DETECTING ACTIVE GALACTIC NUCLEI USING MULTI-FILTER IMAGING DATA. II. INCORPORATING ARTIFICIAL NEURAL NETWORKS

As described in Réfrégier (2003), the Cartesian shapelets basis functions are Gaussian-weighted Hermite polynomials. The one-dimensional dimensionless basis functions are defined as

$$\begin{equation} \Phi _n(x)=[2^n \sqrt{\pi } n!]^{-\frac{1}{2}} H_n(x) e^{-\frac{x^2}{2}} \end{equation} \tag{ 1 }$$

in which H_n is an nth order Hermite polynomial. The dimensional basis functions are defined as

$$\begin{equation} B_n(x;\beta)=\beta ^{-\frac{1}{2}}\Phi _n \left(\frac{x}{\beta }\right) \end{equation} \tag{ 2 }$$

in which β is a characteristic size parameter that relates to the size of the object to be decomposed. The basis functions of shapelets are orthogonal, i.e.,

$$\begin{equation} \int _{-\infty }^{\infty } B_m(x;\beta) B_n(x;\beta) dx = \delta _{mn}\beta ^{-1} \end{equation} \tag{ 3 }$$

The profile f(x) of an object of interest can be decomposed into a series of shapelets basis functions B_n, and expressed as

$$\begin{equation} f(x)=\sum _0^{\infty } f_n B_n(x;\beta) \end{equation} \tag{ 4 }$$

where the shapelets coefficients are given by

$$\begin{equation} f_n=\int _{-\infty }^{\infty } f(x)B_n(x;\beta) dx. \end{equation} \tag{ 5 }$$

The two-dimensional shapelets basis functions are products of the one-dimensional basis functions. They are defined as

$$\begin{equation} B_{n1,n2}({\bf x};\beta)=\beta ^{-1} \Phi _{n1,n2} \left(\frac{{\bf x}}{\beta } \right), \end{equation} \tag{ 6 }$$

where x is the position vector. An image can be decomposed into two-dimensional shapelets basis functions by

$$\begin{equation} f({\bf x})=\sum _{n1,n2=0}^{\infty } f_{n1,n2} B_{n1,n2}({\bf x};\beta) \end{equation} \tag{ 7 }$$

where the shapelets coefficients are

$$\begin{equation} f_{n1,n2}=\int f({\bf x}) B_{n1,n2}({\bf x};\beta) d{\bf x}. \end{equation} \tag{ 8 }$$

Any astronomical image is the raw image (intrinsic image) convolved with the PSF, which primarily reflects the diffraction limit of the telescope's primary mirror and atmospheric turbulence. A PSF of the image can be obtained by studying shapes and sizes of stellar profiles. If the PSF image (P) and the galaxy image (H) are similarly pixelized, the intrinsic image (F) of the galaxy is the deconvolution of the observed image and the PSF image, i.e., H = F*P. The deconvolution can be realized by the inverse of the PSF matrix, but the process can be very slow and numerically unstable. In the IDL package, SHAPELETS, Massey & Réfrégier (2005) suggested convolving the basis functions with the PSF matrix and comparing this matrix with the observed image using least squares fitting. Without the PSF matrix inverse, shapelets coefficients can be calculated by minimizing χ² = (H − FP)^TV⁻¹(H − FP), where V is the noise map.² The intrinsic image of the galaxy can be reconstructed using shapelets coefficients but without PSF convolution.

In theory, an image can be reconstructed completely if the shapelets decomposition is completely sampled or oversampled, i.e., if the order of basis functions is infinity. In practice, only limited orders of basis functions can be used for decomposition; the shapelets decomposition is truncated at a maximum order of decomposition, n_max. In principle, the higher the n_max, the more faithful the reconstructed image is. In reality, higher n_max tends to pick up background noise around the object. n_max is also constrained by the degree of pixelization. This is because the sizes of some oscillations of the higher order basis functions are smaller than the pixel size, and therefore cannot be sampled properly.

