SINGLE PARAMETER GALAXY CLASSIFICATION: THE PRINCIPAL CURVE THROUGH THE MULTI-DIMENSIONAL SPACE OF GALAXY PROPERTIES

PCA (Pearson 1901; Jackson 1991; Jolliffe 2002), also known as Karhunen-Loève transform, is a widely used method for dimensionality reduction and classification. It can be seen as a transformation involving a translation, linear scaling, and rigid rotation of a collection of N p-dimensional data points onto a new coordinate system. The new orthonormal axes, or principal components $\lbrace {\mathbf{PC}}_{i} \rbrace _{i=1}^{p} \in \mathbb {R}^{N \times 1}$, are constructed such that the projections of the data points on the PC_i are uncorrelated. PC₁ is selected as the axis on $\mathbb {R}^{p}$ which has the highest possible variance of the points projected onto it. The next PC_i are ordered in descending value of the variance, PC_p being the lowest. Thus, dimensionality reduction is attained by describing the data in terms of the most important principal components (Hastie et al. 2009). This can be obtained by considering only the space spanned by the first q ⩽ p variance-ranked eigenvectors whose cumulative variance reaches above a high enough threshold.

In practice, the PCs and their variances can be found using singular value decomposition (SVD) of the covariance matrix C of the data points (Golub & Van Loan 1996). SVD allows us to factorize it in the form C = VΣV^T. Here, Σ is a diagonal matrix with the eigenvalues (variances), and V contains the eigenvectors (principal components) in the respective columns. Thus, V contains the expansion coefficients of the transformation ${\mathbf{PC}}_{i} = \Sigma _{j=1}^{j=p} \mathbf {V}_{ji} \mathbf {x}_{j}$ (i = 1, ..., p) from property space to PC space.

In WPCA, the covariance matrix is calculated in a weighted schema. Many times we are confronted with noisy or missing data points. As a solution, we can assign a weight w_i > 0 to each ith data point in order to account for the noisy or missing data points. In this context, WPCA involves considering these weights in the calculation of all averages and covariances between the p properties. In general, the properties might have different units; therefore, they first have to be made unitless by standardization of the data points (subtract from each property its (weighted) average and then divide it by its [weighted] standard deviation).

Principal curves (P-curves) and surfaces (P-surfaces) (Hastie 1984; Hastie & Stuetzle 1989; Tibshirani 1992; Gorban et al. 2008) go one step farther than PCA, providing a low-dimensional curved manifold that passes through the middle of the data points. In this paper we consider a one-parameter (called l) principal curve $\mathbf {f}(l)=[f_{1}(l), \ldots,f_{p}(l)] \in \mathbb {R}^{N \times 1}$, where each of the N data points x = [x₁...x_p] is given a unique closest projection f(l_f(x)) onto the curve. As a convention, l_f(x) is chosen to be the arc length from the beginning of the curve to the projection point of x. In this context, the P-curve can be considered by itself the first and only curved principal component, as the dimensionality of the data is reduced from p to one dimension. In practice, the P-curve is composed by N − 1 line segments that connect the projection points.

The principal curve is defined as the average of the data points that project onto it, minimizing the projection distance between x and f(l_f(x)) over all points. This property of self-consistency allows us to follow a series of iterative projection-expectation steps for its construction (Hastie & Stuetzle 1989). In fact, an educated first guess for the P-curve is to make it equal to PC₁. Later on, the jth estimate f^(j)_i(l) of the curve at the jth expectation step is calculated as ${f}^{(j)}_{i}(l) = E(\mathbf {X}_{i}\vert {l}_{\mathbf {f}^{(j-1)}}(\mathbf {X})= l)$. In practice, we compute this expression using a weighted penalized cubic B-spline regression (Silverman 1985; Hastie & Tibshirani 1990; Rupert et al. 2003; Hastie et al. 2009). These splines are calculated on a series of k knots chosen from the data points, while the degrees of freedom (df) of the regression control the degree of smoothing of the P-curve. The jth projection step is performed next, involving the search for the closest perpendicular projection of x onto f^(j)(l), which is composed of the N − 1 line segments. The iterations stop when the cumulative projection distances from the data points to the P-curve do not change significantly with respect to the one in the previous step.

Although P-curves are constructed on the p-dimensional space of properties, we can consider building the P-curve of the data points projected on the first q most important principal components of the WPCA. This minimizes the complexity and computations, especially in the case of p ≫ q, without losing much information. The approximation is of course valid as long as the first q eigenmodes contain as much of the total variance as possible.

