CORRELATIONS BETWEEN NEBULAR EMISSION AND THE CONTINUUM SPECTRAL SHAPE IN SDSS GALAXIES

We studied the emission-line data of galaxies selected from the SDSS DR6 database (Adelman-McCarthy et al. 2008). The SDSS spectroscopic catalog contains galaxies of all types brighter than 17.77 mag in the r band (Strauss et al. 2002) and a roughly volume-limited sample of luminous red galaxies with redshifts ranging up to z ≈ 0.45 (Eisenstein et al. 2001). The spectra are taken using 3 arcsec diameter fibers; thus, in the lowest redshift galaxies the spectroscopy only samples the central region. The data include redshifts, spectral type, measured characteristics of spectral lines, etc. For a technical overview of SDSS, see York et al. (2000).

The spectral line characteristics published in the spectroscopic catalog are evaluated by an automated pipeline. The pipeline is able to identify 48 spectral lines. In order to determine the fluxes, the lines are fitted by Gaussian profiles. The database lists the fit parameters including position, height, width, EW, and spectral continuum flux to each line. We use these values for our analysis.

The SDSS spectroscopic catalog also provides quantities that carry useful information on the spectral shape of galaxies in a compact form. Connolly et al. (1995) showed that the spectra of galaxies form a low-dimensional manifold. Using only a few parameters (say three), the spectra can be described with very good precision (99% accuracy) and an objective spectral classification of galaxies is also possible. The spectroscopic pipeline applies this dimensional reduction technique as detailed in Connolly & Szalay (1999) and Yip et al. (2004) to obtain the parameters, ecoeff_i, i = 0...4 for each galaxy. They are the weights of the first five PCA eigenspectra of the SDSS galaxies. The derived quantity ${\tt eclass}={\it a}\rm{tan}(-{\tt ecoeff}_1/{\tt ecoeff}_0)$ characterizes the shape of the spectral continuum very well. Its increasing value corresponds to the rising blue end of the spectrum and decreasing 4000 Å break, i.e., small/large eclass values indicate early/late types. (For illustration, see the left panel of Figure 12, which shows the correlation of eclass with the color u − r.)

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The sample for the present study was selected as follows. We selected objects that were classified by the SDSS photometric pipeline as galaxies. We restricted our investigations to the strongest 11 emission lines listed in Table 1. In order to include only reliable data, we required the median S/N in the g* band to be more than 10, sn₀>10, in the SpecObj data table of the SkyServer⁶ catalog science archive. For all galaxies included in our data set, we required that all lines listed in Table 1 are measured (125,832 objects). In order to exclude measurements with extremely large errors, we set an upper cut in the errors of EWs at 5 Å; this affected 5% of the parent sample. Our typical errors are between 0.2 and 0.4 Å, depending on the line. The cleanness of data manifests, e.g., in correct ratios of the known doublet lines. In some cases, the line profile fit does not resolve the broad Hα λ6565 and its two close neighbors [N ii] λ6550 and [N ii] λ6585 correctly. In order to remove this systematic error, we excluded objects with blended Hα λ6565. An undesired effect of this sampling is the exclusion of galaxies that have some broad component around Hα, e.g., Seyfert I galaxies. Our criterion to have all 11 lines reliably measured raises a concern about completeness in very low and very high metallicity ranges. In the former group, [N ii] lines might be unmeasurable, whereas in the latter group [O iii] might be very weak or absent. We address this question later in Section 3.4. We also required sigma < 2 for the [O ii] λλ3727,3730, in order to exclude cases where the fit did not resolve the two lines but captured the blended doublet instead. We found these particular constraints the most suitable for obtaining both a clean and representative data set.