The finite order of basis functions that can be used for decomposition and the discretization of the basis functions limit the ability of shapelets to reconstruct images. As mentioned in Bosch (2010), shapelets cannot properly reconstruct galaxies with high Sérsic indices because the shapelets basis functions are Gaussian weighted, which gives a profile softer than a galaxy profile (n = 1/2), especially for earlier type galaxies with n ∼ 4, and a sharper cut-off at larger radii than a Sérsic profile. A shapelets decomposition with a higher n_max can better represent a larger radius because the oscillations of higher orders have larger amplitudes at larger radii. However, such decomposition cannot represent the steeper profile of galaxy centers. There are also difficulties decomposing high-inclination or high-ellipticity objects. Because the traditional shapelets basis functions are circular, higher-order basis functions are needed to describe significantly elongated galaxies.

Although shapelets is a powerful method for image reconstruction, it cannot reconstruct images faithfully under some conditions because of the finite order of the basis functions. Thus, there have been many efforts to improve the versatility of shapelets. For example, Bosch (2010) proposed a compound shapelets decomposition by combining two shapelets. Melchior et al. (2010) showed that elliptical basis functions are more suitable for decomposing highly elliptical objects. Most of their works focused on maintaining the uniqueness of the shapelets coefficients because they relied on these coefficients to calculate photometric parameters such as flux and ellipticity. Since we use shapelets to optimize the PSF, the coefficients will be used only for image reconstruction. To this end, we propose a two-step process based on the SHAPELETS IDL package. The photometric parameters of a galaxy are then be measured on the reconstructed, PSF-optimized image.

The details of the two-step process are as follows.

• 1.  
  Apply shapelets decomposition with n_max = 20.³
• 2.  
  Generate the galaxy light profile by measuring the flux within a series of radii from the center to the edge of the galaxy, flux(r).
• 3.  
  Compare the light profiles between the observed galaxy (oflux) and the reconstructed galaxy (rflux) by calculating the relative flux difference Δflux = 1 − rflux/oflux.
• 4.  
  If the difference is larger than 2%, conduct another shapelets decomposition on the residual image (observed image minus the reconstructed image).
• 5.  
  Combine both reconstructed images and convolve with the optimized PSF (two-dimensional Gaussian with FWHM = 3 pixels).

We tested the two-step shapelets routine on galaxies randomly selected from the W4 field of the CFTLS (see below). In general, the relative flux difference between the observed and the reconstructed images is within 2%. Figure 1 shows the relative flux difference between the observed and the reconstructed images for the same galaxy. The blue line is the result after a single shapelets decomposition; the red line is the result after two shapelets decompositions. The gray area represents the first seeing disk. The two vertical red dash–dotted lines are R50 (radius contains 50% of the galaxy flux) and R80 (radius contains 80% of the galaxy flux), respectively. As mentioned before, shapelets might have problems dealing with high-Sérsic-index galaxy profiles. It is clear from Figure 1 that a single shapelets decomposition left a core in the galaxy center, which can be recovered by applying a second shapelets decomposition.

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A two-step shapelets can improve the shapelets decomposition of high-Sérsic-index galaxies. It is especially useful because we are particularly interested in AGNs that possess a steeper profile in the center. However, a two-step shapelets with circular basis functions cannot improve the decomposition of a high-inclination or high-ellipticity galaxy. As a result, we are only interested in lower-inclination objects (inclination <60°).

The CFHTLS data release includes the observed images, the SExtractor catalog for each image, the merged ugriz catalog, etc. Each SExtractor catalog contains 10⁵ entries. We only selected objects with the SExtractor parameter FLAGS = 0 (FLAGS≠0 indicates possible image imperfection) in each filter. We also rejected objects with parameters ISOAREA_IMAGE = 0 (isophotoal area above the analysis threshold is zero), FLUX_AUTO <0 (flux within a Kron-like elliptical aperture is negative), or MU_MAX = 99 (peak surface brightness above the background is not measured). All remaining objects are cleanly measured. In the merged ugriz catalog they are either stars with FLAG = 1 or galaxies with FLAG = 0.

