Dwarfs or Giants? Stellar Metallicities and Distances from ugrizG Multiband Photometry

Figure 3 provides a schematic overview of the algorithm that we have developed. This algorithm only uses the photometric data described in the previous section to first disentangle giants from dwarfs using a Random Forest Classifier (RFC; Breiman 2001), as detailed in Section 3.2. Once this classification is done, two sets of artificial neural networks (ANNs), one for the stars identified as dwarfs (Section 3.3.1) and another for the stars identified as giants (Section 3.3.2), are used to determine the photometric metallicity. Finally, another two sets of ANNs are used to determine the absolute magnitude of the two groups of stars in the G band.

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To calculate the uncertainties on the different parameters estimated by our algorithm (P_Dwarf, P_Giant, [Fe/H], and M_G) caused by the photometric uncertainties of the different bands, we generate 20 Monte Carlo realizations for each star. We also conducted 100 and 1000 Monte Carlo realizations for a subsample of 50,000 randomly selected stars. For these, we obtained final uncertainties that were typically <1% different than what we obtained using only 20 realizations. As such, we proceeded with 20 Monte Carlo realizations for the entire sample in order to save expensive computational time. For each realization, we select a magnitude in each band from a Gaussian distribution centered on the quoted magnitude and with a standard deviation equal to the uncertainty on the magnitude. The uncertainties on the derived parameters are set equal to the standard deviation for each parameter from these 20 realizations.

The CFIS-u footprint, which defines the spatial coverage of this study, includes a large number of stars observed by SDSS/SEGUE (Yanny et al. 2009b) and for which spectroscopic data are available. We use those stars with good-quality spectroscopic measurements as training sets for the first two components of our algorithm. It is advantageous to use as large a training set as possible; therefore, we select 74,442 SDSS/SEGUE stars present in the CFIS-u footprint that have a spectroscopic signal-to-noise ratio (S/N) ≥ 25. This threshold was chosen because at lower S/N the distribution of the uncertainties on the parameters given by the SEGUE Stellar Parameters Pipeline (SSPP) as a function of the S/N is irregular, indicating that the parameters are poorly defined. Moreover, more than 96% of the stars with S/N < 25 have a parallax measurement with poor precision (>20%), and these would not be used to calibrate the photometric distance relation even if they passed the S/N cut. For the third component of our algorithm (determination of the absolute magnitude), we use parallax information from Gaia (discussed later).

We first perform a color–color cut to remove A-type stars (which lie in the "comma-shaped" region in the color–color diagram of Figure 4) and white dwarfs, which is defined by inspection of Figure 4 and shown as the orange box. The presence of the Balmer jump in the u band for these stars means that they have a more complex photometric behavior compared to the other stars present in this color–color diagram, and the algorithms are significantly simplified if we remove them from consideration. We note that A-type stars in CFIS have been studied extensively in previous works (Thomas et al. 2018, 2019), and an analysis of the white dwarfs is in preparation (N. Fantin et al. 2019, in preparation). Imposing this color–color selection¹¹ on the SDSS/SEGUE spectra led to a catalog of ∼42,800 stars for which we have astrometric, photometric, and spectroscopic information. This color–color cut represents the first step of our procedure and is applied to both the test/training set and the final main catalog (see Section 4.4).

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Here we describe the method used to disentangle red giant branch (RGB) stars from MS stars.¹² To perform this classification, we use an RFC, whose inputs are the (u − g)₀, (g − r)₀, (r − i)₀, (i − z)₀, and (u − G)₀ colors normalized to have a mean of zero and a standard deviation of one and the outputs are the probability of each to be a dwarf, P_dwarf, or a giant, P_giant ≡ 1 − P_dwarf. It is worth noting here that RFC does not necessarily request a normalization of the inputs, but it is strongly suggested for an ANN. Therefore, to be consistent with the different steps of the algorithm, we use the normalized inputs for the dwarf/giant classification.