Our final sample contains 40,312 galaxies. Since our selection criteria did not include any constraints on the sign of the emission-line fluxes, our sample includes both emission-line galaxies and objects with mainly absorption features. Some characteristics of the sample are shown in Figure 1, namely, distribution of the data in redshift, absolute magnitude, color, and eclass. Redshift ranges up to z = 0.3, with an average value of z = 0.07. In both color u − r and continuum-type eclass there are two groups of objects visible: red, early types at higher u − r and low eclass and blue, late types at smaller u − r and larger eclass. As described by Strateva et al. (2001) based on a study of SDSS galaxies, the two underlying groups of the bimodal distribution in color space can be separated by a single cut at u − r = 2.22. According to this criterion, 41% of our galaxies are red and 59% are blue. The corresponding distribution of spectral types is manifested in eclass too. For more details on the distributions of galaxy spectral types in SDSS, see Yip et al. (2004). We can distinguish between SF galaxies and active galactic nuclei (AGNs) using emission-line diagnostics based on the line ratios $N2=\log (\hbox{[N\,{\sc ii}]} \ \lambda 6585/{\rm H}\alpha)$ and $O3=\log (\hbox{[O\,{\sc iii}]} \ \lambda 5008/{\rm H}\beta)$, which was introduced by Baldwin et al. (1981; BPT). (For the distribution of our data on the BPT diagram see Figure 13; however, the explanation of the different symbols can be found in Section 3.4.) In order to identify the two groups in our sample we adopt the AGN/SF separator of Kauffmann et al. (2003):

$$\begin{equation} O3=0.61/(N2-0.05)+1.3. \end{equation} \tag{ 1 }$$

Objects with O3 values above this line are classified as AGNs; the rest are classified as SFs. Based on this cut, nearly 50% of the sample are SF galaxies, 18% have an AGN-like emission pattern, and over 32% of the objects cannot be classified as either because at least one of the four lines of Equation (1), in most cases Hβ, has a non-positive EW. These are objects with overall weak emission.

Figure 2 shows the distribution of the EW values of our sample plotted against the type parameter eclass (the smaller the redder). The data show the well-known tendency of emission lines becoming stronger from early to late types; see Figure 3 for a few examples. Star formation enhances both the blue color of late-type galaxies and the strength of the emission lines. (This also makes it possible to use certain lines as measures of star formation; see, e.g., Kennicutt 1998; Kewley et al. 2004.) This trend is obvious for all 11 analyzed lines. Early-type galaxies located at negative eclass values show mainly absorption features and almost no emission independent of eclass. This group can be distinguished from late-type galaxies situated mostly at positive eclass values whose emission tends to rise with eclass. However, this tendency is not the same for the different lines, which originate in different physics of formation. We also see galaxies that have seemingly strong absorption at Hγ λ4342 but not at Hα λ6565, which is puzzling as typically one expects the absorption EW at Hα λ6565 to be about 60% of that at Hγ λ4342. Since our EWs are the sum of emission and absorption, this probably means that the Hα λ6565 absorption is filled up by the emission of these galaxies. This effect is enhanced by measurement uncertainties as well. Seemingly strong absorption values at [O iii] and [S ii] are also present due to measurement errors.

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We try to quantify the common trends and differences in the variation of the nebular emission pattern using PCA of the EW data. They characterize the emission strength physically as they do not depend on distance and the effect of the galactic reddening is canceled by normalization. At the same time, EWs are affected by the intrinsic reddening caused by an inhomogeneous distribution of star, gas, and dust components (Calzetti et al. 1994; Charlot & Fall 2000). Because we are interested in data as they are observed in photometric measurements, we choose not to correct for the intrinsic reddening.

PCA is a linear transformation of data vectors to the eigensystem of their correlation matrix. It results in an uncorrelated representation of the data, and makes it possible to identify the most relevant directions of variation by ranking them according to their information content.

PCA of our data was carried out as follows. We represented the EWs of each galaxy with an M = 11 dimensional vector $\mathbf y$, with the average $\overline{\mathbf y}$ subtracted. Diagonalizing the covariance matrix, we obtained the orthonormal set of M eigenvectors or principal components (PCs) $\mathbf e^k$. We ordered them by their eigenvalues λ^k (normalized to unit trace) since they express the relative information content of each eigenvector. For each galaxy, the expansion coefficients of the vector $\mathbf y$ form the new M-dimensional vector $\mathbf c$:

$$\begin{equation} \mathbf {y} = \sum _{k=1}^{M} c_k \, \mathbf {e}^k. \end{equation} \tag{ 2 }$$

The transformation of vector $\mathbf y$ to $\mathbf c$ corresponds to a simple rotation of the data vectors to the basis where their correlation matrix is diagonal. Inverting the transformation, we obtain the original vectors again. However, if we truncate the expansion coefficients at some m < M, the data will not be exactly restored. The effect of omitting the kth PC is the reduction of the variance of the truncated EW estimator

$$\begin{equation} \mathbf {y} ^{(m)} = \sum _{k=1}^{m} c_k \, \mathbf {e}^k \end{equation} \tag{ 3 }$$

by λ^k with respect to the original distribution of the data. Hence, the eigenvalues are actually a measure of the importance of each eigenvector to reconstructing the real distribution of the data.