Our intention is to use shapelets to adjust the PSF, i.e., each image has the same and the sharpest PSF. Although shapelets can reconstruct a galaxy image faithfully (with a relative flux difference less than 2%), shapelets does have some limitations as discussed in Section 2. Therefore, we limited our objects to those with inclinations smaller than 60° in r', but the inclination constraint is not based solely on shapelets limitations; it is also because we want to avoid high internal extinction in galaxies. As mentioned in Section 2, the IDL package SHAPELETS optimizes the basis function coefficients by minimizing χ². For a CFHTLS Wide survey u filter image, when u* > 22 mag, the algorithm that searches for the minimum χ² becomes insensitive, even with S/N ∼ 5. Since the u filter image is essential for this study, and its magnitude is usually the faintest among ugriz magnitudes, we selected only objects with u* < 22 mag.

Following the discussion above, there are about 8300 galaxies within W4's 25 fields, referred to as "W4" throughout the rest of the paper. To enlarge the training sample, we add galaxies from CFHTLS W3. There are 49 fields released by T0005. They are selected the same way as W4, but only include galaxies that are also in the SDSS DR7 and are classified as an AGN or an SF/SB galaxy, and whose redshift range is between 0.0075 and 0.1. The upper redshift limit ensures that each galaxy image is well resolved (i.e., its 50% flux radius is at least at the FWHM of the image). As we discuss in later sections, this study relies on parameters that are measured mostly from the galaxy light profile, which becomes more difficult to measure with increasing distance.

As discussed in Paper I, optical aperture photometric measurements require that the PSFs of all filters have the same FWHM, and the sharper the PSF the better. In order to optimize the PSF and ensure that all PSFs have the same FWHM, we applied a two-step shapelets decomposition on each object in W4 and W3. The two-step shapelets can reconstruct galaxies fairly well; in 60% of all galaxies, the relative flux difference is less than 2%, and 90% of all galaxies have a relative flux difference of less than 5%. The problematic decompositions are usually caused by image imperfections (for example, bad pixels) and close-by neighbors. We rejected galaxies whose flux differences are larger than 5%. The final W4 sample contains 8166 galaxies, and W3 sample contains 440 galaxies. Each galaxy has the same optimized PSF among each of the ugriz fitters.

After applying the two-step shapelets for each galaxy, we generated a light profile on the reconstructed galaxy by measuring fluxes within a series of radii, from the center to the edge of the galaxy. The interval between each radius is 1 pixel. Photometric parameters measured from each galaxy light profile are flux radii, concentration index, and color gradients.

In order to calculate a galaxy's physical size and its luminosity, a distance estimate is required. Since CFHTLS is a purely imaging survey, distance estimates must rely on photometric redshifts. The template-matching technique estimates photometric redshifts that rely on measuring galaxy colors in three or more filters, which give a rough approximation of the spectral energy distribution (SED). This method requires a set of SED templates covering a range of galaxy types, luminosities, and redshifts that represent the populations of the sample to be studied. The redshift of a target galaxy is chosen by minimizing the χ² between the target galaxy and the template galaxy. The accuracy of the photometric redshift depends on the template set and how closely it represents the population of the target sample. Another method, the empirical method, is more practical for a large data sample with both photometric parameters and spectroscopic redshift. It involves fitting the redshifts as a function of the photometric parameters such as colors. The coefficients of the function are estimated by minimizing the χ² between the predicted and measured redshifts. The redshift of a galaxy without a spectrum can be estimated by applying the function to its photometric parameters.

