THINK OUTSIDE THE COLOR BOX: PROBABILISTIC TARGET SELECTION AND THE SDSS-XDQSO QUASAR TARGETING CATALOG

Large samples of quasars are important tools in astrophysics and cosmology for several reasons. They can be used to study the properties of quasars and their co-evolution with galaxies, e.g., through measurements of quasar clustering (e.g., Arp 1970; Hawkins & Reddish 1975; Osmer 1981; Shen et al. 2007; Ross et al. 2009), or they can be used to study the intervening intergalactic medium (e.g., Chelouche et al. 2007; Ménard et al. 2010). Recently, it has been shown that the Lyα forest at z ≈ 2.5 as revealed by background quasars can be used to detect the baryon acoustic feature at this redshift and thus provide an important measurement of the angular diameter distance at a distance that cannot currently be studied with galaxies (McDonald & Eisenstein 2007). SDSS-III's Baryon Oscillation Spectroscopic Survey (BOSS; Schlegel et al. 2007; Eisenstein et al. 2011) aims to do just that by obtaining spectra for ≈160,000 quasars (≈20 deg⁻²) in the 2.2 ⩽ z ⩽ 3.5 redshift range. However, only about 40 fibers deg⁻² are available to the quasar survey such that BOSS quasar target selection must approach an efficiency of 50%.

Quasar target selection in this redshift range is notoriously inefficient (e.g., Fan 1999; Richards et al. 2002) because at z≈ 2.8 the quasar and stellar loci cross in color space. This problem is exacerbated by the fact that medium-redshift 2.2 ⩽z ⩽ 3.5 quasars are relatively rare, such that to reach a density of 20 deg⁻², objects must be targeted down to g ≈ 22 mag, where photometric uncertainties in the relatively shallow Sloan Digital Sky Survey (SDSS) imaging can become substantial. Fortunately, since the BOSS quasars' primary use is to provide lines of sight through the intervening intergalactic medium, they do not have to be selected in a uniform manner. However, in order to allow for ancillary quasar science such as the measurement of the quasar luminosity function at the peak of quasar activity, the BOSS has decided to select half of the targets in a uniform manner. Therefore, a quasar target-selection technique that is 50% efficient down to g ≈ 22 mag at 20 targets deg⁻² is required.

The last decade has seen the advent of statistical studies with quasars using purely photometric samples. The ≈95% efficient photometric UVX catalog of Richards et al. (2004) was used to measure the Integrated Sachs-Wolfe (ISW) effect (Giannantonio et al. 2006, 2008) and cosmic magnification bias (Scranton et al. 2005a), and to study the clustering of quasars on large (Myers et al. 2006, 2007a) and small scales (Hennawi et al. 2006; Myers et al. 2007b, 2008). However, with the latest catalog of ≈1,000,000 photometric quasars in SDSS Data Release 6 (Adelman-McCarthy et al. 2008) brighter than i = 21.3 mag (Richards et al. 2009a), the kernel-density-estimation (KDE) technique has reached its limit, since it does not take into account the photometric uncertainties when evaluating quasar probabilities. It is already the case that ignoring the photometric uncertainties results in an incompleteness of the KDE-selected catalog that is not currently understood. This makes statistical studies and follow-up of interesting sources (e.g., binary quasars; Hennawi et al. 2010; Shen et al. 2010) difficult. The technique described in this paper for quasar target selection can also be used to construct a photometric quasar catalog down to faint magnitudes that does take photometric uncertainties into account.

Since the first attempts at optical selection of quasars (Sandage & Wyndham 1965), broadband quasar target selection has mostly relied on the "UV excess" that characterizes low-redshift quasars (Schmidt & Green 1983; Marshall et al. 1984). However, this selection technique cannot be used by the BOSS, precisely because of the Lyα absorption in the optical that the BOSS aims to study at these redshifts, which reddens the quasars so that they are no longer UV excess objects. Quasar target selection in SDSS-I and SDSS-II consisted essentially of two color-selection algorithms using ugri and griz for low- and high-redshift quasars, respectively (Richards et al. 2002). This color-selection algorithm aimed to avoid the stellar locus and therefore it cannot be used to efficiently target the z ≈ 2.5–3 range that is of interest to the BOSS: the SDSS-I/II quasar target selection in this range is ≲25% efficient and this is at relatively bright magnitudes, viz., i < 19.1 mag (Richards et al. 2002, 2006).

In this paper, we describe a probabilistic target-selection technique that uses density estimation in flux space to assign quasar probabilities to all SDSS point sources. As our density-estimation tool we use extreme deconvolution (XD; Bovy et al. 2009), which uses the photometric uncertainties of the data and can therefore be used to obtain reliable target probabilities down to g ≈ 22 mag. By targeting the highest probability quasars, an efficient and uniform medium-redshift quasar survey such as the BOSS can be performed. We show, using early BOSS data, that this approach achieves approximately 50% efficiency, as required for the BOSS's uniform quasar target-selection algorithm.

We apply this target-selection technique to all SDSS point sources and construct the SDSS-XDQSO quasar targeting catalog. This catalog can be used to define star, low-redshift (z < 2.2), medium-redshift, and high-redshift (z>3.5) target classes. We show that this selection algorithm performs well at low redshift and that it outperforms all other flux-based methods for medium-redshift quasar targeting.

We describe the general density-estimation-based target-selection technique in Section 2. In Section 3 we discuss the data used to train our probabilistic classifier, and in Section 4 we present the specific implementation of the general method that we use for BOSS target selection. In Section 5 we describe the targeting catalog, and in Section 6 we discuss its basic properties. In Section 7, we show results for medium-redshift targets based on early BOSS data. In Section 8, we compare our target-selection technique to other methods and describe various extensions to the basic method described in this paper. We conclude in Section 9.