As the MGS is a magnitude-limited sample, not all galaxy types are sampled equally in the survey volume. As a consequence, we used WPCA and a weighted principal curve of the galaxy population to get an unbiased result.

In detail, at higher redshifts we sample mostly the brightest galaxies, neglecting the faint ones (Malmquist bias). On the other side, at low redshifts the SDSS spectrograph fails to take the spectra of very bright galaxies (see Section 2).

As a solution, we use the V_max weighting method (Schmidt 1968) to account for this incompleteness. Here, each ith galaxy is assigned a weight w_i = V_S/V_{max, i} ⩾ 1, where V_S is the volume of the survey. Here we note that, given the particular [z₁, z₂] and [m₁, m₂] intervals for the survey, the ith galaxy found at z_i could be observed only within a maximum comoving volume V_{max, i} ⩽ V_S. If the ith galaxy of apparent magnitude m_i, k-correction k_i = k(z_i), and at a luminosity distance D_L(z_i) were to have limiting apparent magnitudes m_{1, 2}, then it should be moved to a limiting luminosity distance D_{L, i}(m_{1, 2}) given by

$$\begin{eqnarray} D_{L,i}(m_{1,2}) &\equiv& D_{L,i}(z_{\rm lim};m_{1,2}) \nonumber \\ &=& D_{L}(z_{i}) \times 10^{(m_{1,2} - k(z_{\rm lim}) - m_{i} + k_{i})/5}. \end{eqnarray} \tag{ 1 }$$

Hence, the maximum volume is defined by the biggest interval of D_L inside which a galaxy can appear in the survey:

$$\begin{eqnarray} V_{{\rm max},i} & = & [V(\min (D_{L}(z_{2}),D_{L,i}(z_{\rm lim};m_{2}))) \nonumber \\ &&- V(\max (D_{L}(z_{1}),D_{L,i}(z_{\rm lim};m_{1})))] \times F_{A}. \end{eqnarray} \tag{ 2 }$$

As Equation (1) defines z_lim in an implicit way, we solve for it iteratively. We calculated the V_max values directly inside the database using an integrated cosmological functions library (Taghizadeh-Popp 2010).

The PCA, P-curve, and calculations related to volume densities in this paper (such as histograms) are all V_max weighted.

As a measure to avoid skewing the PCA, we clipped off visually the outliers in each of the p = 7 galaxy properties in order to dismiss artifacts or incorrect measurements. We also used only galaxies which have all seven properties measured, simplifying the calculations and avoiding using Gappy PCA (Connolly & Szalay 1999). This left us with a total of N = 171, 698 galaxies (99% of the initial ones).

Figure 1 and Table 1 present the results from computing WPCA on the seven galaxy properties. From Table 1, we can note that most of the information (97% of the total variance) is contained in the first four principal components. In Figure 1, each PC_i i = 1, ..., 7 can be viewed as a linear combination of properties, with the expansion coefficients V_ji of the jth property stored in the jth row. Coefficients with stronger color show a higher importance of the property for the given PC. The sign of the coefficient shows correlations/anticorrelations between the properties and the final value of the PC.

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Table 1. WPCA Variances

	PC1	PC2	PC3	PC4	PC5	PC6	PC7
σ2PC	4.493	1.115	0.842	0.363	0.090	0.068	0.030
$\sum \frac{\sigma _{\mathrm{PC}}^2}{p}$	0.642	0.801	0.921	0.973	0.986	0.996	1.000

Notes. Variance for each principal components and its associated cumulative variance. Since the data have been standardized, the sum of the variances is equal to the number of dimensions (p = 7).

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For PC₁, the strength (absolute magnitude) of its expansion coefficients V_j1 in the basis of the galaxy properties is shared mostly evenly between these properties, u − r, g − r, SFR/M_*, and M_*/L_r being the most important. The correlations show that high values of u − r, g − r, M_*/L_r, and R90_r/R50_r, together with low values of μ_{50, r}, SFR/M_* and SFR, will produce a high PC₁ value. We might therefore expect that PC₁ is a good separator between the young, blue population of spirals/irregulars and the old population of red old ellipticals.

For PC₂, the SFR and μ_{50, r} are the most important, having opposite signs. Thus, we expect galaxies with bright surface brightness and high star formation to show high values of PC₂.

For PC₃, the most important property is R90_r/R50_r, with an opposite correlation with respect to the next important properties of mostly equal strength (u − r, g − r, SFR, M_*/L_r, and μ_{50, r}). We can expect that big and bright star-forming spiral galaxies with reddish colors (probably from a red core) should have high PC₃.