In our analysis, ANN training replaces the traditional function fitting. The photometric parameters of the training sample, such as color and concentration index, are used as inputs to the network. The network output is the photometric redshift. Instead of adjusting a function's coefficients to minimize the χ² between the photometric redshift and the spectroscopic redshift, ANN adjusts the weights between each neuron to minimize the cost function between the photometric redshift and the spectroscopic redshift. The cost function is defined as

$$\begin{equation} c=\frac{1}{N}\sum ^N_{k=1}(o_k-t_k)^2 , \end{equation} \tag{ 9 }$$

where t_k is the spectroscopic redshift, o_k is the photometric redshift, and N is the number of galaxies in the training sample.

4.1.1. The Training Sample

4.1.2. Training the Network

4.1.3. Results

The neural nets used for photometric redshift estimate are summarized in Table 1. The regression plot of the spectroscopic redshift versus the ANN estimated photometric redshift of the training set is shown in Figure 2. The photometric redshift is the median value among the 10 trained nets. The regression R-value is 0.92, and the rms is 0.026. Ball et al. (2004) estimated redshifts (up to 0.4) with an rms of 0.020 using ∼10⁵ objects from SDSS DR1. Collister & Lahav (2004) (redshift up to 0.7) received an rms of 0.023 using 30,000 objects from SDSS EDR. Our estimation, using only 1428 objects, has an rms of 0.026. In principle, the larger the training sample, the better the net will perform. Both Ball et al. (2004) and Collister & Lahav (2004) used samples 10 times larger than our sample, achieving only a slightly better rms. The only different input parameters we used compared with Ball et al. (2004) were parameters measured from the shapelets-reconstructed image. This shows that the PSF optimization improves the accuracy of the estimation. It might also indicate that 0.02 is close to the accuracy limit of the photometric redshift estimation using an ANN. We applied the trained nets on the $\overline{{\rm W4}/{\rm S7}}$ sample to obtain the photometric redshift.

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Table 1. ANN for the Photometric Redshift

Architecture	10:10:1
Input parameters	Galaxy colors: u − g, g − r, r − i, i − z
	Flux radius: R05, R20, R50 in r measured from the SHAPELETS-reconstructed image
	Mean surface brightness within R50 in r from the SHAPELETS-reconstructed image
Output	The photometric redshift
Training sample	W4/S7, consisting of 1428 objects
Training algorithm	Levenberg–Marquardt
Result	R-value:0.92, rms:0.026

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Galaxy morphological classification is traditionally done visually by trained observers. With large surveys, automated classification methods that rely on empirical relations between the galaxy morphology and a variety of its photometric and spectroscopic parameters have become more popular. The most often used parameters include concentration index, Sérsic index, colors, color gradients, bulge-to-disk ratio, effective radius, and disk scale length. Although automated methods are more efficient, visual classification still gives more accurate results. One of the latest visual classifications for a large sample of galaxies was performed by Nair & Abraham (2010). Nair & Abraham's catalog provides a dataset including morphological T types for over 14,000 galaxies. With their results, we built and trained the neural nets to estimate T type based on photometric parameters.

4.2.1. A Secondary Morphological Classification

Nair & Abraham (2010) selected galaxies from the SDSS DR4 spectroscopic main sample (Strauss et al. 2002). The sample is limited to extinction-corrected g magnitudes brighter than 16 and redshift range 0.01–0.1. Table 2 shows Nair's classification scheme. Nair's classification is consistent with the Third Reference Catalog of Bright Galaxies (de Vaucouleur et al. 1991), RC3. The mean deviation between these two classifications is about 1.2 T types, with Nair's classification slightly later. Nair's classification is fairly consistent with that of Fukugita et al. (2007). The mean deviation between the two classifications is about 0.8 Fukugita's class bins, with Nair's classification slightly earlier. There are 14,034 galaxies in Nair's sample, but only 20 galaxies appear in either the W4 or W3 samples, making it too small a sample to train a neural net. Therefore, we developed a secondary classification. First we used Nair's sample to train a neural net, MorphNet1, and applied it to samples W4/S7 and W3. Then we used samples W4/S7 and W3 to train another neural net, MorphNet2, and applied it to the remaining samples W4 and $\overline{{\rm W4}/{\rm S7}}$.