In what follows, AB magnitudes (Oke & Gunn 1983) are used throughout. Where dereddened fluxes and magnitudes are required we have used the reddening maps of Schlegel et al. (1998).

Target selection is essentially a classification problem: based on a set of attributes, we must classify an object into one of a discrete set of classes. This set of classes can be as simple as target/non-target, but more classes might be beneficial in circumstances in which different targets are assigned different priorities (see below). We assume that we have a set of objects with class assignments available on which we can train the classification algorithm. This is a classical problem in data analysis/machine learning, and many classification methods are available (e.g., neural networks (NNs), Bishop 1995; support vector machines, Schölkopf & Smola 2002; Gaussian process classification, Rasmussen & Williams 2006). However, object attributes in astronomy are rarely measured without substantial and heterogeneous measurement uncertainties, and the training sets are rarely as noisy as the test sets that are to be classified. Most classification algorithms are poorly equipped to deal with these complications in a statistically well-posed manner. They do not naturally degrade the probability of an object being in a certain class if the measurement uncertainties imply that the object overlaps several classes.

A simple way to think about classification is to compare the number densities of the various classes in attribute space: if the number density of class A is higher than that of class B evaluated at an object's attributes, then the probability that the object belongs to class A is higher than that it belongs to class B, with probabilities proportional to the number densities (e.g., Richards et al. 2004). The advantage of this density view of classification is that general density-estimation techniques exist that can handle all of the complications that necessarily arise with astronomical data, e.g., very noisy attribute measurements or missing attributes (Bovy et al. 2009). It also provides a clean probabilistic interpretation of the class assignments.

Consider an object O with attributes {a_j} that we wish to classify into class A or class B, and that these classes are an exhaustive set. In the context of quasar target selection, the attributes could be the ugriz fluxes of a point-like object and the classes "quasar" and "star." We use Bayes' theorem to relate the probability that object O belongs to class A to the density in attribute space

$$\begin{equation} P(O \in A| \lbrace a_j\rbrace) = \frac{p(\lbrace a_j\rbrace | O \in A)\,P(O \in A)}{p(\lbrace a_j\rbrace)}\,, \end{equation} \tag{ 1 }$$

where

$$\begin{eqnarray} p(\lbrace a_j\rbrace) &=& p(\lbrace a_j\rbrace | O \in A)\,P(O \in A)\nonumber\\ &&+ p(\lbrace a_j\rbrace | O \in B)\,P(O \in B)\,, \end{eqnarray} \tag{ 2 }$$

since A ∪ B contains all of the possibilities. Note that we distinguish between discrete probabilities P(·) and continuous probabilities p(·). The first factor in the numerator of the right-hand side of Equation (1) is the density in attribute space evaluated at the object's attributes {a_j}, the second factor is the total number of A objects in an unbiased sample, and the denominator is a normalization factor. It is easy to see that this probability is a true probability in the sense that it always lies between zero and one and that it sums to one.

The common situation in which the attributes of object O are measured with substantial measurement uncertainty is easily handled in this framework through marginalization over the "true" attributes $\lbrace \tilde{a}_j\rbrace$ based on the observed attributes {a_j} and measurement-uncertainty distribution $p(\lbrace \tilde{a}_j\rbrace |\lbrace a_j\rbrace)$ (which can be as simple as a Gaussian distribution for Gaussian uncertainties)

$$\begin{equation} P(O \in A| \lbrace a_j\rbrace) = \int d\lbrace \tilde{a}_j\rbrace \, P(O \in A| \lbrace \tilde{a}_j\rbrace)\, p(\lbrace \tilde{a}_j\rbrace |\lbrace a_j\rbrace)\,, \end{equation} \tag{ 3 }$$

where the first factor in the integral is given by Equation (1). This is again a well-defined probability.

The target-selection problem then becomes the task of training good number-density models for both the target population and the contaminants population to maximize the efficiency and completeness of the survey. In regions of attribute space where the efficiency is low, for example, z ≈ 2.8 quasars where the quasar and stellar loci cross, the challenge is to develop the best possible density models for the most efficient target selection.

For the purpose of the SDSS-XDQSO quasar targeting catalog we need to classify point-like objects into classes "star" and "quasar." This assumes that star–galaxy separation is perfect for extended galaxies, such that the additional class "galaxy" can be ignored. If this were not the case then the class "galaxy" would be included and morphological attributes relevant to star–galaxy separation would be added to the attribute list {a_j}. Contamination from point-like galaxies, which is small in the apparent magnitude range of interest, is automatically handled in what follows since the training set for the "star" class consists of non-varying sources, which includes both stars and point-like galaxies.

The SDSS (York et al. 2000) has obtained u,g,r,i, and z CCD imaging of ≈10⁴ deg² of the northern and southern Galactic sky (Gunn et al. 1998, 2006; Stoughton et al. 2002). SDSS-III has extended this area by approximately 2500 deg² in the southern Galactic cap (Aihara et al. 2011). All the data processing, including astrometry (Pier et al. 2003), source identification, deblending and photometry (Lupton et al. 2001), and calibration (Fukugita et al. 1996; Hogg et al. 2001; Smith et al. 2002; Ivezić et al. 2004; Padmanabhan et al. 2008) are performed with automated SDSS software.

This section describes the data used to train the density classification model. The data used to create the SDSS-XDQSO catalog are presented in Section 5.