For PC₄, all the properties have the same correlations, μ_{50, r}, R90_r/R50_r, and SFR being the most important. Thus, concentrated (and possibly star-forming) galaxies of faint surface brightness have high values of PC₄. As the variance along PC₄ is much smaller than along the previous PCs, this is a rare combination of correlation for these properties to be observed at the same time.

Furthermore, the last three PCs (PC₅, PC₆, and PC₇), which account for less than 2% of the total variance, are less obvious to interpret. They might trace either special cases of galaxy populations or just artifacts and wrong/noisy measurements of the properties.

We decided to construct the principal curve in the four-dimensional space defined by {PC₁, ..., PC₄}, since their combined cumulative variance (0.973) is close to unity (see Table 1). Although the computation for the number of dimensions and data points involved is not too intensive, we think of this as a pedagogical example that can be used for other extreme cases when N ⩾ 10¹⁰ objects with p ⩾ 100 dimensions, for instance. In fact, our election does not change significantly the results compared to using p = 7.

In the expectation step for creating the principal curve, each PC_i is fitted with penalized B-splines of 5.4 degrees of freedom (df), defined at a sequence of k = 211 unique knots chosen at equally spaced quantiles of arc-length values. Principal curves with df ≳ 7 make the curve oscillate excessively, turning back and forth across and along PC₁, whereas with df ≃ 4 it resembles more closely a straight line along the PC₁ direction.

Figure 2 shows the result of fitting the principal curve to {PC₁, ..., PC₄}. The four-dimensional cloud of properties presents two density maxima placed mainly along the PC₁ direction, corresponding to the blue and red galaxy populations. The principal curve closely resembles the letter "W," presenting clearly four different regimes or branches separated by three turning points (T-points).

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We created 20 equal number density galaxy groups (in Mpc⁻³) labeled as {L_i}^{i = 20}_{i = 1} by placing population separators at fixed arc-length values along the P-curve, as shown in Figures 2 and 3. Galaxies are grouped together into the same L_i group when the arc-length values measured at their projection points on the P-curve are placed between two consecutive separators. These separators are positioned in such a way that the (V_max-weighted) number density (in Mpc⁻³) of the galaxies belonging to each of the 20 L_i groups amounts to 1/20th of that from the whole sample of galaxies. This allowed us to study the four principal curve branches in detail. We chose the arc length to increase in the same direction of increasing PC₁, with growing values of arc length as we progress from L₁ to L₂₀. Thus, the P-curve's first branch is comprehended in {L₁, ..., L₆}, the second branch in {L₇, ..., L₁₄}, the third branch in {L₁₅, ..., L₁₈}, and the fourth branch in {L₁₉, L₂₀}. Table 2 shows some statistics of these groups.

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Within each L_i group, we further created five subgroups of galaxies along the arc length naming them {λ_i}^{i = 100}_{i = 1}, also of equal number density in Mpc⁻³ as explained before. We further partitioned these groups similarly, now using several radial separators in the direction perpendicular to the curve, defining 10 concentric cylinder-like separating surfaces. In this way, the groups defined by this finer partitioning have all the same number density (in Mpc⁻³), equal approximately to 1/1000th of the number density of the whole sample. This allowed us to identify and extract localized galaxy populations positioned very close to the spine of the cloud of properties; we study them in Section 6.5.2.

Figure 3 shows the probability density distribution of the arc-length l values as well as the population separators. The curve has a length of l_max = 20.24, and the variance of the arc-length values is σ²_l = 7.79, measured with respect to the center of the curve at 〈l〉 = 7.91. Note that Table 2 shows that the quadratic mean (root mean square) of all the projection distances from the data points to the P-curve takes a value of d_⊥ = 1.31, which is small compared to the length of the curve. The blue and red peaks of maximum density are clearly visible, as well as a small green peak located between the previous two. The first turning point (at L₆) lies closely with the blue maximum (L₇), whereas the red maximum (L₁₇) is a little behind the third T-point (L₁₈), after which we find a hump defining the red sequence of galaxies. We find a green maximum (L₁₆) standing between the second T-point (L₁₄) and the red maximum.