Table 2. Nair & Abraham's (2010) Classification Scheme

T	Class
−5	c0, E0, E+
−3	S0-
−2	S0, S0+
0	S0/a
1	Sa
2	Sab
3	Sb
4	Sbc
5	Sc
6	Scd
7	Sd
8	Sdm
9	Sm
10	Im
99	?

Notes. "?" and "T = 99" are of an unknown class (see Nair & Abraham 2010 for details). Our classification only used galaxies from T = −5 to T = 6. It has no effect on our classification.

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Before using Nair's sample, we removed galaxies whose classification is doubtful, highly uncertain, or "peculiar," or those with features such as shells or tidal tails. We also removed galaxies that are in pairs or flagged "interaction" and galaxies whose inclinations are >60°. The number of remaining galaxies was 8013. The input parameters used for MorphNet1 training were the following.

• 1.  
  Petrosian magnitudes in the u, g, r, i, z bands.
• 2.  
  Galaxy magnitudes in the u, g, r, i, z bands.
• 3.  
  Petrosian radius in r; radii of 50% and 90% Petrosian flux in r.
• 4.  
  Effective radii for de Vaucouleurs and exponential profiles in the r band.
• 5.  
  Concentration index in the r band.
• 6.  
  Mean surface brightness within 50% Petrosian flux radius in r.
• 7.  
  Spectroscopic redshift.

The architecture of the neural net contains three hidden layers with 10 neurons on each layer, denoted as 10:10:10:1. We followed the training procedures described in the previous section and Figure 3 shows the regression plot. The x-axis is the T type given by Nair & Abraham (2010). The y-axis is the mean value and the standard deviation of each T type estimated by the ANN. The regression R-value is 0.93, and the rms is 1.1 Nair's T parameter. MorphNet1 was then applied to the W4/S7 and W3 samples.

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W4/S7 was used as the training sample for MorphNet2. The input parameters were very similar to those for MorphNet1 but with some replacements. Radii of 50% and 90% Petrosian flux were replaced by radii of 50% and 80% galaxy flux, as given by the CFHTLS SExtractor catalogs. Effective radii for de Vaucouleurs and exponential profiles were replaced by flux radii measured from shapelets-reconstructed images. The redshift was also removed from the input because the $\overline{{\rm W4}/{\rm S7}}$ sample lacks spectroscopic redshifts. The architecture was 10:10:10:1. The regression plot is shown in Figure 4. The x-axis is the MorphNet1 T type and the y-axis is the media value of the MorphNet2 estimated T type. While training MorphNet2, we also renumbered Nair's T type (from −5 to 6) with 1–10 digit numbers. The R-value is 0.91, and the rms is 1.2 in Nair's T parameter.

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The formal rms of the estimated T type is about 1.6 in Nair's T parameter considering both MorphNet1 and MorphNet2, just less than 2 T types. Using an ANN, Ball et al. (2004) predicted a galaxy classification with 1.5 T types. Lahav et al. (1996) used ANN on ESO-LV galaxies with an rms around 1.8 T types. Considering our two-step method, uncertainties are not unreasonable. We applied MorphNet2 on sample $\overline{{\rm W4}/{\rm S7}}$ to assign T types.

AGNs can be detected or confirmed via spectroscopic surveys by measuring emission-line flux ratios. AGNs can also be detected via image analysis, which usually involves decomposing either the two-dimensional galaxy image or the one-dimensional galaxy light profile to a bulge, disk, and a possible central component (point source). The identity of the point source is determined by its colors. In Paper I, we showed that, in principle, AGNs, SF/SBs, and passive galaxies may be distinguished in a color–color diagram. We also demonstrated how an AGN can be separated from a passive galaxy by aperture photometric analysis. AGNs can also be distinguished from passive galaxies using parameters that reflect the concentration of light near the nucleus (e.g., the inverse concentration index versus color or the color gradient versus color) as showed in Choi et al. (2009). All these analyses were carried out on two-dimensional parameter planes, yet an analysis in a three-dimensional or multi-dimensional space should yield a clearer separation for each class.