3.1. Stellar Data

3.2. Quasar Data

To estimate the density of stars and quasars in flux space, we use XD⁹ (Bovy et al. 2009). At the faint end (i ≳ 20 mag) of the magnitude range of interest here the flux uncertainties in the training set are substantial, even though they are much smaller than the single-epoch uncertainties used to evaluate quasar probabilities for the "star" class (see below). Deconvolution is therefore necessary to avoid adding in uncertainties twice at the density-evaluation phase. While XD can handle missing data as well, we do not need this feature here. However, this capability of XD is crucial when we want to extend the current framework to include near-infrared (NIR), ultraviolet (UV), or variability information, since these data will not be available for every object in the training set (see Section 8.2).

XD models the underlying, deconvolved, distribution as a sum of K Gaussian distributions, where K is a free parameter that needs to be set using an external objective. It assumes that the flux uncertainties are known, as is the case for point-spread function (PSF) fluxes for point sources in SDSS (Ivezić et al. 2003, 2007; Scranton et al. 2005b). XD consists of a fast and robust algorithm to estimate the best-fit parameters of the Gaussian mixture. For example, we were able to fit the color distribution of the full stellar catalog of 701,215 objects in only a few hours using 30 four-dimensional Gaussians. It is robust in the sense that even a poor initialization quickly leads to an acceptable solution. We are interested not so much in the true underlying distribution function as in finding a good fit to the observed density (after convolving the model with the data uncertainties) without overfitting, so it is not absolutely necessary to find the exact best fit in the complicated likelihood surface. Therefore, we use the simplest version of XD that does not use the heuristic search extension or priors on the parameters (Bovy et al. 2009).

The XD method works by iteratively increasing the likelihood of the underlying, K Gaussian model given the data. It is an extension of the Expectation Maximization (EM) algorithm for fitting mixtures of Gaussians in the absence of noise (Dempster et al. 1977) to the case where noise is significant or there are missing data. The algorithm basically iterates through an expectation (E) and a maximization (M) step. During the expectation step, the data and the current estimate of the underlying density are used to calculate the expected value of (1) indicator variables that for each data point indicate which Gaussian it was drawn from, and (2) the true, noiseless value of each data point and its second moment. In the maximization step, these expected values are used to optimize the amplitude, mean, and variance of each of the K Gaussians. The E and M steps are iterated until the likelihood ceases to change substantially. The algorithm has the property that after each EM iteration the likelihood of the model is increased.

4.1. Construction of the Model

The full model consists of fitting the flux density in a number of bins in i-band magnitude for the various classes of objects. We describe the model in a single bin first for a single example class. Throughout we will use the "star" class as the example.

We opted for a binning approach because the true five-dimensional distribution of fluxes has a dominant power-law shape corresponding to the number counts as a function of apparent magnitude. However, most of the information for discriminating between quasars and stars is not in this power-law behavior, but in the behavior of colors or relative fluxes. While the latter can be represented well by mixtures of Gaussian distributions (see below), the power-law behavior cannot without using large numbers of Gaussians. For this reason, we chose to take out the power-law degree of freedom, i.e., the overall behavior of the density as a function of apparent magnitude. Neither the color distributions of quasars or that of stars is a strong function of apparent magnitude, such that the binning described below does not introduce strong assumptions about the behavior of the color distributions. Any weak magnitude dependence of the color distributions is captured in our model since we use narrow bins in i-band magnitude (see below).

The model only includes the fluxes of an object and not its position on the sky. Since quasars are uniformly distributed on the sky, while stars are concentrated near the Galactic plane, the position on the sky of an object is a potential discriminant. Because the fluxes are much more informative about an object's type than its position, we do not include the position in the model. We discuss below how the model could be extended to include the position of an object as part of the model.

In a single bin in i-band magnitude, we separate the absolute flux from the flux relative to i in the likelihood in Equation (1) as follows:

$$\begin{eqnarray} p(\lbrace f_j\rbrace |O \in \mathrm{``star^{\prime \prime }}) &=& p(\lbrace f_j/f_i\rbrace | f_i, O \in \mathrm{``star^{\prime \prime }})\nonumber\\ &&\times p(f_i| O \in \mathrm{``star^{\prime \prime }})\,, \end{eqnarray} \tag{ 4 }$$

where we now specify that the attributes {a_j} are the ugriz fluxes {f_j}, {f_j/f_i} are the fluxes relative to i, and f_i is the i-band flux. We model these two factors separately.

We model the first factor using XD in narrow bins in i-band magnitude described in detail below. We use relative fluxes rather than colors—logarithms of relative fluxes—since the observational uncertainties are closer to Gaussian for relative fluxes than they are for colors. Except for the absence of a logarithm, relative fluxes are similar to colors and have the same number of degrees of freedom, viz., four. The fact that fluxes must be larger than zero while the Gaussian mixture model does not contain any such constraints, which is one reason to model the logarithm of the fluxes rather than the fluxes themselves, does not matter greatly here because most of the objects in the training set are at least 5σ detections. However, for z>3 quasars the u band has zero flux and the flux measurement can be negative, in which case magnitudes are badly behaved; relative fluxes remain well behaved in this case. To evaluate the XD probabilities during training, we always convolve the underlying model with the objects' uncertainties. We assume that the relative-flux uncertainties are Gaussian—which is a good assumption because the i-band magnitude is always measured at a reasonably signal-to-noise ratio—such that the convolution of the Gaussian mixture with the uncertainties results in a Gaussian mixture, with an object's uncertainty variance added to the model variance for each of the components.