According to Lee et al. (2008), the typical internal uncertainty obtained by the SSPP on the adopted surface gravity (loggadop) is ∼0.19 dex. Therefore, we keep only the SDSS/SEGUE stars that have uncertainties δ log(g) ≤0.2. The final SDSS/SEGUE catalog used to train/test the dwarf−giant classifier contains 41,062 stars. This catalog is shown as a Kiel diagram in Figure 5, where we define the stars lying in the orange polygon as giants and all other stars as dwarfs. Note that we could in principle use Gaia DR2 parallax measurements to classify the stars as dwarfs or giants; however, as we will show later, the parallax measurements of Gaia DR2 are not precise enough, especially for stars with [Fe/H] < −1.0, many of which are generally located at large distances and have poor parallax precision.

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We note that our "giant" selection contains a large majority of RGB stars but also subgiant stars, with an effective temperature between 5000 ≤ T_eff (K) ≤ 6000 and surface gravity between 3.2 ≤ log(g) ≤ 3.9. Even on a Kiel diagram, it is hard to define a strict limit between dwarfs, subgiants, and giants, especially when we include the uncertainties on the surface gravity, which act to blur any boundaries we adopt. The addition of a subgiant class adds to the complexity of the algorithm and does not resolve the underlying problem of the "fuzzyness" between classes. For this reason, we do not consider a specific class of subgiant stars. We also note that our definition of giants does not include the majority of asymptotic giant branch (AGB) stars. However, there are very few AGB stars in the SDSS/SEGUE catalog, and so we are unable to train the algorithm to identify them. Like all supervised machine-learning algorithms, we are ultimately limited by the representative nature of our training set. AGB stars are, however, very rare, and so their absence from our training set does not have a major statistical effect on our results.

We create a training and a test set from the SDSS/SEGUE catalog, composed of a randomly selected 80% and 20% of the sample, respectively. The training set is used to find the best architecture by a k-fold cross-validation method with five subsamples. It is then used to find the best parameters of the RFC that are then applied to the test set to check that the statistics of the two samples are similar. This technique prevents overfitting to the training set. We use the sklearn python package to find the best parameter of the RFC.

The completeness and the purity of the dwarf and giant classes (populated by stars with P_dwarf/P_giant > 0.5) for the test set are shown in Table 1. The values for the two classes are similar to those for the training set, which indicates that there is no overfitting of the data. The RFC classifies correctly the large majority of the dwarfs (96%), with less than 7% contamination by giant stars (note that since dwarfs make up 86% of our training/test set, this implies a factor of 2 improvement over random chance). For the giants, slightly more than half are correctly classified, with relatively low contamination, ∼30%. Thus, our completeness is approximately 4 times better than "random," and our contamination is nearly 3 times lower than "random." Figure 6 is a Kiel diagram of the expected/predicted probability for each star to be a dwarf, and we see that most of the contamination of the dwarfs is from subgiants, which are preferentially classified as dwarfs instead of giants.

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It is worth nothing here that, in Figure 6, the surface gravity from SDSS/SEGUE decreases with the temperature for the MS stars with an effective temperature lower than 4800 K. This is unexpected when compared to theoretical predictions, and it is likely a consequence of the poor determination of the surface gravity by the SSPP in this region. However, this has no impact on our classification, since all the stars in this region are correctly identified as dwarfs by the algorithm.

The detailed performance of our classification scheme as a function of metallicity, surface gravity, effective temperature, and (u − G)₀ color is shown in Figure 7. The completeness and contamination are mostly constant in the range of temperature and color covered by the giants. The completeness and contamination are also constant with metallicity up to [Fe/H] ≃ −1.2, after which the completeness drops rapidly (from 70% at [Fe/H] = −1.3 to 20% at [Fe/H] = −1.0), correlated with a dramatic increase in the contamination. We conclude that the classification works well for giants with metallicities below [Fe/H] = −1.2 but fails to identify the most metal-rich giants. There is also a drop in the completeness of the giants at a surface gravity of log(g) of 3.3, which corresponds to the subgiants being preferentially classified as dwarfs.

