Finding Quasars behind the Galactic Plane. I. Candidate Selections with Transfer Learning

Pan-STARRS1 (PS1; Chambers et al., 2016) has carried out a set of synoptic imaging sky surveys including the 3$\pi$ Steradian Survey and the Medium Deep Survey in 5 bands $(grizy_{P1})$. The mean 5$\sigma$ point source limiting sensitivities in the stacked 3$\pi$ Steradian Survey in $(grizy_{P1})$ are (23.3, 23.2, 23.1, 22.3, 21.4) and the single epoch 5$\sigma$ depths in $(grizy_{P1})$ are (22.0, 21.8, 21.5, 20.9, 19.7). For better astrometry in the crowded Galactic plane field, we use mean coordinates from the PS1 MeanObject table. Mean PSF magnitudes are used for all bands $(grizy_{P1})$, and mean Kron magnitudes (Kron, 1980) are used for $i_{P1}$ and $z_{P1}$ bands. The Galactic extinction coefficients for $(grizy_{P1})$ are $R_{g},R_{r},R_{i},R_{z},R_{y}=3.5805,2.6133,1.9468,1.5097,1.2245$. These coefficients are calculated using $R_{\lambda}=A_{\lambda}/A_{V}\times R_{V}$, where $A_{\lambda}/A_{V}$ is the relative extinction value for band $\lambda$ given by a new optical to mid-IR extinction law (Wang & Chen, 2019), and $R_{V}=3.1$.

We set a few constraints on the PS1 data to ensure the data quality. All sources should be: (i) detected in all PS1 bands ($grizy_{P1}>0$) and significantly detected in $i_{P1}$ (error in PSF mag of $i_{P1}$ band $i\_err<0.2171$, equivalent to $i_{P1}$-band SNR larger than 5); (ii) not too bright in $i_{P1}$ to avoid possible saturation ($i>14$); (iii) measured with Kron magnitude (Kron, 1980) in the $i_{P1}$ and $z_{P1}$ bands ($i_{\mathrm{Kron}}>0\ \&\ z_{\mathrm{Kron}}>0$). For simplification, we use ($g,r,i,z,y$) to represent the PSF magnitudes of PS1 bands ($grizy_{P1}$) in color indexes (e.g. $g-r$, $g-W1$) and derived quantities $i-i_{\mathrm{Kron}}$ and $z-z_{\mathrm{Kron}}$. The $z_{P1}$ PSF magnitude does not appear alone and will not be confused with the redshift symbol $z$.

The AllWISE catalog is built upon the work of the Wide-field Infrared Survey Explorer mission (WISE; Wright et al., 2010) by combining data from the WISE cryogenic and NEOWISE (Mainzer et al., 2011) post-cryogenic survey. WISE has 4 bands at 3.4, 4.6, 12, and 22 $\mu$m (W1, W2, W3, and W4). The 5$\sigma$ limiting magnitudes of the AllWISE catalog in W1, W2, W3, and W4 bands are 19.6, 19.3, 16.7, and 14.6 mag. The Galactic extinction coefficients for W1, W2, W3 used in this study are $R_{W1},R_{W2},R_{W3}=0.1209,0.0806,0.124$. These coefficients are also calculated with relative extinction $A_{\lambda}/A_{V}$ values from Wang & Chen (2019).

We cross-match the PS1 sources with AllWISE using a radius of $1^{\prime\prime}$ to avoid source confusion in the dense fields of the Galactic plane. We also set a few constraints on the AllWISE data. All sources should be: (i) AllWISE point sources ($ext\_flg=0$); (ii) not too bright to avoid possible saturation ($W1>8\ \&\ W2>7$); (iii) significantly detected in $W1$ and $W2$ bands ($W1snr>5\ \&\ W2snr>5$); (iv) unaffected by prioritized image artifacts in each band ($cc\_flags=$“0000”); (v) unblended with nearby detections, so that only one component is used in each profile-fitting for each source ($nb=1$).

Gaia DR2 (Gaia Collaboration et al., 2016, 2018b) contains celestial positions and the apparent brightness in $G$ band for approximately $1.7$ billion sources. For $1.3$ billion of those sources, parallaxes and proper motions are available. Broad-band photometry in the $G_{\mathrm{BP}}$ (330–680 nm) and $G_{\mathrm{RP}}$ (630–1050 nm) bands are available for $1.4$ billion sources. We use the proper motions and their uncertainties from the Gaia DR2 catalog (columns pmra, pmra_error, pmdec, and pmdec_error) to find quasars.