We model the four-dimensional relative fluxes {f_j/f_i}, which each come with an individual, non-diagonal, four by four uncertainty covariance, using 20 Gaussian components. The number 20 was decided upon as follows. For a few bins, we performed XD fits with 5, 10, 15, 20, 25, and 30 components. While fits with less than 20 components overly smoothed the observed distribution, models with more than 20 components used the extra components to fit extremely low significance features in the observed distribution. Because of the scale of the full model (see below) no bin-by-bin objective method for setting the number of components was pursued, although we did verify that all of the resulting fits were reasonable.

We model the second factor, p(f_i|i ∈ ``star''), by first combining it with the number factor P(i ∈ ``star''). This combined factor becomes the number density as a function of apparent magnitude. This quantity will always be expressed in units of deg⁻². For the "star" class, we model the number density directly using the number counts of the training data, by spline interpolating the histogram of i-band magnitude number counts per square degree. For the "quasar" class, we use a model for the quasar luminosity function (Hopkins et al. 2007) to calculate the number density of quasars as a function of apparent i-band magnitude; we multiply these theoretical number densities with a simple model for the SDSS incompleteness of point sources

$$\begin{equation} I(i) = \left(1 + \exp \left(\frac{i-21.9}{0.2}\right)\right)^{-1}\,, \end{equation} \tag{ 5 }$$

designed to reproduce the incompleteness as given in Abazajian et al. (2003). The p(f_i|i ∈ class) factors for the various target classes are shown in Figure 1. The total number densities for the various classes are given in Table 1.

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The full model consists of 47 bins of width 0.2 mag between i = 17.7 and i = 22.5, spaced 0.1 mag apart such that adjacent bins overlap. We further divide the "quasar" class into three subclasses corresponding to low-redshift (z < 2.2), medium-redshift (2.2 ⩽ z ⩽ 3.5; the BOSS quasar redshift range), and high-redshift (z>3.5) quasars. The XD fits for all but the brightest bin for a given class are initialized using the best-fit parameters for the previous bin. There are typically ≈20, 000 objects in each bin for the stellar training data; for the quasars there are 85,998 low-redshift, 14,060 medium-redshift, and 3519 high-redshift quasars in each bin.

In each of 4 × 47 bins, we fit 20 four-dimensional Gaussians, yielding a total of 4 × 47 × (20 × 15 − 1) = 56, 212 parameters.

4.2. Comparison of the Model and Observations

In this section, we assess the performance of the XD technique for modeling the relative-flux distributions of stars and quasars, and provide examples which demonstrate that the algorithm produces excellent fits to the data.

The XD method implicitly produces an error-deconvolved model of the relative-flux distribution, which would correspond to the true parent distribution of our training set in the limit of infinite signal-to-noise ratio. We refer to this as the "underlying" model of the data. An important test of our approach is to apply the XD technique to the noisier single-epoch data of the stars in Stripe-82, and compare our determination of the underlying model to the much higher signal-to-noise ratio Stripe-82 co-added photometry of the same sources. This is a stringent test since the underlying model fit is based on much noisier single-epoch data than the co-added photometry.

If the co-added photometry were perfect, then the co-added data would represent samples of the underlying model with zero noise. In practice, even the Stripe-82 co-adds are noisy at the faint end, thus the relative-flux distribution of the co-adds will be correspondingly smoothed. To make a sensible comparison, we sample the underlying model at the same number of points as are in the co-added catalog in a given magnitude bin, and convolve these samples with uncertainties of the co-added data. As we do not have a model for the noise in the co-added data, and our samples are in general at different locations in relative flux space, we simply match each sample from the underlying distribution to the closest object in the co-added catalog and use that co-add object's uncertainties.

The results for this exercise for a single bin are shown in Figures 2 and 3. We see that the agreement between the model and the co-add truth is excellent.

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After fitting the relative-flux distribution of the training data, we can also compare the best-fit model with the training data by convolving the model with the data uncertainties. This uncertainty convolution is again performed by matching a sampled point to the nearest object in the training set and drawing from that object's uncertainty distribution. Some results of these tests are shown in Figures 4–6.

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Table 1. Total Number Counts in 17.75 ⩽i < 22.45

	z < 2.2	2.2 ⩽ z ⩽ 3.5	z>3.5	Star
	Quasar	Quasar	Quasar
Number counts (deg−2)	140.72	50.70	6.13	5209.38

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Figure 4 shows flux–flux and color–color diagrams in one of the bins in i-band magnitude for medium-redshift quasars. The first and third columns show a sampling from the best-fit XD model to the relative fluxes. The second and fourth columns show the data that these fits were based on: these data are quasars from the SDSS DR7 quasar catalog resampled according to the quasar luminosity function as explained in Section 3.2. Similar figures for the other apparent magnitude bins are very similar as the distribution of quasar colors does not vary much with apparent magnitude. The same comparison for low- and high-redshift quasars was performed for quality assurance, but these figures are not shown here.

Figure 5 shows the comparison between the XD fit to the co-added point-source data from SDSS Stripe-82 and the data itself for a relatively bright apparent magnitude bin. Figure 6 shows the same for a fainter bin in i-band apparent magnitude. In this bin, the photometric uncertainties are significant such that the stellar locus is severely smoothed. We see that the underlying deconvolved XD model is good in the sense that after convolution with the data uncertainties it reproduces the observed distribution.

The resampled error-convolved fits to the data in all of these cases are essentially indistinguishable from the real data.