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The relative importance of each photometric color in the classification scheme, computed using the feature importances method implemented in the sklearn package, is shown in Figure 8. This method uses the weight of each feature in each node of the different trees of the RFC to measure the relative importance of such a feature for the classification. The most important feature is the (r − i)₀ color, with around one-fourth of the information used to classify the stars coming from it. This is not surprising since this color is a good indication of the effective temperature; therefore, we assume that this color is being used to select the temperature range that preferentially contains giants (4900 K ≤ T_eff ≤ 5500 K). The second most important feature is (u − g)₀, which has a tight correlation with the metallicity of MS stars (Ivezić et al. 2008; Ibata et al. 2017b). A similar correlation exists for the giants, albeit one with a different zero-point (Ibata et al. 2017b). The other colors, which account for ∼50% of the relative importance, presumably give additional minor complementary information (on the metallicity, temperature, and surface gravity) to disentangle the dwarfs from the giants, using the full shape of the SED.

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From Figure 8, we conclude that the dwarf/giant classification primarily uses photometric features that trace temperature and metallicity. This becomes clearer in Figure 9, where the locus of giants is obvious on the effective temperature–metallicity diagram (left panels), or on a color–color diagram using the two most important photometric features, the (r − i)₀ and the (u − g)₀ colors (right panels). The top two panels of Figure 9 show clearly that the SDSS/SEGUE sample does not contain a large number of metal-poor dwarfs in the temperature range that overlap with the majority of giants (4900 K ≤ T_eff ≤ 5500 K). The selection criteria for SDSS/SEGUE are generally complex (Yanny et al. 2009a). However, the absence of metal-poor dwarfs is exacerbated by the relatively shallow depth of the SDSS/SEGUE data set, which does not contain stars fainter than G ≃ 18 mag. This means that dwarfs are generally quite close and so are preferentially selected from the disk, which is much more metal-rich on average than the halo.

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As a consequence of this, the fraction of true dwarfs misidentified as giants increases drastically with the metallicity in this temperature region, as shown in Figure 10. The majority of true dwarfs with [Fe/H] < −1.5 are classified as giants by the algorithm. This should be compared to the overall sample, in which the fraction of misidentified dwarfs never exceeds 0.2 (and which is almost constant for metallicity lower than [Fe/H] = −1.0). Intriguingly, it is fascinating to note that even in the temperature range including the giants, more than 50% of true dwarfs (and true giants) are correctly identified between −1.45 ≤ [Fe/H] ≤ −1.1. This demonstrates that additional features, not just those relating to temperature and metallicity, are being used by the algorithm to classify dwarfs and giants. It seems reasonable to suppose that these features relate directly to the surface gravity of the stars, such as the Paschen lines present in the i, z, and G bands or the Ca H and K absorption lines in the u, g, and G bands (see Starkenburg et al. 2017).

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Our finding partially contradicts Lenz et al. (1998), who show that it is not possible to simultaneously separate cleanly stars by temperature, metallicity, and surface gravity using the SDSS filter set (which are broadly similar to the CFIS and PS1 filters), with the notable exception of A-type stars. However, their analysis did not take into account possible nonlinear relations between photometric colors and the relevant stellar parameters. By construction, our RFC accounts for nonlinearity in these relations, allowing us to use photometry to better predict the probability of a star being a dwarf or a giant. We also note that we experimented with different methods to classify dwarfs and giants, including ANNs and a principal component analysis (PCA). We found that the ANN gives similar results to the RFC, but the results were less easy to interpret; the PCA did not produce good results owing to its requirement of linearity.

With the advent of new spectroscopic survey in the northern hemisphere, such as WEAVE and SDSS-V, the number of metal-poor dwarfs with an effective temperature between 4900 K ≤ T_eff ≤ 5500 K that have spectra will likely increase. These future data will be excellent training sets to improve the dwarf/giant classification. Indeed, we will discuss later other issues with the training/test sets for which future spectroscopic data sets will likely provide essential improvements.