In this study, we employ a non-parametric aperture photometric analysis using parameters such as flux radii, colors within each flux radius, and color gradients as input. An ANN for pattern recognition is used to test whether AGNs, passive galaxies, and SF/SB galaxies can be distinguished from one another.

4.3.1. Training Sample Selection

The training sample is selected by cross-identifying the MPA/JHU value-added galaxy catalog (Tremonti et al. 2004) DR7 and sample W4. There are 1420 galaxies, referred to hereafter as W4/M7; the remaining galaxies in W4 are referred to as $\overline{{\rm W4}/{\rm M7}}$. Following SDSS's spectroscopic classification criteria,⁶ an object is:

• 1.  
  An AGN, if

  $$\begin{eqnarray} &&{\mathrm{\log }_{10}\frac{F([\mathrm{O}\,\scriptsize{{III}}])}{F({\mathrm{H}\beta })}> 0.7-1.2 \times (\log _{10}\frac{F([\mathrm{N\,\scriptsize{II}}])}{F(\mathrm{H\alpha })}-0.4)}. \nonumber\\ \end{eqnarray} \tag{ 10 }$$
• 2.  
  An SF galaxy, if

  $$\begin{eqnarray} &&\log _{10}\frac{F([\mathrm{O\,\scriptsize{III}}])}{F(\mathrm{H\beta })}< 0.7-1.2 \times (\log _{10}\frac{F([\mathrm{N\,\scriptsize{II}}])}{F(\mathrm{H\alpha })}-0.4). \nonumber\\ \end{eqnarray} \tag{ 11 }$$
• 3.  
  An SB galaxy, if (11) is satisfied and the EW(Hα) >500 Å.

Of 1420 objects, MPA/JHU lists 74 AGNs, 278 SBs, and 443 SFs. The remaining 625 objects are classified only as "GALAXY." We expect that some of these are passive galaxies. Generally, passive galaxies do not have significant nuclear emission lines, such as [O ii] λ3727 Å, [O iii] λ5007 Å, Hα, and Hβ, which are contributed either by AGN activity or star formation. Balogh et al. (2004) show that the EW of Hα is close to 0 for galaxies without ongoing star formation; EW(Hα) varies from 4 Å to >40 Å for ongoing star formation. Therefore, we consider an object a passive galaxy when its emission-line flux ratio satisfies (11), and EW(Hα) <3 Å. Out of 625 galaxies, 297 can be considered passive galaxies. Among these, only 5% show EW([O iii] 5007 Å) >1.5 Å, and 3% show EW([O ii] 3727 Å) >1.5 Å. Thirty-two galaxies that satisfy (10) are added to the AGN group. For the remaining 296 objects, we drew the emission-line flux ratio of [N ii] to Hα versus EW(Hα) in Figure 5. It is clear that these galaxies (black circles) are closer to SF galaxies (blue dots) than to passive galaxies (red dots). Since we already have sufficient SF galaxies in the training sample, we removed these galaxies from consideration. We also added galaxies from sample W3, which included 220 AGNs and 140 passive galaxies. The remaining galaxies in sample W3 are SF and SB galaxies, which are no longer used. The final training sample contains 326 AGNs, 437 passive galaxies, 278 SB galaxies, and 433 SF galaxies.

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4.3.2. The Network Training

Sample W4 is separated into two subsamples: W4/S7 which includes SDSS spectroscopic redshifts, and $\overline{{\rm W4}/{\rm S7}}$ which only has ANN-estimated photometric redshifts with an rms of 0.026. Although the W4/S7 and $\overline{{\rm W4}/{\rm S7}}$ subsamples are taken from the same W4 sample, they are not necessarily from the same underlying distribution. The two-sample Kolmogorov–Smirnov test (K-S2 test) indeed rejects the null hypothesis that the two subsamples' redshifts share a similar distribution at a level of probability of less than 5%.