Figure 4 shows the results of PCA of the EW vectors. The average EWs for each of the 11 lines (vector $\overline{\mathbf y}$) are plotted in the top panel. Below these are the first five eigenvectors $\mathbf e^k$ ordered by their eigenvalues. Their information content is 89.1%, 7.8%, 1.8%, 0.7%, and 0.2% of the total variance, respectively. The numerical results are summarized in Table 2. The meaning of each PC can be easily interpreted by comparing the weight of the ith line in the kth eigenvector e^k_i with the mean EW of the ith line $\overline{y}_i$.

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The first eigenvector e¹ is very similar to the average vector $\overline{\mathbf y}$. It means the most important variation in emission-line EWs is simply a constant multiplicative factor in the amplitude that varies from galaxy to galaxy. A larger c₁ indicates stronger nebular emission. The PC is dominated by the strongest line Hα λ6565.

The second eigenvector e² represents mostly the [O iii] λλ4960,5008 and nitrogen lines. The coefficients of [O iii] λλ4960,5008 and [N ii] λλ6550,6585 have opposite signs. The same holds for [O ii] λλ3727,3730 and [N ii] λλ6550,6585 in the eigenvector e³. This enables the nitrogen emission lines to vary independently of oxygen in the reconstructed emission-line spectrum. The EW data (Figure 2) show that [N ii] emission grows continuously, slowly from early to late types, while the oxygen lines have a stronger type dependence, becoming steep, especially for the extremely blue galaxies. Due to the higher ionization degree, the behavior of the [O iii] doublet as a function of type is different from the other lines: while showing no significant emission at low and moderate eclass values, there is a steep rise at eclass >0.5. Nitrogen in these components is relatively strong compared to the Balmer lines; thus, both e₂ and e₃ influence the [N ii]/Hα λ6565 ratio.

These eigenvectors do not significantly change the ratios of lines in the same doublet since their weights normalized by their average values are nearly equal. The constant ratios have physical reasons and it is a strong effect which persists in these components.

In e⁴, the two lines of the [O ii] λλ3727,3730 doublet are represented with opposite signs. The effect of this component is to change their ratio. This eigenvector reflects the measurement errors of the [O ii] λλ3727,3730, which is difficult to deblend with the resolution of SDSS spectroscopy.

Vector e⁵ contains mostly [N ii] λλ6550,6585, with some weak Balmer and oxygen lines. Thus, we expect that it might influence the precise reconstruction of [N ii] lines.

Doublet lines in some higher PCs often appear with an opposite sign, which is mainly the effect of errors. We do not detail the further components as their variance is less than 0.2% for each; they are dominated by noise.

The distribution of the data in the subspace of the first three PCs is shown in Figure 5. The range occupied by galaxies is approximately a two-dimensional curved manifold. Most of the objects form a triangular region that is closely parallel to the axes e¹ and e² and closely perpendicular to e³, having small c₃ coordinates. The distribution also has a "head" at the lower end in c₁ and a "tail" having large c₁ values. The galaxies in the "tail" also have significant contribution from the third PC but are still located on the curved surface which is a continuation of the main locus described above. The fact that PCA overestimates the dimensionality of the data is a known limitation of the method. It is because PCA is a linear transformation while the physics of emission lines generates a nonlinear structure (see Figure 2).

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As shown in Figure 6, the points in the triangular main locus embedded in the subspace of the first three eigenvectors can be generated by two vectors originated in a point C (Chan et al. 2003). Their coordinates in (c₁, c₂, c₃) space are: C = (−22, 5, 0), u = (102, − 45, 15), and u = (102, 15, − 15). The origin and vectors correspond to a certain emission pattern recovered from three PCs according to Equation (3). As the figure shows, the origin has almost no emission. Thus, since the region is situated approximately in the (e¹,e²) plane, the EW values in these two vectors alone can give us some idea of physical interpretation of the first two PCs.