Using the models of quasar and stellar fluxes described in the previous section, we create a quasar targeting catalog. For every point source (objc_type = 6) in the SDSS Data Release 8 (DR8) with a reasonable detection¹⁰ in at least one band that is primary and has a dereddened i-band magnitude 17.75 ⩽i < 22.45 we calculate the probability that the object is a quasar or star using Equation (1). Specifically, we use the models from the previous section as follows. For an object with dereddened i-band magnitude i, we first find the bin for which this magnitude is contained within the central 0.1 mag of the bin (this is always possible since neighboring bins overlap by 0.1 mag). We use this bin to evaluate the relative-flux density for this object's dereddened fluxes. We convolve the underlying 20 Gaussian mixture model with the object's uncertainties as in Equation (3). This uncertainty convolution is simply adding the object's uncertainty variance to the intrinsic model variance for each component.

We further evaluate the number density as a function of apparent magnitude for the object's i-band flux. We do this for each of the classes ("star," and the three "quasar" classes) and normalize the probabilities as in Equation (2). Code that can be used to reproduce this information is made publicly available and is described in Appendix B.

For each object, we also calculate the bitmask used for BOSS quasar target selection. This bitmask is calculated using version bosstarget v2_0_10 of the BOSS quasar target-selection code. For the convenience of the user, we summarize this bitmask into a simple good flag. When this flag is zero the object passes all of the BOSS flag cuts; when the flag is one, the object fails one of the basic BOSS targeting flag criteria (interp_problems, deblend_problems, or moved). An object with a good flag equal to two fails one of the more esoteric BOSS quasar selection cuts. No objects with a good flag >0 would be targeted by the BOSS. These flag cuts are discussed in more detail in Appendix A.

In addition to the quasar and star probabilities and flag-logic described above, we also report each of the factors in Equation (4) for all of the target classes for each object in the catalog. This allows the user of the catalog to substitute different number-count or relative-flux-density models for any of the components.

A description of the catalog is given in Table 2. The full catalog contains 160,904,060 objects. Summing the probabilities, we expect this catalog to contain about 5,058,716 quasars, 2,551,146 of which are low redshift, 1,410,405 are mid redshift, and 1,097,165 are high redshift. However, as we show below, this is probably a slight overestimate, especially for the high-redshift class.

We stress that this quasar targeting catalog is entirely probabilistic and that the majority of objects are certainly stars. To create high-confidence quasar samples, it is necessary to select those objects with the largest P(``quasar''). As an example, we show in Figures 7 and 8 color–color plots of those objects in the SDSS-XDQSO catalog with P(``quasar'') ⩾ 0.8 and P(``star'') ⩾ 0.95. For the photometric quasars in Figure 7 a sparse sampling of z ⩾ 2.5 quasars from the SDSS DR7 quasar catalog (Schneider et al. 2010) and a model for the quasar locus from Hennawi et al. (2010) is shown for comparison. For the photometric stars in Figure 8, a model for the stellar locus from Hennawi et al. (2010) and some representative classes of stars along the stellar locus are shown. While we cannot be sure for any of the objects in either of these photometric samples whether they are truly quasars or stars, the color distribution of the ensemble does resemble that of a set of quasars and stars, respectively.

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5.1. Catalog Availability

While there are many detailed considerations in constructing a specific quasar sample from the SDSS-XDQSO catalog, for the purposes of testing we use an arbitrarily chosen cut of P(``quasar'')>0.5 to define a "quasar" sample and P(``star'')>0.95 to define a "star" sample. We show the sky distribution of quasars and stars in Figure 9. As expected for a quasar catalog, the distribution is mostly flat, with only minor increases in density near the Galactic plane where the density of stars and hence potential contaminants is extremely high. This expected behavior is satisfying since our model did not include a prior depending on Galactic longitude and latitude. The stars selected by the star cut do show the expected strong dependence on Galactic latitude. If we had included a better prior depending on the Galactic latitude and longitude of targets, there would have been less quasar targets in regions closer to the Galactic plane, since objects would then get a higher probability of being stars. Quasars are targeted predominantly at high Galactic latitude, such that the inclusion of such a prior does not lead to great improvements to the targeting efficiency; therefore, we have not pursued this here.

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In Figure 10, we show the XDQSO probabilities of known, i.e., spectroscopically confirmed, quasars, and stars. These come from a compilation of known quasars and stars from various surveys, including some early BOSS data. We split the sample of known quasars into our three redshift bins. We see that both known low-redshift quasars and known stars are assigned very high probabilities for the correct class, with no significant populations of low-probability objects for a given class. The distribution of mid-redshift probabilities for known mid-redshift quasars shows that most of these quasars have high correct probabilities, but that there is a population of objects around P ≈ 0. Inspection of the probabilities of being stars for these objects shows that not all of them get high star probabilities, such that at least some of these are "misclassified" into the wrong redshift range. Most of the known high-redshift quasars are assigned high high-redshift probabilities, with only a small population being "misclassified" as stars. The color–color distributions of "misclassified" quasars show that many of these are z ≈ 2.8 quasars that overlap the stellar locus, such that it is unsurprising that color selection fails. The sky distribution of "misclassified" quasars shows that many of these objects lie in SDSS Stripe-82, where deeper photometry for quasar selection is available than the single-epoch fluxes used in this comparison. These Stripe-82 quasars were probably targeted based on higher signal-to-noise co-added photometry, whereby they could be better distinguished from stars.