It is important to keep in mind that the giant sample produced by the algorithm may contain a nonnegligible fraction of actual metal-poor dwarfs, since dwarfs generally outnumber giants in any survey. However, in the critical temperature range (4900 K ≤ T_eff ≤ 5500 K), the difference in absolute magnitude between a true dwarf and true giant at the same color is at least ≃3 mag, equivalent to an incorrect distance of at least ≃150% (this error being even larger for redder stars). Thus, many of the misidentified dwarfs in our survey can be easily identified by consideration of their Gaia proper motions.

The next step of the algorithm is to determine, independently for each of the two classes, the photometric metallicity and the absolute magnitude (and thus the distance) of each star.

There have been many studies that estimate the photometric parallax of stars, especially MS stars (Laird et al. 1988; Jurić et al. 2008; Ivezić et al. 2008; Ibata et al. 2017b; Anders et al. 2019). This is possible because the MS locus has a well-defined color–luminosity relation. Using this property, Jurić et al. (2008) derived the distances of 48 million stars present in the SDSS DR8 footprint out to distances of ∼20 kpc using only r- and i-band photometry. However, this study did not take into account the effects of metallicity, which shifts the luminosity for a given color, more metal-rich stars being brighter than the metal-poor ones (Laird et al. 1988; but also see Figure 3 from Gaia Collaboration et al. 2018b). Age has less impact on the photometric parallax because its effect is to depopulate the bluer stars while maintaining the shape of the MS locus for the redder stars.

It is well known that it is possible to derive the photometric metallicity of a star by measuring its UV excess (Wallerstein & Carlson 1960; Wallerstein 1962; Sandage 1969), since metal-poor star have a stronger UV excess than metal-rich stars. Carney (1979) shows that is is possible to measure the metallicity of a star with a precision of 0.2 dex for stars with photometry better than 0.01 dex in the Johnson UBV filters. More recently, Ivezić et al. (2008) showed that it is possible to obtain a similar precision with the ugr SDSS filters and used it to derive the photometric parallax of 2 million F/G dwarfs up to 8 kpc, where the distance threshold is limited by the precision of the SDSS u band (see also Ibata et al. 2017b).

In this section, we build on this body of literature and present a data-driven method to estimate the metallicity and distances of dwarf stars (Section 3.3.1) and giants (Section 3.3.2).

3.3.1. Dwarf Stars

We first determine the photometric metallicity of the dwarfs before computing the distance of these stars. To determine the distance, we prefer to use the absolute luminosity derived using the Gaia parallaxes, rather than the parallaxes themselves, considering that a large number of Gaia parallaxes are negative owing to the stochasticity of the survey (Luri et al. 2018). By construction, using the absolute magnitude leads to derived distances that are always positive.

A degeneracy exists between metallicity, luminosity, and colors, especially for stars of high metallicity (Lenz et al. 1998). As pointed out by Ibata et al. (2017b), neglecting the impact of metallicity on the derived absolute magnitude can lead to important errors, rendering the derived distances invalid. In principle, it should be possible to obtain the metallicity and absolute magnitude of the dwarfs simultaneously, in a single step. However, as described below, only about half of the dwarfs that have good metallicity measurements also have good enough parallax measurements to estimate their absolute magnitudes. In order to use the maximum amount of information available, we decide to use two steps, the first to determine the photometric metallicity and the second to estimate the absolute magnitude.

To evaluate the photometric metallicity of the dwarfs, we construct a set of five independent ANNs, whose inputs are the same colors used for the dwarf−giant classification. Using a set of independent ANNs rather than only one is preferable because it allows us to estimate the systematic errors on the predicted values generated by the algorithm. Using a subsample of the training set, we found that five independent ANNs give similar systematic errors to a set with 10 or 15. Moreover, it also prevents any eventual overfitting. The five ANNs, constructed using the Keras package (Chollet 2015), have different individual architectures and are composed of between two and five hidden layers. As for the RFC, the training set is used to find the best architecture of each ANN with a k-fold cross-validation method with five subsamples, where we impose that each ANN has an independent architecture. The parameters used to train/test¹³ the ANNs are the adopted spectroscopic metallicities from the SSPP (FeHadop) and their uncertainties for ∼35,000 dwarfs from the SDSS/SEGUE data set. The techniques mentioned earlier for estimating metallicity from photometric passbands have typical uncertainties of δ[Fe/H] ≃ 0.2 (relative to the spectroscopic measurement). A quality cut is applied on the spectroscopic data set to only use dwarfs with adopted uncertainities on the spectroscopic metallicity of δ[Fe/H]_spectro ≤ 0.2. We note that this criterion has only a very small impact on the SDSS/SEGUE dwarf catalog, since it removes ∼100 stars.

