({b/a|}_{\mathrm{ref}}\right)}, \end{equation} \tag{ 9 }$$

where b/a|_ref is a reference inclination. The wave number $\tilde{\nu }$ is the inverse of wavelength, in the unit of inverse micron, μm⁻¹. The coefficients a_j are listed in Table 4 of Paper I, where b/a|_ref = 0.17. In presenting this formalism we use the extinction curve and its inclination dependence from Paper I, which apply to the inner 0.5 half-light radius of the disk galaxies. Extinction curves that are applicable to other parts of the galaxies, e.g., the whole galaxies, naturally can be used when required.

As presented above, the machine learning approaches give photo-z which do not show prominent bias with inclination. This result is not entirely surprising, if it is seen as the success of the methods in segregating the photometry of the disk galaxies by their inclination (see also Section 3). Here we consider the Random Forest approach, the power of which in estimating the photo-z is discussed in detail in Carliles et al. (2010). Basically, this method builds an ensemble of randomized regression trees and computes regression estimates as the average of the individual regression estimates over those trees. The trees are built by recursively dividing the training set into a hierarchy of clusters of similar galaxies. The procedure minimizes the resubstitution error in the resultant clusters (Equations (1)–(3) of Carliles et al. 2010, and references therein). We train a forest of 50 trees on 100,000 randomly selected galaxies from the SDSS spectroscopic sample and regress on our disk galaxy sample to obtain the photo-z estimates shown in Figure 1(c).

Given the inclination of disk galaxies to be a parameter which modulates the variance in the photometric sample (Section 3.2), we deduce that the implicit inclusion of the inclination during the training procedure of the Random Forest method would give even better photo-z estimates than the case where only the SDSS colors are used (i.e., Figure 1(c)). Indeed, the photo-z estimates improve, with the resultant bias being 0.0004 and the rms being 0.017. The error in the photo-z versus inclination for this case is shown in Figure 5. It would be interesting to see if other machine learning approaches give similar improvement. For example, although the inclination of a galaxy was not included as a training parameter in the work by Oyaizu et al. (2008), their Artificial Neural Network approach in principle allows for multi-parameter training. It would also be worthwhile to train on other galaxy parameters for comparison, e.g., the morphological parameters as considered in Way et al. (2006, 2009).

Zoom In Zoom Out Reset image size

Another important question is whether the magnitude corrections (Equations (1)–(5)) are applicable to deriving photo-z that are unbiased with inclination. Using the Random Forest approach, we select from the above training sample face-on only (b/a = 0.9–1.0) galaxies as our new training sample (about 50,000 objects). We train on the uncorrected u−g, g−r, r−i, i−z colors of this new sample and regress on the corrected colors of our disk galaxy sample. The color corrections are done using Equations (1)–(5) (see the Appendix). If the corrections give correct face-on colors of the disk galaxies, these colors should be fully described by those of the face-on galaxies in the training sample, and the resultant photo-z should be unbiased with inclination. Indeed, we find no inclination dependency in the photo-z error, as shown in Figure 6. The bias and rms are, respectively, 0.002 ± 0.0003 and 0.018 ± 0.0005.

Zoom In Zoom Out Reset image size