We compare the normalized redshift distributions of the W4/S7 sample (the red bars) and $\overline{{\rm W4}/{\rm S7}}$ (the blue bars) in Figure 6. The bin size is 0.025. The number within the parentheses is the size of each sample. The gray background is the normalized redshift distribution of the entire W4 sample, which mostly consists of photometric redshifts. There are two "spikes," one around z = 0.05 in sample W4/S7 and another around z = 0.12 in $\overline{{\rm W4}/{\rm S7}}$. These significant differences between subsamples W4/S7 and $\overline{{\rm W4}/{\rm S7}}$ might indicate a possible systematic error.

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To rule out the possibility that some galaxies with z = 0.05 were misidentified as z = 0.12, we took a closer look at the W4/S7 sample. We selected galaxies whose spectroscopic redshifts are in the z ∼ 0.05 bin and plot the distribution of their ANN redshifts in Figure 7(a). It is clear that z ∼ 0.05 galaxies are sometimes misidentified as z ∼ 0.075 galaxies, but the few galaxies that are misidentified in the z ∼ 0.12 bin cannot contribute to the z ∼ 0.12 "spike," In order to gain some insight about what might contribute to the z ∼ 0.12 "spike," we selected another group of galaxies from the W4/S7 sample, whose ANN redshift is within the z ∼ 0.12 bin. We plot the spectroscopic redshift distribution of these galaxies in Figure 7(b). It is clear that galaxies with larger redshifts can be misidentified as z ∼ 0.12. This might be because some parameters (such as flux radii) used to train the neural net rely on the angular resolution, and their usefulness will be diminished as the distance increases. Since we are only interested in galaxies within z ⩽ 0.1, this will not affect our analysis.

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The "spike" or the lack of z ∼ 0.05 galaxies in the $\overline{{\rm W4}/{\rm S7}}$ sample appears to be real. Blanton et al. (2003) calculated that the characteristic i magnitude of the SDSS main galaxy sample is −21 mag. If a galaxy's redshift is 0.05, its apparent magnitude is about +16 mag. In sample (W4/S7)_0.1, there are 86 out of 289 galaxies whose i magnitude is less than +16. In the $(\overline{{\rm W4}/{\rm S7}})_{0.1 }$ sample, there are only 74 out 1388 galaxies whose i magnitude is less than +16. It seems that the "spike" may be explained by a relatively larger fraction of brighter galaxies in the (W4/S7)_0.1 sample than in $(\overline{{\rm W4}/{\rm S7}})_{0.1}$.

Galaxy apparent magnitudes are adopted from the CFHTLS. The absolute magnitudes are calculated and corrected for Galactic extinction and the K-correction. The mean ugriz magnitudes and the standard deviations are calculated for the W4/S7 and $\overline{{\rm W4}/{\rm S7 }}$ samples. The results are shown in Table 4. The mean absolute magnitude of galaxies in sample (W4/S7)_0.1 is about 0.5 mag brighter than sample $(\overline{{\rm W4}/{\rm S7}})_{0.1}$. Not unexpectedly, because of the superior image quality and longer exposure times, CFHTLS can probe deeper into the galaxy luminosity function than SDSS.