Figure 4 shows the density maps of the scatter of each {PC₁, ..., PC₄} as a function of the arc length. The different shapes that this scatter presents depend evidently on the contortions or twists of the principal curve along the PCs. As the four branches of the curve mostly turn left and right along PC₂, the scatter in PC₂ shows the same "W" shape as the P-curve. On the other hand, the curve increases its length in the PC₁ direction, so the scatter shows a mostly linear relation between PC₁ and arc length. The same analysis applies to the scatter of the next PCs, which is boomerang shaped for PC₃ and mostly constant with respect to arc length for PC₄ (although with little wiggles).

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Figure 5 shows the most representative galaxy morphologies and average spectra for the {L_i}^{i = 20}_{i = 1} groups.

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The most evident feature is the change in color and the slope of the spectra (from blue to red), as well as an overall weakening of emission lines (e.g., Balmer series of hydrogen and forbidden lines, such as O iii, O ii, N ii, etc.) and an increase of metallic absorption lines and bands (Na, Mg, H, K, G) as we reach high arc-length values. In the same way, morphological types include various types of blue galaxies at the beginning and middle of the curve, whereas red ellipticals dominate the end of it. This bimodality is expected and agrees with PC₁ in Figure 1, also appearing in other studies as the change along the first principal component (e.g., Yip et al. 2004a; Coppa et al. 2011). We can, however, identify as well more subtle populations along the arc length, indistinguishable in PC₁ alone. These distinct population are defined on each of the four branches of the principal curve, connected by the three turning points.

With respect to morphologies, we see that the arc length correlates very well with the Hubble galaxy type. We do, however, miss the distinction between barred/non-barred spiral galaxies due to the lack of properties able to separate them. Blue irregulars and blue compact dwarf (BCD) galaxies (Papaderos et al. 2006; Corbin et al. 2006) appear in the first branch of the principal curve. Some of these types of BCDs were identified as the green pea galaxies at higher redshift (Cardamone et al. 2009). These morphologies change then into low surface brightness galaxies (LSBGs) with spiral and irregular shapes, which dominate the first turning point and blue maximum. Bright spirals with strong blue star-forming arms appear in the second branch, which by the second turning point show sizable bulges. A dramatic change happens in the third branch, where reddish big-bulged spirals and lenticulars dominate, forming part of the green and red maxima. A new transition happens at the third turning point, at which the big bright red ellipticals (CDs) and brightest cluster galaxies (BCGs) dominate at the end of the P-curve's fourth branch.

Emission lines, such as the forbidden O ii, O iii, S ii, and Ne ii, as well as the Balmer series of hydrogen (e.g., H_α, H_β, H_γ), are strong in the violently star-forming blue galaxies at the first branch. These lines weaken as we transition into LSBGs, but interestingly H_α and H_β become stronger in the second branch, reaching maximal values in the star-forming spirals at the second turning point. After this, they weaken again to become imperceptible in the bright ellipticals in the fourth branch. N ii follows the same pattern as H_α, but somehow remains still visible in CD galaxies, as seen in many spectral atlases (e.g., Dobos et al. 2012). On the other hand, O iii declines steadily through the arc length, disappearing after the red maximum.

Absorption lines, such as Na, Mg, and the G band, become evident in the star-forming spirals by the end of the second branch (as the bulge increases in size), and appear strong in the ellipticals at the fourth branch. Although the H and K lines of calcium are always visible, the 4000 Å break increases steadily with arc length, turning into a striking feature in bright ellipticals.

Figure 6 shows the evolution of the galaxy properties as a function of arc length in the principal curve, whereas Table 3 contains the average values of the properties at each {L_i}^{i = 20}_{i = 1}.

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Looking at the seven properties on which the WPCA was built, the most important feature is that they present the same shape as the PC_i in which they have the greatest leverage. On the other hand, the properties not present in the WPCA show similar shapes or behaviors, depending on their individual correlations with the initial seven properties. Generally, their behavior (as a function of arc length) is defined by the four branches and three turning points, resembling in most cases a distorted W of the P-curve.

Thus, we can group all the properties with respect to which PC_i they most resemble. For example, Figure 1 shows that log M_*/L_r, log SFR/M_*, u − r, and g − r have the greatest leverage in PC₁ from the first four PCs. This makes these properties resemble the shape of PC₁ in Figure 4, where it is mostly a linear relation with respect to arc length, with a scatter modulated by the turning points. Other properties that correlate with PC₁ are r − i, g − z D_n(4000), some Lick indices, and [O iii]. Note that [O iii] has a strong linear dependence on PC₁ and behaves differently with the other emission lines (such as Balmer series) due to the higher ionization degree. The arc length also correlates linearly with eclass, which is the classification parameter derived in the PCA of galaxy spectra in Yip et al. (2004a), defined as a function of the expansion coefficients in the bases of the first two eigenspectra. In general, these properties can be expressed as a linear combination of each other, as seen in astrophysical use, e.g., M_*/L = a × Color + b (e.g., Baldry et al. 2004).