The loss (or cost) function used to train the ANNs is a modified rms function that includes the uncertainties on the metallicity:

$$\begin{eqnarray}&&{ \mathcal L }=\sqrt{\displaystyle \frac{1}{n}\sum _{i=1}^{n}\displaystyle \frac{{({y}_{\mathrm{true},i}-{y}_{\mathrm{pred},i})}^{2}}{\delta {y}_{i}^{2}}},\end{eqnarray} \tag{ 1 }$$

where y_true,i and δy_i are the spectroscopic metallicity and its uncertainty for the ith star, respectively, and y_pred,i is the corresponding metallicity predicted by the algorithm.

Once each ANN is trained, we define the metallicity of the dwarfs ([Fe/H]_Dwarf) as the median of the outputs of the five ANNs and the systematic error as their standard deviation. The difference between the photometric and the spectroscopic metallicity is shown in the top two panels of Figure 11 and is σ_[Fe/H] = 0.15 dex. This is a moderate improvement on the method of Ibata et al. (2017b) (σ_[Fe/H] = 0.20 dex). Note that the residuals do not show any significant trend with the metallicity, except for the most metal-poor stars ([Fe/H] < −2), for which the photometric metallicity tends to be higher than the spectroscopic measurement. We remark that the residuals increase with decreasing metallicity, indicating that the predicted photometric metallicities are less reliable for stars with metallicity lower than [Fe/H] < −2.0. This will be partly due to the lower number of stars present in the training set at this metallicity than at higher metallicity. The average uncertainty on the spectroscopic metallicity of the dwarfs is δ[Fe/H] = 0.04, and the systematic error is δ[Fe/H]_Dwarf,sys = 0.02. Thus, most of the scatter on our metallicity measurement is due to the intrinsic apparent color variation of the dwarfs of similar metallicity.

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Once the metallicity is determined, it is used in combination with the colors to estimate the absolute magnitude of each dwarf and therefore its distance. As for the metallicity, a set of five ANNs is constructed, where the inputs are the same colors used previously in addition to the derived metallicity. The output is the absolute magnitude in a given band. We decide to use the Gaia G band as a reference since this is the filter with the lowest photometric uncertainty at a given magnitude.

The absolute magnitudes of the dwarfs in the SDSS/SEGUE catalog are computed from Gaia parallaxes (ϖ) according to

$$\begin{eqnarray}&&{M}_{{G}_{{gaia}}}={G}_{0}+5+5{\mathrm{log}}_{10}(\varpi /1000).\end{eqnarray} \tag{ 2 }$$

It is worth noting that Gaia tends to underestimate the parallaxes, and we therefore correct all parallaxes by a global offset of ϖ₀ = 0.029 mas, as suggested by Lindegren et al. (2018). Luri et al. (2018) show that the inversion of the parallax to obtain the distance (and so the absolute magnitude) is only valid for the stars with low relative parallax uncertainties, typically ϖ/δϖ ≥ 5 (a relative precision of ≤20%). In these cases, the probability distribution function (pdf) of the absolute magnitude can be approximated by a Gaussian centered on ${M}_{{G}_{{gaia}}}$ with a width $\delta {M}_{{G}_{{gaia}}}$, such that

$$\begin{eqnarray}&&\delta {M}_{{G}_{{gaia}}}=\delta G+\left|\displaystyle \frac{\delta \varpi }{\varpi \mathrm{ln}(10)}\right|.\end{eqnarray} \tag{ 3 }$$