SDSS (York et al., 2000) has mapped the high Galactic latitude northern sky and obtained imaging as well as spectroscopy data for millions of objects including stars, galaxies, and quasars. The 14th data release of the SDSS Quasar Catalog (SDSS DR14Q; Pâris et al., 2018) contains 526,356 quasars. We cross-match the DR14Q catalog with PS1 and AllWISE both with a radius of $1^{\prime\prime}$. To ensure the data quality, we use the same constraints in Section 2.1 and Section 2.2 to retrieve a subset of DR14Q. This subset has 289,271 sources and is denoted as $GoodQSO$ hereafter. As can be seen from the HEALPix (Górski et al., 2005) density map of $GoodQSO$ (Figure 1), very few sources of $GoodQSO$ are located at $|b|\leq 20^{\circ}$.

In order to compare high-$b$ sources with Galactic plane sources that we use, a sample of stars and a sample of galaxies are extracted from SpecPhotoAll table of SDSS Data Release 15 (Blanton et al., 2017; Aguado et al., 2019). We cross-match both the star and galaxy sample with PS1 and AllWISE with a radius of $1^{\prime\prime}$. The SDSS star sample has 23,693 sources. We also apply quality constraints in Section 2.1 and Section 2.2 to select galaxy subset with good photometry for later use. The resulted subset of galaxy (denoted as $GoodGal$ hereafter) has 1,635,053 sources. Most SDSS stars and galaxies are located at high Galactic latitude ($|b|>20^{\circ}$).

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST, also called the Guoshoujing Telescope) is a special reflecting Schmidt telescope, the design of which allows both a large effective aperture of 3.6 m–4.9 m and a wide field of view of $5^{\circ}$ (Wang et al., 1996; Su & Cui, 2004; Cui et al., 2012). The LAMOST spectral survey (Zhao et al., 2012; Luo et al., 2012, 2015) consists of two major components, i.e. the LAMOST Experiment for Galactic Understanding and Exploration (LEGUE; Deng et al., 2012), and the LAMOST ExtraGAlactic Survey (LEGAS). The LEGUE observes stars in different sky regions with different magnitude ranges, including the Galactic halo with $r<16.8$ mag at $|b|>30^{\circ}$, the Galactic anti-center with $14.0<r<17.8$ mag at $150^{\circ}\leq l\leq 210^{\circ}$ and $|b|<30^{\circ}$ (Yuan et al., 2015), as well as the Galactic disk with $r\lesssim 16$ mag at $|b|\leq 20^{\circ}$ with uniform coverage along Galactic longitude. The LEGAS mainly identifies galaxies and quasars that are within the SDSS footprint but complementary to the SDSS spectroscopic samples (e.g. Shen et al., 2016; Yao et al., 2019). Nevertheless, extragalactic objects in the LEGUE plates are also targets of the LEGAS. The LAMOST spectral survey has obtained the largest stellar spectra sample to date. We retrieve star sample from LAMOST general catalog from DR1 to DR7v0. A total number of 3,940,076 LAMOST stars meet the the same constraints in Section 2.1 and Section 2.2. From this LAMOST star sample, we select 1,334,577 Galactic plane stars with $|b|\leq 20^{\circ}$ (denoted as $T_{Star}$ hereafter). Most $T_{Star}$ sources are from the LEGUE survey and are brighter than 18 mag in $i_{P1}$ band.

The Million Quasars (Milliquas) Catalog (Flesch, 2019) is a compilation of quasars and quasar candidates from the literature. The Milliquas v6.4c update includes 758,908 type-I QSOs and AGN up to 31 December 2019. We use this catalog to extract extant GPQ sample within PS1 footprint. There are 4,344 quasars (with “Q” label in the “Descrip” column) located in $|b|\leq~{}20^{\circ}$ in Milliquas v6.4c. Cross-matching these 4,344 known GPQs with PS1 and AllWISE both with a radius of $1^{\prime\prime}$ gives 2,757 sources. After applying same constraints as in Section 2.1 and Section 2.2, we get a subset of 1,853 sources. This Galactic plane subset of Milliquas quasars, denoted as $MLQSUB$, will be used later for candidate validation.

The task of quasar selections can be described by classification problems in machine learning. Here we look into the three-class classification for stars, galaxies, and quasars with photometric data. The learning process requires two independent datasets for model training and model validation respectively. Training and validation sets can be two nonoverlapping subsets from a common parent sample with both features (colors and/or magnitues) and class labels (star, galaxy, and quasar). Usually the class labels are given by spectroscopic identifications. The classification algorithm learns a mapping relation from features to class labels with the training set. Often, the trained classification model (classifier) is applied to another dataset without class labels (i.e. no spectroscopic identifications), which is called application set or test set. The classifier takes features from the test set as inputs $X$ (a.k.a. covariates) and gives class labels as outputs $Y$.