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As indicated in Section 3.3, the overall strength of the emission is manifested in the first coefficient c₁. We define the relative emission-line flux fraction μ to be the total emission-line flux normalized by the continuum flux within the range 3728–6733 Å. If we plot this quantity for each galaxy in the plane (e¹,e²), we can see the monotonic growth of μ with c₁ (Figure 7). It is apparent that the gradient of μ is almost parallel to the axis e¹. The inset plot shows a linear relation between μ and c₁; the "head" of the distribution has an emission flux fraction of less than 1%, while the largest c₁ galaxies have μ ≈ 0.5, i.e., equal flux contribution from continuum and emission lines in the analyzed wavelength interval.

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Another striking effect in Figure 6 is that while oxygen lines are relatively strong, the nitrogen emission is weak in vector $\mathbf v$; v − u, the difference, shows negative [N ii] lines. This indicates a difference in metallicity. We estimated the ratio of oxygen and hydrogen abundances from the ratio of [N ii] λ6585 and Hα fluxes using the empirical formula of Pettini & Pagel (2004):

$$\begin{equation} 12 + \log \,(\rm{O/H}) = 8.9 + 0.57 \, \log \left(\hbox{[N\,{\sc ii}]} \ \lambda 6585/H\alpha \right). \end{equation} \tag{ 4 }$$

As this estimator is calibrated for H ii galaxies, we excluded objects dominated by non-thermal emission from this analysis by requiring $\log (\hbox{[N\,{\sc ii}]} \ \lambda 6585/{\rm H}\alpha) < -0.3$. Figure 8 shows that the vector v − u points approximately in the direction of the negative metallicity gradient: at a fixed contribution from vector $\mathbf u$ the objects tend to have smaller metallicities if the mixing ratio of $\mathbf v$ is larger, i.e., in the upper part of the diagram. The "head" has large metallicity values which indicate old galaxies.

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In Figure 6, data are plotted only up to c₁ = 300; however, the "tail" of the distribution reaches nearly c₁ = 700. The vector w together with the previous two vectors generates the long tail. Its endpoint E[Å] = (500,250,80) is out of the range of the figure, so only a fifth of the vector, w/5 is plotted. Point E represents galaxies with extremely strong nebular emission. Since vector w carries very strong EW values compared to vector v (or u), the spectrum of the point E, as well as the points near the large c₁ end of the distribution, is very similar to that of vector w. Very strong Balmer lines and [O iii] λλ4960,5008, weak [O ii] λλ3727,3730, and nitrogen deficiency can be observed. Even though our data are not corrected for reddening, the large [O iii]/[O ii] ratio is a genuine strong effect. It implies a large ionization parameter of the emitting gas, which increases with c₁ at large c₁ values. In summary, the galaxies of the "tail" have extremely strong emission, very low metallicities, and high ionization parameters, which indicate that they are young bursting objects.

Figure 9 shows the color u − r in the subspace of the first and the second PC. A red "head" and a blue "tail" can be seen, and the color becomes continuously bluer toward large c₁ values. This indicates a close relation of the distribution in the emission-line PC space to the bimodality seen in color. The histograms at the bottom of the diagram indicate the distribution of red and blue galaxies defined by the u − r = 2.22 cut of Strateva et al. (2001). The transition between the two distributions is around c₁ ≈ −15 where we have an equal number of red and blue galaxies in our sample. By a cut at c₁ = −15, we can select 93%/86% of blue/red galaxies, with around 10%/11% contamination from the other group, respectively.

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The color–c₁, c₂ correlation indicates that c₁ must be strongly correlated to the continuum shape of the spectral energy distribution (SED). Figure 10 shows the relation between eclass and the first PCs. The c₁–eclass relation is very similar to the relation of c₁ and −(u − r). This is not surprising as they both measure the same effect: the difference between the intensity of the blue and the red ends of the spectrum. Their relation is illustrated in Figure 11 and resembles two linear relations, one for the early-type objects and the other for the late-type objects. The u − r = 2.22 cut in color corresponds to a cut at eclass ≈−0.05. As shown by the histogram in Figure 11 (bottom), it also roughly corresponds to the inflection point of the eclass distribution. The color distribution is also shown projected to the right margin of the diagram. In fact, the separator lines lie somewhat blueward from the inflection points in both eclass and color, which might be the effect of undersampling of early types by our selection. If we consider eclass =−0.05 as the separator of early and late spectral types, we can see that the separation is slightly clearer than in the case of colors. The cut selects 93%/88% of late/early spectral types with a fraction of 8%/11% of early/late-type objects misclassified by the cut. The correlation of eclass and the relative emission parameter μ is shown in the right panel of Figure 11. We will discuss the issue of the emission-line PCs and spectral type in more detail later in Section 4.