In the same way, log SFR and μ_{50, r} resemble strongly the "W" shape of PC₂. Some emission lines belonging to this group are N ii and H_α, the latter being a well-known proxy for SFR (Kennicutt 1998). Also, M_r is a good proxy for log M_* (Bell et al. 2003), both of which are related to R90_r and μ_{*, r}, and seemingly correlate with μ_{50, r}. Interestingly, the shape of the average Lick Na D index is similar to the "W" shape of PC₂ (and SFR), but the larger 1σ dispersion in the average makes it not very significant. Lick Na D is expected to represent a strong absorption feature in old stellar populations, as shown in the ellipticals at high arc lengths. We can, however, see that it presents also a relatively high average at the second turning point. This is related to the fact that Na absorption is not only present in stars, but also in the interstellar medium as a consequence of outflows or winds present in high star formation spiral galaxies (Chen et al. 2010).

Note that the boomerang shape of R90_r/R50_r is almost identical to PC₃. Also correlating with the concentration index is fracDeV_r (Stoughton et al. 2002), which determines the mixing in the modeling of the light profiles galaxies between an exponential disk and a de Vaucouleurs r^1/4 law for the elliptical bulge.

6.2.1. The BPT Diagram

Figure 7 shows the emission-line ratios BPT diagram (Baldwin et al. 1981) of the MGS galaxies, where active galactic nucleus (AGN) identification can be done easily. We considered galaxies presenting Hα, Hβ, [N ii], and [O iii] emission lines with well-measured equivalent widths, of fractional errors smaller than 0.33 and velocity dispersion smaller than 500 km s⁻¹. These cuts reduce the L₁ − L₁₅ groups to ∼90% of their size, going down to 35% for the remaining groups at higher arc length. Since none of these emission lines where included in the building of the WPCA, we overplotted the average location of the {L_i}^{i = 20}_{i = 1} groups.

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The average locations of the groups can be seen to be connected by a two-branched track in line-ratio space. The left branch covers the region of star-forming galaxies, whereas the right branch crosses the separator of Kauffmann et al. (2003a) into the region filled by AGNs. Interestingly, the joining point between the two branches happens at L₁₄, which contains the second turning point in the principal curve. This is a striking feature, as it shows that the P-curve is powerful enough to describe galaxy properties beyond the ones included in its construction.

Figure 8 shows the LFs corresponding to the {L_i}^{i = 20}_{i = 1} equal number density groups. They can be directly compared to the evolution of M_r as a function of arc length in Figure 6. The LFs were computed with the V_max method of Schmidt (1968) explained in Section 5.1, where the estimated LF value at each magnitude bin is the sum of the weights of all galaxies in that bin, with w_i = V⁻¹_{max, i}. Table 4 contains the fitting parameters, with the fitting functions as follows:

• 1.  
  Double power law

  $$\begin{equation} \Phi ({L})d{L} = \phi ^{*} \left(\frac{{L}}{{L}_{*}}\right)^{\alpha } \left(1+ \xi \left(\frac{{L}}{{L}_{*}}\right) \right)^{-\frac{1}{\xi }} \frac{d{L}}{{L}_{*}}, \end{equation} \tag{ 3 }$$

  granted that 1 + ξ(L/L_*) ⩾ 0. This double power-law fit (Alcaniz & Lima 2004) collapses when ξ = 0 into the Schechter fit (Schechter 1976) Φ(L)dL = ϕ*(L/L_*)^αexp (− L/L_*)dL/L_*. The ξ parameter can be related to the tail index in extreme value statistics (e.g., Gumbel 1958; Galambos 1978), defining for the brightest luminosities an infinite reaching power-law tail (ξ > 0), exponential tail (ξ = 0), or a cutoff at a finite maximum luminosity L_max = L_*/|ξ| (ξ < 0).
• 2.  
  Lognormal

  $$\begin{equation} \Phi ({L})d{L} = \frac{1}{ {L} \sigma _{L} \sqrt{2\pi } } \exp \left(-\frac{ (\ln {L} - \mu _{L})^2}{ 2\sigma _{L}^2 }\right)d{L}. \end{equation} \tag{ 4 }$$

  Note that a lognormal distribution in luminosity space is equivalent as a normal-Gaussian distribution in magnitude space.