A basic assumption for traditional machine learning is that training and test data follow the same probability distribution (Bishop, 2006; Hastie et al., 2009; Vapnik, 2013). However, this assumption no longer holds if we use high-$b$ data for model training and low-$b$ data for application, because the joint distribution of inputs and outputs $P(X,Y)$ differs between training and test data (i.e. dataset shift; Quionero-Candela et al., 2009).

For our GPQ selections, the dataset shift includes changes in both source colors and prior probabilities of different classes. Sources in the Galactic plane become fainter and redder than those at high Galactic latitude due to greater reddening, which changes the distribution of input features and the conditional probability of the output labels given the inputs $P(Y|X)$ (i.e. covariate shift; Shimodaira, 2000; Sugiyama & Kawanabe, 2012). Prior probabilities of stars are much higher than those of quasars (and galaxies) in the Galactic plane, which means the marginal probability ${P(Y)}$ differs from that at high Galactic latitude (i.e. class-balance change; Saerens et al., 2002; Du Plessis & Sugiyama, 2014). Moreover, class ratio between extragalactic objects and stars may vary significantly from one place to another in the Galactic plane, which we refer to as “internal” class-balance change of the test data.

Transfer learning can be applied to improve the learning performance under dataset shift from a source domain to the target domain (see a review in Pan & Yang, 2009), where domain is a set $\mathcal{D}$ that consists of a feature space $\mathcal{X}$ and a marginal probability distribution $P(X)$, $\mathcal{D}=\{\mathcal{X},P(X)\}$. For our classification task, the source domain data are from high Galactic latitude ($|b|>20^{\circ}$) and the target domain data are from the Galactic plane ($|b|\leq 20^{\circ}$). In this study we only care about areas at $\delta>-30^{\circ}$ due to the limit of PS1 survey coverage. Comparison of some properties of the source and target domains are listed in Table 1.

As large numbers of stars, quasars and galaxies have been spectroscopically identified at $|b|>20^{\circ}$, labels for these three classes are available in the source domain. Since spectroscopically identified samples of quasars and galaxies are significantly lacked at $|b|\leq 20^{\circ}$, labels for these two classes are unavailable in the target domain. Nevertheless, labels of many stars in the target domain are available with the help of LAMOST spectroscopic survey.

According to the classification scheme for different settings of transfer learning by Pan & Yang (2009), the set-up of classification in the Galactic plane can be categorized into Transductive Transfer Learning, where source domain labels are available and target domain labels are unavailable. A popular approach to Transductive Transfer Learning is Feature-based Transfer (e.g. Blitzer et al., 2006; Argyriou et al., 2006), which reduces the difference between the source and target domain through feature transformation in either one or both of the domains.

To solve the dataset shift problem of classification in the Galactic plane, we borrow the idea of Feature-based Transfer Learning. Using the mapping relation between the features of high-$b$ and low-$b$ objects, we can generate mock samples of quasars and galaxies in the Galactic plane to simulate the covariate change of their colors and magnitudes. The LAMOST Galactic plane stars also contribute to a more accurate probability distribution of data in the target domain. To reduce the effect of class-balance change, we manually go through two binary classification steps rather than running a three-class classification algorithm only once.

As data of LAMOST Galactic plane stars are available, we only focus on reducing the differences in features of extragalactic objects between training and test data. For our classification problem, all features will be constructed with photometric data from PS1 and AllWISE. We assume that the differences in photometric properties between extragalactic objects in the Galactic plane and those off the plane are only caused by different extinctions/reddening along their sight-lines. In this way, we can simply generate mock extragalactic objects behind the Galactic plane with data obtained at high Galactic latitude, using the mapping relation determined by the Galactic extinction law and Galactic dust map.

The covariate change can then be shown as color change of extragalactic objects on a set of color-color diagrams. Our classification will perform better by adding mock samples of quasars and galaxies behind the Galactic plane into the training set.

With the data-level improvements above, the covariate change can be reduced. Efforts on the algorithm level are required to handle the class imbalance and class-balance change. During the GPQ selections, instead of performing a star-quasar binary classification, we additionally take galaxies into account and perform a three-class classification.

Many machine learning software packages support multi-class classification jobs, by transforming the task into multiple binary classification problems. However, the built-in treatment is often inflexible and sometimes destructive when dealing with class imbalance problems. For example, in the scenario of using one-vs-rest (also known as one-vs-all) strategy for multi-class classification, at some stages, samples of one class are regarded as the positive samples while all samples of other classes are regarded as negative samples. Even if all the classes in the training set have a same sample size, the binary classification situation is imbalanced as the positive class (the “one”) has less samples than the negative class (the “rest”). In our case, the GPQ set (in both training and test set) has significantly less samples than the sets of galaxies and stars, thus severe class imbalance will happen.