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The absolute magnitude in the r band is shown in Figure 12. Apart from the large scatter, the objects are generally fainter at larger c₁. At the largest c₁ values, only low-luminosity objects are present, with typical M_r ≈ −17. This is in concordance with the earlier studies of SDSS galaxies by Tremonti et al. (2004) who found that the most metal-deficient galaxies are faint.

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Figure 13 shows the connection of the first two PCs to the AGN/SF diagnostic diagram. The line ratios N2 versus O3 are plotted in the left panel, together with the AGN/SF separator line of Equation (1). The points are colored by the first PC, using a cut at c₁ = −5, which appears to be the most reliable c-cut for separating the AGN from the SF. Red symbols denote c₁ < −5; blue symbols indicate c₁> − 5. This separator lies blueward of the color or spectral-type separator, as a significant fraction of AGNs has a bluer/later spectral type and has more contribution from emission lines than do the average red galaxies. Beside the mixing at the lower edge of the BPT diagram, the two c₁ regions roughly agree with the AGN/SF separation: the c₁ cut selects 82%/84% of SFs/AGNs with 6.5%/37% contamination from the other group, respectively. We note that the c₁ = −5 cut selects 92%/58% of SF/AGN as defined by Stasinska et al. (2006), with 34%/10% contamination from the other group, respectively. The cut of Stasinska et al. (2006) classifies more objects as AGNs and is therefore more consistent with a higher cut, about c₁ ≈ 0. The separation of the types is not well defined partly because of the mixing of SF and AGN activity in some low emission galaxies. In the right panel of Figure 13, AGN (red) and SF galaxies (blue), selected by Equation (1) are plotted on the c₁:c₂ diagram. The vertical line is at c₁ = −5. The plot shows the same subset of objects as in the left panel, which means all galaxies where Equation (1) is not applicable are excluded. They are low emission objects having non-positive Hα, Hβ, [N ii] λ6585, or [O iii] λ5008. Nearly all of them (99.9%) are at c₁ < −5. Note that low nebular emission galaxies are underrepresented due to the selection criteria. If present, the missing galaxies would populate the "head" too. We also note that AGNs with broad Hα emission lines are excluded from our sample. With our present sample, the "head" (at c₁ < −5) consists of low emission objects, 31% of which could be classified as AGNs. The plot confirms that the "tail" consists of SF galaxies. The classification using c₁ and c₂ is also reminiscent of the (W(Hα), N ii/Hα) diagram proposed by Cid Fernandes et al. (2010) which distinguishes among SFs, AGNs, and LINER-like galaxies, since c₁ is closely connected to the Hα EW. However, because of our selection criteria on equivalent width, we have very few LINER-like galaxies in our sample; these are situated at the lowest c₁ values.

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Now we return to the question of missing low- and high-metallicity objects, rejected because of the EW criterion. According to a check carried out in both a low (7.5–7.6) and a high (8.6–8.8) metallicity bin, indeed, both ranges are underrepresented by about one-third in our selection, if compared to the rejected group. However, there is still a sufficient number of both low- and high-metallicity galaxies in the selected sample for correlation analyses. Although for the relative occurrence of the emission patterns our sample might not be conclusive, it is still suitable for studying the relations between emission lines and other observables.

We can conclude that PCA isolates two groups of objects: red, early-spectral-type, low-emission, high-metallicity, bright galaxies, with a significant fraction of AGNs in the "head" and the rest of the objects which are blue, late-spectral-type, high-emission, lower metallicity, fainter, SF galaxies. In the main locus, there is a gradient of all quantities listed above. All these characteristics get continuously more prominent and reach extreme values toward the end of the "tail."

The low effective dimensionality of the emission pattern found by PCA is not surprising. As already discussed in the Introduction, the reason is the dominance of only a couple of mechanisms. Whether the dominant source of emission is star formation or AGNs has the largest impact on the emission-line flux; the second-most important influencing factor is metallicity. These effectively generate a two-dimensional locus of galaxies in the emission-line space.