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The LF of the whole sample is well fitted by the Schechter fit, except for the bumps at the high-luminosity tail and at the low-luminosity end starting at M_r ∼ −18.8, also observed by Blanton et al. (2003, 2005). We fitted it right before the bump.

The changes in the behavior of the LF along the P-curve are determined also by the three turning points. In summary, M* becomes fainter in the first branch, and then brighter afterward. On the other hand, the slope of the faint tail behaves similarly to PC₂. In fact, it is steep in the first branch, becomes shallower in the second branch, then again steep in the third branch. The fourth branch contains an extremely shallow slope, where the LFs resemble mostly a lognormal distribution. We can see that we recover the LF shapes shown in Binggeli et al. (1988), which are based on morphological types and vary between Schechter fits (as in L₁ to L₁₅) and bell-shaped LFs fitted by lognormal fits (such as the L₁₉ and L₂₀).

As we progress along the first branch of the principal curve (L₁ to L₆), the BCDs become less luminous on average, having M* dimmer in about ΔM* ≃ 2, with a mostly constant steep faint end (α ∼ −1.55). Note that these galaxies, and mostly the low surface brightness spirals at the first turning point, create the bump seen in the overall LF. In fact, L₆ contains the faintest galaxies in our sample (Figure 6). At this point, α = −1.69 gives the steepest power-law slope at the faint luminosity tail. Note that this slope is expected to reach α ≃ −1.5, as noted in Blanton et al. (2005).

In the second branch (L₇ to L₁₃), the star-forming spirals present a faint end that flattens dramatically and starts to drop continuously, with an increase of Δα ∼ 1.6 from L₇. At the same time, they start becoming much more luminous, with M* brightening in ΔM* ∼ −2.2 (from L₆).

In the third branch (L₁₄ to L₁₇), for the red spirals and lenticulars M* continues becoming brighter (ΔM* ∼ −2.5), but at the same time the faint-end slope starts becoming steep again (Δα ∼ −1.6), back to the values of α ∼ −1.6 found at the end of the first branch. Note that L₁₆ (green maximum) and L₁₇ (red maximum) show long power-law faint ends with a sharp cutoff at the bright end. They are better fitted by a double power-law fit, and since ξ < 0 they present bright-end finite cuts at M_r ∼ −23.0 and −23.3, respectively.

The third turning point (L₁₇) is a unique case. The LF presents three power-law-like sections, the faintest one being flat. We attempted to fit it with a Schechter profile.

In the last two groups (L₁₉ and L₂₀), the faint-end tail has dropped enormously. We attempted a lognormal fit for the luminosities, since the LFs look more bell shaped, especially the ones belonging to the {λ_i}^{i = 5}_{i = 1} groups that track the spine.

In this section, we investigate the second moment of the galaxy distribution as a function of the arc length, the spatial distribution, quantified by the clustering. We explore here not only the dependence of the galaxy clustering as a function of L, but also the relative distribution of galaxies as a function of L, which can be quantified by the cross-correlation function.

Following common practice, we compute first the redshift space correlation function as a function of the distances parallel (π) and perpendicular (r_p) to the line of sight. We use a generalized version (Szapudi & Szalay 1998) of the Landy & Szalay (1993) estimator

$$\begin{equation} \xi (r_p, \pi) = \frac{D_aD_b - D_aR_b - D_bR_a + R_aR_b}{R_aR_b}, \end{equation} \tag{ 5 }$$

where the subscripts a and b refer to the two samples we are considering when measuring the cross-correlation function. We use the same methods as Heinis et al. (2009) to compute the correlation functions. In brief, for each sample we generate random catalogs following the SDSS footprint defined by its sectors. We use 50 times more random objects than galaxies. We reproduce the selection function by randomly drawing redshifts from the current sample. We correct from fiber collision using the method described in Heinis et al. (2009). Note that the fiber collision correction applies only to the D_aD_b term in Equation (5).

As ξ(r_p, π) is sensitive to redshift distortions, we consider the projected spatial correlation function afterward, which is free from such effects:

$$\begin{equation} w_p(r_p) = 2\int _0^{\pi _{{\rm max}}}\xi (r_p, \pi)d\pi, \end{equation} \tag{ 6 }$$

where we use π_max = 25 Mpc for convergence purposes.

We compute error bars on w_p(r_p) from jackknife resampling. We build jackknife samples using the SDSS stripes, which are defined to be 25 wide great circles on the sky, following the survey latitude. In practice, we consider 23 jackknife samples built from the stripes.