To reduce the disadvantage of one-vs-rest strategy which is commonly used in machine learning algorithms, we convert this three-class classification problem into two binary classification problems manually. In the first step, the Galactic plane sources are classified into two classes: stars and extragalactic objects. Extragalactic objects are then classified into quasars and galaxies in the second step. By combining the two minority classes of quasar and galaxy into one, we would expect the class imbalance to be better controlled in the first step. The physical basis for merging the quasar and galaxy classes is that quasars are a special type of galaxies. For the second step, we expect that the quasar-to-galaxy ratio to be nearly constant across different locations in the Galactic plane. Thus the variable quasar-to-star or galaxy-to-star ratio is avoided and the internal class-balance change is lessened in the learning process.

Let $E$ be a set of extragalactic objects ($E$ can be $GoodQSO$ or $GoodGal$). The synthesis process consists of following steps.

1. Correcting for extinctions. Extinctions of objects in set $E$ are corrected according to a two-dimensional dust map provided by Planck Collaboration et al. (2014, hereafter Planck14), and the optical to mid-IR extinction law from Wang & Chen (2019) with $R_{V}=3.1$. The $E(B-V)$ values are retrieved using a Python module, dustmaps (Green, 2018).

2. Assigning new locations. We generate a random sample of points that are uniformly distributed on the sky with $|b|\leq 20^{\circ}$. The number of these random points is equal to the sample size of $E$. Coordinates of these points are randomly assigned to objects of $E$ as their new locations. Now we get a new set $E_{m}$ ($MockGPQ$, $MockGal$) without line-of-sight extinctions.

3. Adding new extinctions. We add extinctions to the $E_{m}$ sample using the Planck14 dust map based on their new (mock) locations.

4. Setting limiting magnitudes. We obtain a subset of $E_{m}$ by choosing sources brighter than the PS1 single epoch 5$\sigma$ depths in all PS1 passbands: $(grizy_{P1})<(22.0,\ 21.8,\ 21.5,\ 20.9,\ 19.7)$. This subset, denoted as $E_{gm}$ ($GoodMockGPQ$, $GoodMockGal$), represents “good” mock sample that can be detected by PS1 survey in all bands. However, we don’t apply similar constraints to AllWISE bands as the magnitude which corresponds to a 5$\sigma$ sensitivity varies with location. Also, this extinction-selection effect relies more on the optical survey depth than the IR survey depth. Factors such as observation strategies and source confusions in dense fields are not taken into consideration in this step. Therefore we may overestimate the detection rate of GPQs (and galaxies) through this synthesis. We select sources within the PS1 footprint (i.e. $\delta\geq-30^{\circ}$) and obtain set $E_{gm\textnormal{-}\mathrm{PS1}}$ ($GoodMockGPQ\textnormal{-PS1}$, $GoodMockGal\textnormal{-PS1}$).

5. Constructing training sets with mock and real data. For mock quasars that are not included in $GoodMockGPQ$-PS1, their original counterparts (high-$b$ quasars in the input set $GoodQSO$; denoted as $C_{QSO}$) are also added to the training and validation sets along with $GoodMockGPQ$-PS1. For mock galaxies that are not included in$GoodMockGal$-PS1, 25% of their original counterparts (high-$b$ galaxies in the input set $GoodGal$; denoted as $C_{Gal}$) are added to the training and validation sets. The resulted quasar and galaxy samples for training and validation are denoted as $T_{QSO}$ and $T_{Gal}$, respectively. $T_{QSO}$, $T_{Gal}$, and the LAMOST Galactic plane star sample $T_{Star}$ form the training and validation sets for machine learning classification.

By adding good mock samples and real data ($C_{QSO}$ and $C_{Gal}$) together instead of using only good mock samples as training data for quasars or galaxies, we increase the data diversity as well as sample size of the training set. This data diversification ensures that the training set can provide more discriminative information for the machine learning model (Gong et al., 2019). In addition, more training data can help reduce overfitting.

The flowchart of the synthesizing procedures is displayed in Figure 2.

In the synthesizing process, we adopt the Planck14 dust map because it detects dust at a greater depth and better estimates the 2D extinctions in the Galactic plane than do the dust maps constructed with stellar photometry (e.g. Green et al., 2018, 2019). We assume a uniform $R_{V}=3.1$ although $R_{V}$ varies slightly in the Galaxy with a dispersion of about 0.18 (Schlafly et al., 2016). Such minor variations in $R_{V}$ can lead to small uncertainties of magnitudes and colors of individual mock quasars (galaxies), but have limited impacts on the statistical properties of the training sample because mock sources with large extinctions (and thus large uncertainties caused by $R_{V}$ variations) are removed by the magnitude limits, as we shall see in Section 4.2.