A quality cut is performed on the SDSS/SEGUE dwarfs to keep only those stars with a relative Gaia parallax measurement better than 20%. With this criterion, the mean relative parallax precision of the spectroscopic dwarf sample is ∼10%, corresponding to an average uncertainty on the absolute magnitude of $\delta {M}_{{G}_{{gaia}}}=0.22$ mag. The spectroscopic dwarf data set used to train/test the set of ANNs is composed of 18,930 stars and is a good representation of the distribution of metallicities and absolute magnitudes in the initial data set (covering a range of metallicity of −3.0 dex < [Fe/H]_spectro < 0.5 dex, a range in absolute magnitude 3 < M_G < 7.5, and ≃600 stars with [Fe/H]_spectro < −2.0 with good parallax precision).

The set of five ANNs used to derive the absolute magnitude has a different structure than the set used for the estimation of the metallicity, with four or five hidden layers and a higher number of neurons per layer than previously. However, the loss function is the same, where y_true, δy, and y_pred now correspond to ${M}_{{G}_{{gaia}}}$, $\delta {M}_{{G}_{{gaia}}}$, and ${M}_{{G}_{\mathrm{pred}}}$, respectively. We define the predicted absolute magnitude in the G band of the dwarfs (${M}_{{G}_{\mathrm{Dwarf}}}$) as the median of the outputs of the five ANNs and the systematic error (δ${M}_{{G}_{\mathrm{Dwarf},\mathrm{sys}}}$) as their standard deviation. As illustrated in the bottom panels of Figure 11, the predicted absolute magnitude shows a scatter of ${\sigma }_{{M}_{G}}=0.32$ mag compared to the absolute magnitude computed from the Gaia parallaxes. This corresponds to a relative precision on distance of 15%, very similar to the precision found by Ivezić et al. (2008). It is worth noting that the scatter is almost constant over the range of absolute magnitude, except around ${M}_{{G}_{{gaia}}}\simeq 4.2$, where a few stars tend to have a higher predicted absolute magnitude than observed. These stars are probably young stars (<5 Gyr) on the MS turnoff (MSTO). Since CFIS observes at high galactic latitude ($| b| \gt 18^\circ$), their number is negligible, and we expect that this underestimation of the luminosity of younger stars should have a negligible impact on statistical studies of the distance distribution.

3.3.2. Giant Stars

The method to derive the metallicity and the absolute magnitude for the giants is similar to that for the dwarfs, with a first set of ANNs to derive the metallicity and a second set of ANNs to estimate the absolute magnitude. The architecture of the two sets of ANNs is exactly the same as for the dwarfs.

The adopted metallicities and the uncertainties for the spectroscopic giants are used to train/test the first set of ANNs. Again, we apply a quality cut on the metallicity that uses only the 5670 giants with a metallicity precision better than δ[Fe/H]_spectro ≤ 0.2 (this quality cut removes less than 0.5% of the initial giant sample). The procedure used is exactly the same as for the dwarfs, and the predicted metallicity of the giants ([Fe/H]_Giant) is equal to the median of the five ANNs outputs, and the systematic errors (δ[Fe/H]_Giant,sys) are the standard deviation of these outputs. As shown in the top two panels of Figure 12, the residual between the photometric and spectroscopic metallicity is σ_[Fe/H] = 0.16 dex and does not show any trend with metallicity, except for stars with [Fe/H] < −2.0 as for the dwarfs. This is a very significant improvement over previous studies, where the giants were not treated separately from the dwarfs, leading to an overestimation of their metallicity by [Fe/H] = 0.16 dex (Ibata et al. 2017b).

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In contrast to the dwarfs, it is not possible to keep only the giants with a relative Gaia parallax accuracy ≲20%. At a similar luminosity, the giants are more distant than the dwarfs, which leads to a higher uncertainty in their parallax. Adopting the same selection as for the dwarfs creates a data set of only ≃1000 stars, whose overall metallicity distribution is very different from the overall metallicity distribution of the giants. Indeed, a large majority of these stars (∼670) have a metallicity higher than [Fe/H] > −1, while 3/4 of the overall spectroscopic giant data set has a metallicity lower than [Fe/H] < −1 with a peak around [Fe/H] ≃ −1.4 (see the bottom panel of Figure 13). In addition, more than 60% of the giants with a parallax accuracy of ≲20% seem to be subgiant stars, misidentified by our dwarf−giant classifier.