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We have cross-correlated our catalog with the FIRST point-source catalog (Becker et al. 1995) with a matching radius of 05 (McGreer et al. 2009). We expect that most of these objects—optically unresolved point sources with compact radio morphology—are quasars, with only a small fraction being compact radio galaxies (McGreer et al. 2009). In Figure 11, we show the quasar probabilities for the matching FIRST objects. The majority of these sources have high quasar probabilities. Those that do not have high quasar probabilities either overlap with the stellar locus or have very unusual colors. An inspection of FIRST sources targeted as quasars by the BOSS shows that FIRST-selected quasars with small XDQSO quasar probabilities are significantly redder than quasars with high XDQSO probabilities; they are missed by the XDQSO method because it is trained on optically selected quasars which are typically bluer than radio-selected quasars.

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Finally, we show in Figure 12 the quasar probabilities for the uvx objects in the Richards et al. (2009a) photometric quasar catalog. We expect ≈95% of these objects to be z < 2.4 quasars. Only a small fraction of objects are assigned low quasar probabilities, which is unsurprising given the 95% efficiency of the photometric catalog. We discuss the Richards et al. (2009a) technique further in Section 8.1.1 in the specific context of target selection.

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As discussed in the introduction, the BOSS aims to obtain spectra of ≈160, 000 quasars in the redshift range 2.2 ⩽ z ⩽ 3.5 to observe the baryon acoustic feature in the Lyα forest and obtain a percent-level measurement of the angular diameter distance to redshift 2.5. This requires observing at least 15 quasars deg⁻² over the BOSS survey area. Since only about 40 targets deg⁻² will be allocated for quasar targets, target selection must reach 50% efficiency to achieve this goal. Achieving this efficiency is complicated by the substantial overlap between the quasar and stellar loci in color space in the BOSS redshift range and the need to target to g ≈ 22 mag where photometric uncertainties are significant.

While a uniform target selection of the BOSS quasars is unnecessary for the cosmological distance measurement because the quasars are only used to reveal the intervening Lyα forest, the BOSS quasar target selection reserves 20 targets deg⁻² for a "CORE" sample of objects that is selected by a single method over the course of the survey, to allow statistical studies of quasar properties. Since only single-epoch SDSS ugriz fluxes are available for the bulk of the BOSS survey footprint, this CORE sample must be based on these single-epoch fluxes alone.

The SDSS-XDQSO catalog presented in Section 5 allows such a uniform sample to be defined while maintaining a high target efficiency. In practice, we use the medium-redshift quasar probabilities pqsomidz (see Table 2) and target all good objects with dereddened (g < 22 mag or r < 21.85 mag) and i>17.8 mag—the BOSS flux limits—with pqsomidz larger than a threshold set to give 20 or 40 targets deg⁻² over a certain region (e.g., a part of the survey such as Stripe-82 or the entire survey footprint).

Even though the XDQSO targeting technique was not used during the first year of BOSS data acquisition, we can nevertheless evaluate the XDQSO performance by determining how many of the XDQSO quasar targets are spectroscopically confirmed BOSS quasars. This is a conservative test as it presumes that the BOSS quasar sample is highly complete; any incompleteness would lead us to underestimate our efficiency since there would then be real quasars selected by XDQSO  but which did not receive a BOSS fiber. In Figure 13, we show the XDQSO targeting efficiency in an area from the BOSS. This area consists of BOSS observations of Stripe-82. Since over eight epochs are available for most sources in Stripe-82, BOSS quasar target selection was based on deeper co-added data and included variability selection (Palanque-Delabrouille et al. 2011). Therefore, it is highly complete and it is therefore the best test of the XDQSO technique. We only use the ≈205 deg² region of Stripe-82 where at least 15 2.2 ⩽z ⩽ 4 quasars deg⁻² were observed by the BOSS; on average 27 z>2.15 quasars deg⁻² are available. This is close to the total number of quasars in this redshift range expected based on various quasar luminosity functions (Richards et al. 2006; Jiang et al. 2006; Hopkins et al. 2007). We see that at 20 targets deg⁻² the efficiency is about 50%; the efficiency drops rapidly at higher target densities.

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A further test of the XDQSO technique was performed during the fall of 2010. Approximately 20 medium-redshift quasar targets (catalog tag pqsomidz) per square degree were observed as part of the BOSS in a region of ≈200 deg² just north of SDSS Stripe-82. As of 2010 November 22, 4593 objects were targeted and 2194 of these were classified as 2.2 ⩽z ⩽ 3.5 quasars, corresponding to an efficiency of 48%. In Section 8.1, we compare the XDQSO selection to other targeting algorithms which were tested during the first year of BOSS commissioning.

We can also use the early BOSS data to evaluate whether the XDQSO probabilities given to objects are correct in the ensemble sense. That is, we can bin objects in P(2.2 ⩽ z ⩽ 3.5 quasar|{f_j}) and evaluate the fraction of these objects that are quasars. This test again assumes that the test sample of quasars is highly complete. The results from this exercise are shown in Figure 14. We see that the quasar probabilities appear to be overestimated. Several shortcomings of the model for the quasar and stellar populations could be responsible for this result. Since we used relatively small samples of stars in each bin compared to the number of quasars—the proportion being around one to one rather than reflecting the true number densities according to which stars outnumber quasars by a factor of ≈100 down to g ≈ 22 mag—we probe much lower density regions for the quasars than we can for the stars. Thus, in low stellar-density regions, the stellar density is likely to be underestimated with respect to the quasar densities, leading to overestimated quasar probabilities in these regions. This is unlikely to change the ranking of quasar targets significantly—most targets will be affected in the same way. Thus, quasar targeting using the SDSS-XDQSO catalog is robust with respect to this modeling problem. However, this aspect of our analysis does merit caution when taking the specific value of the quasar probabilities serious in an analysis of quasar properties. Our model also does not include variations in the stellar number density with Galactic latitude and longitude, so the stellar density will be underestimated in regions closer to the Galactic plane. Several of the extensions described in Section 8.2 can be used to mitigate these problems with the catalog.