We define the extinction-based selection rate in the Galactic plane as $R=|E_{gm}|/|E|$, where $|E|$ is the cardinality, i.e. number of elements/sources of set $E$. The source numbers of the input samples are $|GoodQSO|=289,271$ and $|GoodGal|=1,635,053$; the source numbers of the output samples are $|GoodMockGPQ|=101,482$ and $|GoodMockGal|=771,392$. Therefore the selection rates for GPQs and galaxies are $R_{GPQ}=|GoodMockGPQ|/|GoodQSO|=0.35$ and $R_{Gal}=|GoodMockGal|/|GoodGal|=0.47$, respectively. The selection rate of galaxies is higher than that of GPQs because the input galaxies are on average brighter than the input quasars. With Step 2 in Section 4.1, sources of $MockGPQ$ and $MockGal$ are randomly and evenly distributed in the Galactic plane ($|b|<20^{\circ}$). But after Step 4, the densities of remaining sources ($GoodMockGPQ$ and $GoodMockGal$) are inversely related to the dust map (Figure 3): more extragalactic sources remain detectable in regions with smaller $E(B-V)$, and voids of detection present at regions with large $E(B-V)$. A sky survey deeper than PS1 might help make up some fraction of the gap in the middle of the Galactic plane. The $GoodMockGPQ$ sample is sparser compared to $GoodMockGal$, simply because the input quasars are fewer than galaxies. Most sources of $GoodMockGPQ$ and $GoodMockGal$ have line-of-sight color excess of $E(B-V)<1.5$, which corresponds to extinction of $A_{V}<4.65$ with $R_{V}=3.1$. The medians of line-of-sight $E(B-V)$ of $GoodMockGPQ$-PS1 and $GoodQSO$ are 0.21 and 0.03, respectively. In general, $GoodMockGPQ$-PS1 sample has significantly larger $E(B-V)$ compared to $GoodQSO$ (see Figure 4 (a)). Therefore covariate change for color indexes from high-$b$ to low-$b$ regions cannot be ignored. In addtion, $GoodMockGPQ$-PS1 sources are fainter than $GoodQSO$ sources (Figure 4 (b)).

A series of color-color diagrams for $GoodQSO$ and $GoodMockGPQ$-PS1 along with SDSS stars and Galactic plane point sources are shown in Figure 5 and Figure 6. In Figure 5, from the left to the middle panels, the covariate change of colors of quasars from high Galactic laititude to the Galactic plane can be directly observed. The Galactic reddening makes the cluster of GPQs in a color-color plane extend towards redder colors (to the upper right along the reddening vector) and scatter more than high-$b$ quasars. The scattering is greater in color indexes of bluer bands, while less at redder bands. This trend is also observable in the quasar evolutionary tracks with $E(B-V)=0,0.75,1.5$. From the top to the bottom panels in Figure 5, the distance between two quasar evolutionary tracks with different reddening decreases.

The covariate change of stellar colors is also evident from the color-color diagrams. The stellar loci are simple and clear for high-$b$ (SDSS) stars (see Figure 5 (1a, 2a, 3a)). However, additional spikes along the direction of increasing $E(B-V)$ appear in the stellar loci of Galactic plane stars due to reddening, as can be seen from Figure 5 (1b, 1c, 2b, 2c). Therefore we expect to better distinguish stars from other sources using Galactic plane stars from LAMOST instead of high-$b$ SDSS stars in the training set.

Since mid-IR bands are less sensitive to extinction and reddening, the covariate change of AllWISE colors are less obvious than that of PS1 colors. For instance, in Figure 6 (a), the quasar evolutionary tracks with $E(B-V)=0,0.75,1.5$ stay very close to each other. Since the AllWISE colors of quasars do not change much from high Galactic latitude to the Galactic plane, we can use the $W1-W2$ versus $W2-W3$ color-color diagram to examine the purity of the final quasar candidates, by comparing probability distributions of the candidates and $GoodMockGPQ$-PS1 sources.

The AllWISE color-color diagram also gives “hardness” information on the classification problems that separating quasars from galaxies is harder than separating quasars from stars. In general, quasars have redder $W1-W2$ and $W2-W3$ colors than stars and galaxies due to the power-law SEDs and hot dust of quasars. From Figure 6 (b), we can recognize stellar locus ($W1-W2\approx 0$; on the lower-left) and galaxy locus ($W1-W2\approx 0.5,\ W2-W3\approx 3.5$; on the middle-right) from the density plot. The quasar reference contour line marks a “quasar region” where most quasars are located in the $W1-W2$ versus $W2-W3$ diagram. Most stars are away from the quasar region, while a large number of galaxies enter the quasar region, indicating that such galaxies can contaminate the mid-IR quasar selection.