We use a volume-limited sample extracted from our main sample (see Section 2). In order to maximize the signal ratio of the clustering measurements, we do not use 20 samples in L, but 8 of them built the following way: We collided L₁ to L₆ in two groups of a similar number of galaxies (L_w1 and L_w2), L₇ to L₁₀ in one group (L_w3), and the remaining L samples two by two (L_w4 = {L₁₁, L₁₂}, L_w5 = {L₁₃, L₁₄}, L_w6 = {L₁₅, L₁₆}, L_w7 = {L₁₇, L₁₈}, and L_w8 = {L₁₉, L₂₀}).

Figure 9 shows the results for the autocorrelation functions of these samples in the diagonal plots. As a reference, we show as a solid line in all diagonal plots the autocorrelation function of the sample with the highest arc length (L_w8). The results from the autocorrelation function show that the amplitude of the correlation function at large scales (r_p ∼ 10 Mpc) increases with the arc length. This implies that the host halo mass also increases with l. This result is expected as l does correlate with u − r and g − r colors, for instance. It is indeed well known that the amplitude of the correlation function increases for redder objects in the local universe (e.g., Zehavi et al. 2005, 2011). There are also some interesting features in the small scales (r_p < 0.1 Mpc) clustering. Indeed, most samples show clustering power at all scales, except our groups 2 and 3 in particular, where there is an indication of a lack of pairs at r_p < 0.1 Mpc, suggesting a population mainly composed of central galaxies.

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In the off-diagonal plots, we show the cross-correlation functions between these samples. It is beyond the scope of this paper to fully interpret all these measurements with Halo Occupation Distribution models (e.g., Cooray & Sheth 2002). We will use simple arguments to highlight the information contained in the cross-correlation function.

We represent as a solid line in all off-diagonal plots the expected cross-correlation function given by

$$\begin{equation} w_{X_{ab}}(r_p) = \sqrt{w_a(r_p) w_b(r_p)}, \end{equation} \tag{ 7 }$$

where w_a(r_p) and w_b(r_p) are the autocorrelation functions of samples a and b. Equation (7) gives the cross-correlation function that is expected in the case where galaxies from the two samples are well mixed in the dark matter halos hosting them (see, e.g., Zehavi et al. 2005). This is of interest at small scales, where the cross-correlation function contains information about close pairs of galaxies that lie within the same dark matter halos. Our results show an interesting trend in this context. Indeed, the cross-correlation function of galaxies with close arc lengths is similar to the expected cross-correlation function from Equation (7). On the other hand, the cross correlation of galaxies more distant in terms of arc length diverges from the expected correlation function. In particular, the measures are overestimated by Equation (7) at scales r_p < 0.1 Mpc. This means that there are less close galaxy pairs in the measures than what is expected in the case of a perfect mix between galaxies of different arc length. Note that there is still a clustering signal at these scales, which means that there are some galaxies belonging to distant L groups in the same dark matter halos. However, our results show that there are fewer pairs than what is expected if galaxies are properly mixed. This suggests that the probability that two galaxies reside in the same dark matter halo decreases as a function of their distance in arc length.

The analysis of galaxy properties with WPCA and P-curve methods allowed us to find and pinpoint some relevant groups that stand aside from the main trends of the whole galaxy sample. In particular, here we pay attention to the small blob of galaxies that clusters apart from the main cloud in Figure 2. Another interesting task is to isolate a pure population of red galaxies in the L₂₀ group whose LF is lognormal, as it appears in Figure 8.

6.5.1. Blob of Small Red Disk Galaxies/Lenticulars of High M_*/L_r and μ_{*, 50}

In Figure 2 we found a small blob of galaxies clustered apart from the main cloud in the PC₃ vs. PC₄ panel. We separated them using the following separating line (built by eye): PC₄ ⩽ −1.3 + 0.55(PC₃ − 2.0). We further constrained these galaxies by choosing an appropriate interval in arc length [l_min, l_max] = [11.6, 12.6], which includes most of the L₁₈ group and part of the L₁₉ group. In fact, the blob appears also in Figure 6 bracketed within this arc-length range in the R90_r/R50_r, μ_{*, r}, and log R90_r panels. Precisely, R90_r/R50_r is the property that dominates in PC₃ and PC₄ with opposite signs, as shown in Figure 1. After the selection cuts, we are left with 136 blob galaxies, the imaging and average spectrum of which are shown in Figure 10.