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8.1. Comparison with Other Target-Selection Methods

Several sophisticated methods already exist to select quasars from photometric SDSS data, especially for the mid-redshift range targeted in the BOSS.

8.1.1. Richards et al.'s (2009a) NBC-KDE

The NBC-KDE photometric quasar catalog of Richards et al. (2009a) also uses density estimation to calculate relative number densities of stars and quasars to determine quasar probabilities. It differs from the technique described in this paper in that it (1) uses colors rather than relative fluxes in Equation (4), (2) assumes a flat probability distribution for p(f_i|i ∈ class), (3) uses a simple KDE technique of the observed distribution of colors, thus ignoring the flux uncertainties of the training data, and (4) ignores the flux uncertainties of the data at evaluation. Since the catalog is limited to i < 21.3 mag, ignoring the flux uncertainties is mostly harmless, although medium- and high-redshift quasars have very low u- and g-band fluxes with large associated uncertainties. This assumption also makes it harder to extend the catalog to fainter magnitudes. The NBC-KDE catalog also does not compute exact kernel-density likelihoods for all of the SDSS data because of computational limitations. The XD technique permits these likelihoods to be computed for all of the data (see above).

We compare the XDQSO catalog with the NBC-KDE catalog by (1) limiting the XDQSO catalog to the SDSS DR6 footprint, since the NBC-KDE catalog only covers this area, (2) limiting the XDQSO catalog to i < 21.3 mag, and (3) limiting the NBC-KDE catalog to i ⩾ 17.75, the bright limit of the XDQSO catalog. We select those targets in the NBC-KDE catalog with good ⩾0 and then select equal numbers of targets from the good entries in the XDQSO catalog in the three shared target classes of the two catalogs: lowz, midz, and highz. Note that the good flags in the two catalogs are different. The results of this comparison are given in Table 3. The confirmed quasars in this table come from a compilation of all spectroscopically confirmed quasars. The bulk of these are SDSS quasars that were selected to be brighter than i = 20.2 mag.

Table 3. Comparison of the NBC-KDE and SDSS-XDQSO Catalogs

Catalog	Confirmed z < 2.2	Confirmed 2.2 ⩽ z ⩽ 3.5	Confirmed z>3.5
	Quasar	Quasar	Quasar
NBC-KDE	77,144	11,485	2802
XDQSO	77,517	12,711	2338
Total no. of targets	569,785	151,860	9086

Download table as:  ASCIITypeset image

We stress that this comparison is significantly biased in favor of the NBC-KDE catalog: the NBC-catalog was allowed to set the target threshold and its good flag includes cuts on Galactic latitude that we did not apply to the XDQSO catalog. Nevertheless, we see that the two catalogs perform similarly for low-redshift quasars, and that the XDQSO technique performs better in the mid-redshift range where the quasar and stellar loci overlap and photometric target selection is difficult.

The NBC-KDE catalog performs better at high redshift than the XDQSO catalog. This is not unexpected as we only used 150 deg² of stellar data to train the XDQSO algorithm. The red-star stellar-locus outliers that contaminate z>3.5 quasar selection are very rare on the sky: if we assume that they are 10 times more abundant than z>3 quasars we expect our stellar training set to only contain about 2000 of these. This is not enough to model their color distribution since they are spread over all magnitude bins. In the following section, we discuss improvements to the XDQSO technique that can improve high-redshift quasar selection.

8.1.2. Yeche et al.'s (2010) Neural Network

The quasar-selection technique of Yeche et al. (2010) uses an NN to select quasars in the BOSS mid-redshift range. This technique uses as the input variables the four colors u − g, g − r, r − i, and i − z, the g-band magnitude, and the five magnitude uncertainties. These 10 variables are propagated through a simple NN that is trained on sets of known quasars and point-like objects from SDSS DR7 to obtain an output parameter xnn on which potential targets can be ranked. A similar NN is used to find a photometric redshift znn for the quasar targets. To select targets in the mid-redshift range, they recommend the cuts znn >2, u − g>0.4, and g − i < 2. While this technique uses the photometric uncertainties, it does this in a black-box manner that is quite different from the probabilistic way in which XD treats the uncertainties; the NN approach cannot work well when the test set is noisier than the training set.

We use the version of the NN technique that is part of the official BOSS quasar selection framework to compare the XDQSO technique and the NN technique at different target densities using early BOSS data. For the NN, we first make the cuts listed above and then rank the targets on xnn until we reach the desired target density. For XDQSO, we rank the good targets based on pqsomidz (see Table 2). The results from this comparison for BOSS year-one observations of Stripe-82 are shown in Figure 13. We see that the NN performs significantly worse than XDQSO at all target densities.

8.1.3. Kirkpatrick et al.'s Likelihood

The Likelihood technique of J. A. Kirkpatrick et al. (2011, in preparation) uses an approach similar to the NBC-KDE catalog. Rather than colors it uses fluxes, such that the apparent magnitude factor also used by XDQSO is automatically taken into account, and rather than a kernel with an optimized bandwidth, Likelihood uses a delta function—corresponding to a model of the underlying density consisting of delta functions centered at the location of each object in the respective training set. These delta functions are convolved with the flux uncertainties at evaluation such that a smooth density is obtained nevertheless. The Likelihood technique uses a similar stellar training data as the XDQSO catalog. The quasar training set is also modeled by resampling the quasar luminosity function, but these are essentially the same set of quasars used to train the XDQSO technique. Rather than simulating relative fluxes only, a full quasar catalog with five-dimensional ugriz fluxes in the relevant magnitude range is simulated.