Some comparisons between mock quasars ($GoodMockGPQ$-PS1) and mock galaxies ($GoodMockGal$-PS1) behind the Galactic plane are also shown in Figure 7. In the color-color diagrams of PS1 bands, galaxies largely overlap with quasars (Figure 7 (a, b, c)); while on the $W1-W2$ versus $W2-W3$ plane, these two classes are slightly more separable (Figure 7 (d)). Except for the colors, the difference between PSF magnitude and Kron (1980) magnitude is often used as a morphological separator (Strauss et al., 2002; Farrow et al., 2014) for galaxies and point sources including quasars. However, separating quasars from galaxies becomes harder with $i_{\mathrm{PSF}}-i_{\mathrm{Kron}}$ at the faint end (Figure 7 (e)), as has been pointed out by Yang et al. (2017). Among all the $\sim 1.6$ million $GoodGal$ sources, $\sim 200$ are point sources with $i_{\mathrm{PSF}}<18$ and $i_{\mathrm{PSF}}-i_{\mathrm{Kron}}<0$, which can also been seen from Figure 7 (e). These point sources include few quasars with “galaxy” labels and may also include some stars that are misclassified as galaxies. We do not pay attention to these point sources because they only contribute to a tiny fraction of the whole galaxy sample.

To sum up, we examine the properties of $GoodMockGPQ$, $GoodMockGal$ and PS1-AllWISE point-like sources in the color-color spaces. For quasar candidates selection, contamination from both stars and galaxies should be taken care of. Simple PS1 color cuts are only capable of selecting quasars that are away from the stellar loci. Using a series of PS1 colors in high-dimensional space might help reduce the overlap between the stellar loci and clusters of quasar and galaxy. Moreover, with AllWISE colors, quasars can be better separated from stars and galaxies. Therefore we expect that the combination of PS1 and AllWISE data will make quasar selection more efficient.

An estimation on the sky density of GPQs will be useful for evaluating the final GPQ candidate sample and the selection method. However, the $GoodMockGPQ$ sky distribution in the middle panel of Figure 3 does not reflect the true density of GPQs due to two reasons: (i) the synthesizing process does not consider the source crowdedness and its effects on the photometric data quality; (ii) the source number of $GoodMockGPQ$ only depends on the size of input $GoodQSO$ sample when the dust extinction map is fixed.

Let the density of $GoodMockGPQ$ be $D_{\mathrm{old}}$, then the relative density of quasars with good photometry across the Galactic plane is:

The predicted marginal probability of GPQs to the PS1-AllWISE sample with good photometry is $D_{\mathrm{new}}/D_{\mathrm{goodph}}$, which ranges from $2\times 10^{-4}$ to 0.17 with a median of $3\times 10^{-3}$. The maximum value of 0.17 is not reliable because it locates at edges of the HEALPix map ($\delta\sim-30^{\circ}$), where source count in a pixel does not correspond to the true number in the sky region.

In the first classification step, the input data for training and validation consist of synthetic quasar sample $T_{QSO}$, synthetic galaxy sample $T_{Gal}$ (see Section 4.1), and LAMOST Galactic plane star sample $T_{Star}$ (Section 2.6). The input data have more than 3 million rows. For binarization, we assign the label EXT (extragalactic object) to all $T_{QSO}$ and $T_{Gal}$ instances, and keep the label for $T_{Star}$ as STAR. Here we regard extragalactic objects as the positive class and stars as the negative class.

We first apply five-fold cross validations with optuna to find the optimal setting of hyperparameters that minimizes the log loss among 500 trials. For a binary classification problem with a true label $y\in\{0,1\}$ and a probability estimate $p=\mathrm{Pr}(y=1)$, the log loss per sample is the negative log-likelihood of the classifier given the true label:

Some fixed parameters in our programs are: objective=binary:logistic; booster=gbtree; tree_method=hist. For hyperparameters that are tuned, the default values, optimal values found by the cross validations, and corresponding metric scores of these parameters are listed in Table 2. The number of boosting rounds (num_boost_round, a.k.a. n_estimators in scikit-learn API of XGBoost) is fixed to 100 and not tuned together with eta (a.k.a. learning_rate), because the effects of increasing num_boost_round can cancel that of decreasing eta, and vice versa. In the training process, we need to lower the learning rate eta and increase the num_boost_round to reduce the generalization error. The Classifier No.1 (CLF-1) is trained using $\mathtt{eta}=0.02$, $\mathtt{num\_boost\_round}=1,200$ with other optimal parameters in Table 2.