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About 2/3 of the spectroscopic giants have a relative precision on their parallax measurement higher than 20%. Therefore, the pdf of their absolute magnitude cannot be approximated by a Gaussian, as was the case for the dwarfs. As shown by Luri et al. (2018), their pdf is asymmetric and the maximum is not centered on the "true" absolute magnitude. Using the maximum of the pdf in these cases leads to an underestimate of the distance and therefore an underestimate of the absolute magnitude. Depending on the relative parallax precision, the "true" absolute magnitude can be more than 1σ away from the maximum likelihood value.

In principle, we could perform a data augmentation of the spectroscopic giant data set to take into account the uncertainties on the different parameters, especially the parallax. This can be done by Monte Carlo sampling the spectroscopic giant data set in a range of 2.5σ–3σ around the maximum likelihood of the observables. In order not to add any bias, this distribution should be symmetric around the maximum likelihood of the different parameters. However, to keep a physical value of absolute magnitude, the parallaxes should be positive. This leads to a data set composed of stars with ϖ − 2.5 ∗ δϖ > 0, reducing drastically the number of stars used to train the model, which also biases the sample to much more metal-rich stars, similar to the effect of cutting on the parallax uncertainties described above.

For these reasons, we use all the spectroscopic giants that have a positive parallax measurement. Following Hogg et al. (2019), we apply a quality selection on the parallax to keep only giants with an uncertainty on the parallax δϖ < 0.1 mas, so that we are not dominated by stars with extremely poor measurements. The final data set used to train/test the set of ANNs is composed of 3497 giants. However, we found that for this giant sample the Gaia parallaxes have to be corrected by an offset of ϖ₀ = 0.033 mas in order to obtain reasonable distances for the globular clusters (GCs; see Section 4.3). This offset is slightly higher than for the dwarfs (of ϖ₀ = 0.029 mas) but lower than the offset of ϖ₀ = 0.048 mas found by Hogg et al. (2019) for red giant stars.

Due to a lack of precise parallaxes, the exact pdf (and hence the uncertainties) of the absolute magnitude (calculated using Equation (2)) cannot be known for most of the giants without adopting a prior on the density distribution of the giants in the Milky Way. We cannot, therefore, estimate the uncertainties on the absolute magnitude using Equation (3). For this reason, the loss function used as a metric for the ANNs is a standard rms that does not take into account the uncertainties on the absolute magnitude.

The scatter on the absolute magnitude shown in the left panel of Figure 12 is ${\sigma }_{{M}_{G}}=0.79$ dex, with a bias of ∼−0.28 dex, similar to the median of the residuals (Me(ΔM_G) = −0.28). This bias is a consequence of the underestimation of the distance to the giants using the values given by the maximum of the pdf. The mean residual is larger $(\langle {\rm{\Delta }}{M}_{G}\rangle =-0.09)$ because it is influenced by those stars with the largest residuals, a result of the limited statistical sample.

Indeed, the Gaia parallaxes of these stars are inaccurate, and distances obtained by inverting the parallax tend to underestimate the true distance of these stars (Luri et al. 2018). Thus, the absolute magnitudes used to train the relation are inaccurate and are likely underestimated. It is therefore not possible to use the scatter between the predicted absolute magnitude and the absolute magnitude obtained from the Gaia parallax to verify that the predicted distances are correct with this method. However, as we will show in Section 4.3 using the GCs present in the CFIS footprint, the distances of the giants thus determined give good estimates of their real distances, despite the lack of precision of the absolute magnitudes used to train the relation.