The Likelihood technique uses the flux uncertainties to smooth the discrete underlying delta-function distribution of its training sets. However, since it does not use an optimized bandwidth, there is the danger that the density might be undersmoothed in certain regions. At the faint end, where the training set has a significant contribution from the uncertainties, the Likelihood method also effectively convolves with the uncertainties twice. This follows because the training set is a sample from the observed distribution of fluxes, rather than the true underlying distribution; the former is the latter convolved with the uncertainty distribution. Finally, as with the NBC-KDE catalog, the calculation of the quasar and star probabilities is extremely slow compared to XDQSO. In our comparison below, we use cached versions of the Likelihood catalog created by the BOSS target-selection team.

In Figure 15, we first compare the XDQSO quasar probabilities in the BOSS mid-redshift range with the probabilities calculated using the Likelihood method for sources in Stripe-82. We select targets at 20 targets deg⁻² and show those targets selected by both techniques or by only one of the techniques. While many of the targets cluster around the one-to-one line, there is a distinct population of targets that receive high Likelihood probabilities, yet low XDQSO probabilities. A similar population of high XDQSO-only targets is absent, indicating that the Likelihood method indeed has problems with undersmoothing.

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In Figure 13, we compare the XDQSO catalog with the Likelihood technique at different target densities. The two methods perform similarly, with a slightly better performance for the XDQSO catalog over the whole range.

A further comparison between the XDQSO and the Likelihood methods for medium-redshift quasar selection was performed during the fall of 2010 using BOSS observations of an ≈200 deg² region just north of Stripe-82 (see Section 7). Both methods were given similar target densities to allow for a direct comparison. As of 2010 November 22, XDQSO was given 4593 targets, while Likelihood received 4853 targets. Of the 4593 XDQSO targets 2194 were classified as 2.2 ⩽z ⩽ 3.5 quasars, while of the 4853 Likelihood targets only 2056 turned out to be medium-redshift quasars. From this test and that in Figure 13, we conclude that the XDQSO technique's performance is about 10% better than that of the Likelihood method for the selection of medium-redshift quasars.

8.2. Extensions

8.3. Decision Theory

The XDQSO method described in this paper is a new general target-selection technique that can be used to design highly efficient surveys for faint objects that are hard to distinguish from contaminants based on their measured attributes. This high efficiency is achieved through the use of the XD density estimation to classify objects based on the number densities of desired objects and contaminants in attribute space. We used this approach to create an input catalog for BOSS quasar target selection, which aims to target ≈160,000 2.2 ⩽ z ⩽ 3.5 quasars over ≈10,000 deg² at 50% efficiency down to g ≈ 22 mag. Quasar colors in this redshift range overlap strongly with stellar colors and classification is made even more difficult by the large photometric uncertainties at these faint magnitudes. We demonstrated that at the 20 targets deg⁻² required for the BOSS "CORE" sample the XDQSO technique is indeed approximately 50% efficient and that it outperforms all other medium-redshift quasar target-selection techniques.

We release the full 160,904,060-object SDSS-XDQSO quasar targeting catalog containing star and low-, medium-, and high-redshift quasar probabilities for all primary objects in the 17.75 ⩽ i < 22.45 dereddened i-band magnitude range in SDSS DR8. We also release code to reproduce the catalog from the SDSS point-source catalog.

The XDQSO target-selection technique can be extended to include low signal-to-noise data in NIR and UV filters and to include other information such as quasar variability. It is the low signal-to-noise ratio regime at the faint edge of surveys that often contains the most interesting objects. XDQSO is the only target-selection technique currently available that can calculate robust quasar probabilities taking the data uncertainties fully into account. Since the most successful photometric quasar catalogs are based on calculating good photometric quasar probabilities (Richards et al. 2009a), the XDQSO technique or similar will be essential to create the best and largest photometric quasar catalogs in upcoming surveys such as Pan-STARRS and LSST.

It is a pleasure to thank Gordon Richards, Michael Strauss, and Christophe Yeche for helpful comments and assistance. J.B. and D.W.H. were partially supported by NASA (grant NNX08AJ48G) and the NSF (grant AST-0908357). D.W.H. is a research fellow of the Alexander von Humboldt Foundation of Germany. J.F.H acknowledges support provided by the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the German Federal Ministry of Education and Research.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.

SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, University of Florida, the French Participation Group, the German Participation Group, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, New Mexico State University, New York University, the Ohio State University, the Penn State University, University of Portsmouth, Princeton University, University of Tokyo, the University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.

As part of the information in our catalog, we include an entry good that facilitates two different sets of useful flag cuts. One set is the BOSS flag cuts, which are appropriate when attempting to use the information in the catalog directly for statistical studies. For objects that do not pass this set of flag cuts, good >0. The second set of flag cuts is less restrictive, and is more appropriate for follow-up observations where high completeness is required (such as, for instance, a search for high-redshift quasars). For objects that do not pass this less restrictive set of flag cuts, good = 1. Objects that pass these cuts but fail other BOSS flag cuts have good = 2.

We also provide a flag photometric, which is not bitwise—it is either true or false–and can be utilized at the user's discretion. This is a standard SDSS flag: photometric refers to objects that were observed under good imaging conditions. Restricting to photometric affects the coverage of the catalog and should only be considered when constructing a highly restrictive sample.

A.1. good = 1

A.2. good = 2