We then classify the PS1-AllWISE point-like sources with CLF-1. To exclude as many stars as possible, we adopt a high threshold on $p_{\mathrm{EXT}}$ (model-predicted probabilities of sources for being extragalactic) to select extragalactic candidates. Sources with $p_{\mathrm{EXT}}>0.99$ are labeled as EXT, and the others are labeled as STAR and removed.

We use $T_{QSO}$ and $T_{Gal}$ samples as input data for training and validation in the second classification step. Here we regard quasars as the positive class and galaxies as the negative class.

The same processes of parameter tuning and training as those of CLF-1 are applied to build CLF-2. We keep some parameters unchanged as: objective=binary:logistic; booster=gbtree; tree_method=hist. For hyperparameters that are tuned, the default values, optimal values found by the cross validations, and corresponding metric scores of these parameters are listed in Table 3. The CLF-2 is trained using $\mathtt{eta}=0.02$, $\mathtt{num\_boost\_round}=1,500$ with other optimal parameters in Table 3.

The optimal scores of the six metrics in Table 3 are all lower than those in Table 2, indicating that the quasar–galaxy classification “hardness” is higher than that of star–extragalactic problem. Here we also use a high threshold of probability to select sources of our target class. We classify the sources labeled as EXT with CLF-2. Sources with $p_{\mathrm{QSO}}>0.95$ are kept as GPQ candidates, where $p_{\mathrm{QSO}}$ is the probability of a source for being a quasar predicted by the XGBoost model.

In the first classification process, we classify all PS1-AllWISE point-like sources to stars and extragalactic objects. We ignore stars in the second classification step. Although the metrics of CLF-1 are high (Table 2), some stars can be misclassified as extragalactic objects, and then be classified either as quasars or galaxies. Faint stars are more likely to be misclassified than bright stars because stars in the training sample ($T_{Star}$) are biased towards the bright end. When using optical and near-IR colors for candidate selection, white dwarfs are major contaminants for low redshift quasars, and M/L/T dwarfs are typical contaminants for high redshift quasars (e.g. Kirkpatrick et al., 1997; Vennes et al., 2002; Chiu et al., 2006). In the mid-IR regime, potential stellar contaminants for quasars are Young Stellar Objects (YSO), Asymptotic Giant Branch (AGB) stars, and Planetary Nebulae (PNe) (Kozłowski & Kochanek, 2009; Koenig & Leisawitz, 2014; Assef et al., 2018).

The YSOs are stars at the early stages of evolution, and are often divided into four subclasses (Lada, 1987): Class I, Class II, Flat spectrum, and Class III. Among them, Class II and Flat spectrum YSOs are the most-likely contaminants since they have optical and mid-IR SEDs similar to those of quasars. Since we require both optical and mid-IR detections for classification, optically faint Class I YSOs are eliminated in the first place. As has been studied by Koenig & Leisawitz (2014), Class III YSOs are clustered around $W2-W3=0$ and $W1-W2=0$, while Class I, II, and Flat spectrum YSOs occupy the region approximately with $W1-W2>0.25$ and $1.0<W2-W3<4.5$ (see their Figure 5). The latter YSO region is overlapped with the quasar region shown in Figure 6, therefore the contamination should be taken care of.

AGB stars are evolved stars with low temperatures and high luminosities. They are surrounded by circumstellar envelopes, and IR excess exists in their broad SEDs. According to Figure 5 from Koenig & Leisawitz (2014), only a minority of AGB stars actually overlap with Class I, II, and Flat spectrum YSOs (and thus quasars) in the $W1-W2$ versus $W2-W3$ diagram. Therefore we expect the contamination from AGB stars is less than that from YSOs.

The PNe have a series of narrow emission lines as well as IR excess. Known PNe can be later removed from the GPQ candidate sample by cross-matching with the Simbad database (Wenger et al., 2000).

In order to remove stellar contaminants such as white dwarfs, M/L/T dwarfs, YSOs, and AGB stars from GPQ candidates, we apply an additional cut based on Gaia proper motion, because the proper motion distribution of quasars is different from that of Milky Way stars. Although quasars should have negligible transverse motions, non-zero proper motions of them are measured by Gaia due to various effects, such as photocenter variability of quasars (see Bachchan et al., 2016, and references therein). In addition, proper motions with large uncertainties are not reliable. Therefore we need a probabilistic cut instead of a cut on the total proper motion. We define the probability density of zero proper motion ($f_{\mathrm{PM0}}$) of a source based on the bivariate normal distribution of proper motion measurements of the source as:

We take the logarithm of $f_{\mathrm{PM0}}$ for better comparison between samples. Figure 10 shows distributions of $\mathrm{log}(f_{\mathrm{PM0}})$ of stars, galaxies and quasars used in this study. For stellar samples, in addition to $T_{Star}$ (LAMOST Galactic plane star sample), a subsample of the SDSS Stripe 82 Standard Star Catalog (hereafter S82 star; Ivezić et al., 2007) that meets the same constraints in Section 2.1 and Section 2.2 is also included for comparison. We choose a $\mathrm{log}(f_{\mathrm{PM0}})\geq-4$ cut that excludes 94.1% of both LAMOST Galactic plane stars and S82 stars, while retains 99.8% of the quasars. Nevertheless, faint stars can be major contaminants even with such strict cut on $\mathrm{log}(f_{\mathrm{PM0}})$.