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According to Table 3, the blob is composed mostly of small red disk galaxies, with minimal star formation and void of gas. In fact, u − r is at least 1σ redder than the average in the L₁₈ group. Furthermore, they are modeled with an important component of an exponential disk (fracDeV_r = 0.00^+0.48_{− 0.00}). The concentration index R90_r/R50_r ≃ 1.95 is low and far from the often used R90_r/R50_r = 2.6 separator between ellipticals and spiral galaxies (Strateva et al. 2001). The average size of R90_r ≃ 1.6 kpc is small compared to the one of L₁₈ (R90_r ≃ 4.0 kpc), lying beyond the 1σ significance level. Interestingly, the small size makes M_*/L_r ≃ 4.07 M_☉ L⁻¹_{☉, r}, μ_{50, r} ≃ 18.83 mag arcsec², and μ_{*, r} = 10^10.06 M_☉ kpc⁻² to be also well above 1σ of their average at L₁₈. With respect to the uncertainty of these values, errors in the estimate of M_* can arise when fitting the spectra, especially when there is a strong component of dust, which these galaxies appear to have. The statistical error in M_* is around 15%. When comparing to a catalog of groups from Tago et al. (2010), we found that at least half of these galaxies are in groups of more than 10 members. We speculate that they were depleted of gas by ram pressure stripping or another mechanism that did not perturb the structure of the disk.

6.5.2. Close-to-Spine Red Galaxies of Lognormal Luminosity Function

The principal curve provides an objective way to order galaxies along its arc length. The success in dimensionality reduction and classification power of the P-curve is related to how much the projections of the galaxies are spread along arc length. In fact, a large variance along arc length gives more room for building separators and discerning different galaxy types. In our case, the arc-length values along the P-curve have a variance of σ²_l = 7.79 (see Table 2), which is bigger than cumulative variance Σ^{i = 4}_{i = 1}σ²_PCi = 6.81 of the principal components from which it was built. This means that the curvature of the P-curve helps to discern information not included in the intrinsic linearity of WPCA. The length of the P-curve cannot be made arbitrarily long or short, due to an evident bias-variance trade-off. The shortest curves (with no curvature) are identical to PC₁, with an average spread across it equal to Σ^{i = p}_{i = 2}σ²_PCi, and a high bias due to the straight P-curve missing the important bends in the structure of the cloud of points. On the other hand, the longest curves possible would be the ones connecting all the data points, which produce a null bias but high variance, as the curve will fit the noise in the structure of the cloud. In fact, as we experimented with values of df ≃ 7, the curve attempts to cover all the space spanned by the cloud, twisting and coiling itself in ways that describe additional detailed features of galaxies, while we are now interested in the global trends. Our election of df = 5.4 is an intermediate case, where the root mean square of the projection distances on the curve is d_⊥ = 1.31, smaller than (Σ^{i = 4}_{i = 2}σ²_PCi)^1/2 = 1.52. The ratio 1σ²_l/d²_⊥ = 0.75 gives us a notion of the amount of information that the P-curve is able to discern. The physical origin in the scatter of the remaining 25% is still to be explored and depends locally on the direction in the eigenspace along which d_⊥ is measured.

In our analysis, the P-curve has been able to recover the well-known bimodality between the blue and red populations. Since galaxy properties are highly correlated, only a few properties should be enough to explain the variations in the zoo of galaxies, namely u − r (from PC₁), SFR (from PC₂), and less importantly R90_r/R50_r (from PC₃). In fact, the variations recovered by the P-curve and its "W" shape depend strongly on SFR. The color u − r, almost linearly correlated with arc length, tracks specifically the 4000 Å break in the continuum, which gives a measure of stellar age and separates the early to late galaxy types. However, this is not enough, as R90_r/R50_r tells about morphology, whereas the SFR tracks the amount of material produced in recent star bursts, which shows as the strength of emission lines such as the Balmer series and eventually correlates with the galaxy size, mass, and luminosity. For example, within the blue population (bluer than the green maximum) we can find the low star formation and surface brightness spiral galaxies separating the bluest star-forming spheroidals/irregulars and the redder star-forming spirals with a prominent bulge. The importance of the color and emission lines from star formation in explaining variations in galaxy populations has also appeared in previous PCA studies (e.g., Yip et al. 2004a; Coppa et al. 2011; Győry et al. 2011).

7.3.1. AGN Activity and Star Formation Quenching

7.3.2. Hierarchical Model of Galaxy Formation