Table 4. Mean ugriz Absolute Magnitudes and the Standard Deviations of W4

Sample	Mu	Mg	Mr	Mi	Mz
(W4/S7)0.1	−18.4 ± 1.1	−19.5 ± 1.2	−19.9 ± 1.3	−20.3 ± 1.4	−20.4 ± 1.5
W4/S7	−19.2 ± 1.2	−20.3 ± 1.3	−20.7 ± 1.5	−21.1 ± 1.4	−21.3 ± 1.4
$(\overline{{\rm W4}/{\rm S7}})_{0.1}$	−17.2 ± 1.4	−18.2 ± 2.7	−18.2 ± 2.2	−17.8 ± 1.9	−19.1 ± 1.8
$\overline{{\rm W4}/{\rm S7}}$	−18.1 ± 1.3	−19.2 ± 1.9	−19.5 ± 2.2	−19.5 ± 1.9	−20.2 ± 1.6

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The galaxy morphological classification is estimated using the MorphNet1 and MorphNet2 ANNs based on Nair & Abraham's classification. Although we applied the neural nets to the entire W4 sample, only galaxies with z ⩽ 0.1 should be considered meaningful because Nair & Abraham's sample is limited to z ⩽ 0.1. Figure 8 shows the normalized T type distribution of sample (W4)_0.1 (1677 galaxies) shown as gray bars; regarding its two subsamples, (W4/S7)_0.1 (in red) includes 289 galaxies and $(\overline{{\rm W4}/{\rm S7}})_{0.1}$ (in blue) includes 1388 galaxies. The bottom histogram is the T type distribution of the training sample selected from Nair & Abraham (2010), which includes about 8000 galaxies. There is a large fraction of galaxies with T = −5 in Nair's sample, which is unusual as elliptical galaxies typically constitute fewer than 10% of field galaxies (Mahtessian 2011). This is likely in part because Nair & Abraham (2010) classified elliptical galaxies as T = −5, including those that are generally classified as T = −6 and −4, and those only classified as E but without an explicit subclass. It also might explain the extra T = −3 galaxies in the (W4/S7)_0.1 sample, which includes some misidentified galaxies from T = −5. The T type distributions of (W4/S7)_0.1 and $(\overline{{\rm W4}/{\rm S7}})_{0.1}$ are very similar except at T = −3. The KS2-test indicates that they might have the same underlying distributions with p > 0.05. Overall, within z ⩽ 0.1, and u apparent magnitude ⩽ + 22, the galaxy morphology is mostly evenly distributed from lenticulars (T = −2) to late-type spirals (T = 5).

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We classified W4 galaxies as AGNs, SF/SBs, and passive galaxies using neural nets SpecNet1 and SpecNet2 based on the classifications from MPA/JHU DR7. As identifying AGNs relies largely on parameters deduced from the galaxy's light profile, our interpretations are also limited to galaxies with z ⩽ 0.1. The (W4)_0.1 sample includes 1677 galaxies; among these there are 186 AGNs, 186 passive galaxies, and 1305 SF/SB galaxies. Table 5 summaries the spectral classification of the (W4)_0.1 sample.

Table 5. Spectral Classifications for the (W4)_0.1 Sample

Type	(W4/M7)0.1	$(\overline{{\rm W4}/{\rm M7}})_{0.1}$	Total
AGN	11%	11%	11%
Passive	19%	8%	11%
SF/SB	70%	81%	78%

Notes. In the (W4/M7)_0.1 sample AGNs and SF/SB galaxies are classified by MPA/JHU; passive galaxies are constrained by emission-line ratios and equivalent widths. Galaxies in the $(\overline{{\rm W4}/{\rm M7}})_{0.1}$ sample are classified by neural nets SpecNet1 and SpecNet2.

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We compared the absolute magnitudes of AGNs, SF/SBs, and passive galaxies in Table 6. The mean magnitudes of AGNs and passive galaxies are very similar, with AGNs slightly brighter. SF/SB galaxies are always ∼1–2 mag fainter. The reason for this is that SF/SB galaxies are more likely to be later-type galaxies, which are generally fainter than early-type galaxies. This is clearly demonstrated in Figure 9 where the T types of AGN, SF/SB, and passive galaxies are compared. AGNs are typically lenticulars (T = −3, −2) or early-type spirals (T = 0–3). SF/SB galaxies are generally later-type galaxies (T > 3). The passive galaxies also tend to be earlier-type galaxies as indicated by their definition: early-type, bulge-dominated, and non-active.

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