We calculate $\mathrm{log}(f_{\mathrm{PM0}})$ for GPQ candidates after cross-matching them with Gaia DR2. For candidates without Gaia DR2 proper motion records, we assign a default value of 99 for $\mathrm{log}(f_{\mathrm{PM0}})$. Sources with $\mathrm{log}(f_{\mathrm{PM0}})\geq-4$ are kept as reliable GPQ candidates.

With our Transfer Learning Framework and aforementioned additional selection criteria, we obtain a reliable GPQ candidate sample with 161,532 sources from PS1 and AllWISE. We cross-match the GPQ candidates with Simbad database (Wenger et al., 2000), and find 2,786 matches. The object types and summary are shown in Table 4.

We categorize all matched sources to four groups: AGN/QSO, star (including PNe and PN candidates), galaxy, and other-type objects. Among all the matches, 53.98% (1,504) are recorded as AGN/QSO (including candidates), 8.97% (250) are recorded as star (including candidates), 4.02% (112) are recorded as galaxy, and 33.02% (920) are other-type objects labeled according to the detection properties (e.g. wavelength). Those other-type objects have higher probabilities to be AGN/QSO than stars, as most (728+27) of them are radio sources, and 64 are X-ray sources (see Table 4). For the 4.02% sources labeled as galaxy, a number of them may also host AGN/QSO as we have applied careful selection criteria to remove possible galaxy contaminants. Among the 250 sources labeled as star, 40 are candidates, and the other 210 are known stars. 101 of the known stars were once selected as QSO candidates using SDSS photometry, and then identified as stars by the 2dF-SDSS LRG and QSO Survey (Croom et al., 2009). From these analyses, we can conclude that, the purity of our GPQ candidates on the small subset of 2,786 Simbad matches can be as high as $\sim 90$%. The true purity may vary at different locations in the Galactic plane.

As another test on stellar contamination, we cross-match our GPQ candidates with LAMOST Galactic plane star sample. This match identifies 29 LAMOST stars, all of which are not recorded in Simbad. Therefore the total number of known stars in the GPQ candidates is 239.

We also examine the fraction of known GPQs that can be recovered with our candidates table. The known GPQ sample is $MLQSUB$ with 1,853 sources, which is retrieved from Milliquas catalog and described in Section 2.7. The $MLQSUB$ is selected with same constraints as those on our application PS1-AllWISE data, to get a consistent analysis result. Cross-matching $MLQSUB$ with our GPQ candidates results in 1,763 matches, meaning that 95.14% of GPQs from Milliquas can be selected with our methods, under same quality constraints on the photometric data. The recent sixteenth data release of SDSS Quasar Catalog (DR16Q; Lyke et al., 2020) has a total of 750,414 sources, in which 3,727 sources are located at $|b|<20^{\circ}$. Only 1,320 of these SDSS GPQs meet the photometric quality constraints in Section 2.1 and Section 2.2. Cross-matching our GPQ candidates with SDSS DR16Q gives 1,292 matches, which corresponds to a recall rate of 97.88% under the same photometric quality constraints, or a recall rate of 34.67% to the whole identified sample. The overall completeness of the sample of candidates is mainly limited by the photometric quality constraints.

We remove 239 known stars (see Section 7.1) from our GPQ candidate sample. We then match the remaining GPQ candidates by coordinates with TOPCAT internally, and found 347 close pairs within $0.2^{\prime\prime}$. These pairs are very likely duplicated sources, because the PS1 survey can not resolve two sources within an angular distance of $0.2^{\prime\prime}$. The median image quality for PS1 $3\pi$ survey is FWHM = (1.31, 1.19, 1.11, 1.07, 1.02) arcseconds for ($grizy_{\mathrm{P1}}$) (Magnier et al., 2016). Therefore we only keep one source for each close pair, and obtain the final GPQ candidates sample with ${{160,946}}$ sources. The GPQ candidate catalog is compiled based on this sample, with photometric data from PS1 DR1, AllWISE, and astrometric data from Gaia DR2. The descriptions for the catalog are displayed in Table 5.