We studied the effect of truncation of the PC basis on the restored EWs. We examined the convergence of the truncated EW estimator (Equation (3)) as a function of the number of eigenvectors used for the reconstruction. The error of the estimation can be characterized using the residuals added in quadrature over all lines:

$$\begin{equation} (\Delta y^{(m)})^2 = \sum _{i=1}^{11}\big(y^{(m)} _i - y _i\big)^2. \end{equation} \tag{ 5 }$$

Figure 14 shows Δy^(m) averaged over the whole sample as well as for three c₁ bins. Remember that larger c₁ values mean stronger emission and later spectral type. For the earliest bin, c₁ < 0, the contribution of the emission lines is so small that estimating by average (m = 0 case) produces larger error than ignoring the emission lines, i.e., setting EW = 0 for all lines. We demonstrate the error of the estimation by zero EWs for eclass <−0.05 galaxies by an arrow at the left margin of the diagram. For early spectral types, ignoring the emission would mean a 10 Å error if summed over all lines.

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If we drop all eigenvectors with eigenvalues less than 1%, we will have the first three PCs. Their total percent variance for the case m = 3 is 98.6%. Using the first three components, we can reconstruct the total emission with 3 Å average precision. The error is type-dependent; its absolute value increases with c₁. For the strongest emission bin, the average Δy⁽³⁾ is 8 Å. However, unlike the absolute error, the relative error defined as

$$\begin{equation} \delta y^{(m)} = \Delta y^{(m)} / | y | \end{equation} \tag{ 6 }$$

is smaller for stronger emission objects. The average relative error is 25% for the whole sample, dominated by the error of objects having small EW values. For the extremely strong emission bin it is only 5%.

We show the errors of the strongest, most important lines individually. In Figure 15, we showed the convergence of the residuals of the individual EWs of [O ii] λ3730, [O iii] λ5008, [N ii] λ6585, and Hα λ6565:

$$\begin{equation} \Delta y _i^{(m)} = \big|y^{(m)} _i - y _i\big|, \end{equation} \tag{ 7 }$$

averaged over the sample. Similar to the overall emission represented by the summed residuals, all the individual lines are reconstructed with an average error not larger than 2 Å using three PCs (or not larger than 4 Å using two PCs). For the early-type objects at eclass <−0.05, we repeated the estimation by zero emission-line flux similar to Figure 14. If we set all EWs to zero, the errors of the individual lines are still below 5 Å for this group, as shown by the errors in the left margin of the plot.

We checked the effect of the truncation on physical quantities such as metallicity and emission flux fraction if these were determined using three-PC-reconstructed EW data. The relative emission-line flux fraction μ is plotted in Figure 16, truncated to the first three PCs versus the original. The rms error of the reconstruction is as small as 0.001 which means an estimation of the emission-line flux fraction within a precision of 0.1%. The lines reconstructed from the first three PCs obey the known ratios of the doublet lines, especially [O iii] λλ4960,5008 and [N ii] λλ6550,6585 with a relatively good precision. The line ratio [N ii] λ6585/[N ii] λ6550 in the reconstruction using the first three eigenvectors is 3.26, whereas the fitted ratio in the original data is 3.23. For [O iii], the fitted [O iii] λ5008/[O iii] λ4960 ratio is 3.06 for the reconstructed data and 3.01 for the original data. These features are so strong that only higher PCs begin to violate them by including noise components. However, the flux ratios of lines that are not in the same doublet are not restored precisely. If we want to use the [N ii] λ6585/Hα ratio for diagnostic purposes to distinguish between thermal emission of H ii regions from the AGN-like emission, we need the first three PCs and the fifth PC as well. The fifth eigenvector is the one that makes possible an efficient fine-tuning of this ratio as it contains Hα and [N ii] with opposite signs. Figure 17 shows that metallicity estimation with the first three PCs has a relatively large error, which can be suppressed by the inclusion of the fifth eigenvector. The rms errors of the reconstructed metallicity are 0.23 and 0.13 without and with e₅, respectively. The reconstruction is less precise at high metallicities. This is because of the weak Hα and Hβ lines, since, as described in Section 3.4, these are typically early-type galaxies.

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