We now apply the algorithm to the full SDSS/SEGUE data set for stars with a spectroscopic S/N ≥ 25. The photometric metallicity distribution function (MDF) from the stars classified by the algorithm as dwarfs (P_Dwarf ≥ 0.5) and giants (P_Giant ≥ 0.5) is compared to the spectroscopic metallicity distribution of the dwarfs and giants for the SDSS/SEGUE in Figure 13.

The expected and predicted MDFs for dwarfs and giants in Figure 13 are generally similar, especially for the dwarfs. For the giants, the agreement at the metal-poor end is good, but the number of giants predicted to have [Fe/H] > −1.0 falls to zero. This is a direct consequence of the misclassification of the most metal-rich giants, as discussed in Section 3.2. Further, these misclassified giants are the origin of the excess of dwarfs around [Fe/H] ≃ −0.5 compared to the number expected from the spectroscopy.

If we consider only stars classified as dwarfs/giants with high confidence (P_Dwarf/Giant ≥ 0.7; red lines in Figure 13), the MDF of the dwarfs is almost unchanged. This means that the large majority of dwarfs are classified by the algorithm with high confidence. For the giants, the MDF of stars classified with high confidence decreases sharply at [Fe/H] = −1.2, instead of [Fe/H] = −1.0 for the stars with P_Giant ≥ 0.5. Thus, the classification of giants is more uncertain at high metallicities than at lower metallicities. Again, this is another manifestation of the difficulty of the algorithm in identifying more metal-rich giants. The overpredicted number of giants around [Fe/H] = −1.8 is a consequence of the trend in the photometric metallicity relation, which tends to overestimate the metallicity of stars with [Fe/H] < −2.0, as explained in Section 3.3.2.

The left panel of Figure 14 compares the absolute magnitude expected using the Gaia parallaxes (M_G,Gaia) with the absolute magnitude predicted by the algorithm (M_G,photo) for stars in SDSS/SEGUE with ϖ/δϖ ≥ 5. The stars predicted as dwarfs and giants by the algorithm are shown in red and blue, respectively. As expected, the predicted absolute magnitude of the large majority (more than 97%) of the stars identified as dwarfs is similar to the Gaia measurements. However, a small population of dwarfs have predicted absolute magnitudes significantly lower than observed (around M_G,Gaia ∼ 3). This population corresponds to metal-rich giants/subgiants that have been misidentified as dwarfs. For those stars identified as giants by the algorithm that are actually giants, the agreement between the predicted and actual magnitudes is good. However, all those stars identified as giants that are actually dwarfs are estimated to be too bright.

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It is not surprising, given the criterion imposed on the relative precision of the parallax, that the fraction of contaminant dwarfs is much higher than the ≃30% measured previously. Indeed, the Gaia catalog is limited at a magnitude of G = 21, and since dwarfs are intrinsically less bright than giants, dwarfs in this catalog are on average closer than giants. Since the parallax method is more precise for the stars at shorter distances, it is natural that most of the stars with relative parallax measurements better than 20% are actual dwarfs. Nevertheless, it is very interesting to see that the actual giants have predicted magnitudes similar to Gaia's. Moreover, the scatter is only slightly larger than for the dwarfs, despite the fact that the data used to train the set of ANNs are less accurate for giants than for dwarfs.

Approximately 480,000 stars from the LAMOST DR3 catalog are present in the CFIS footprint.¹⁴ While this is 10 times more than for SDSS/SEGUE, most LAMOST stars are metal-rich. Less than 3000 LAMOST dwarfs have a metallicity of [Fe/H] ≤ −1.5, and their metallicity accuracy is lower than for the corresponding metallicity range in the SDSS/SEGUE data set. Scientifically, we are more interested in distant stars, at the faint end of the Gaia catalog, so we focused the training of the algorithm on the SDSS/SEGUE data set instead of the LAMOST data set. However, this means that the LAMOST data set can be used to independently test and validate our algorithm.

Figure 15 shows the Kiel diagram of LAMOST stars color-coded by the probability of being a dwarf according to our algorithm. Hotter giants are generally classified correctly by the algorithm, but a large number of giants are misclassified. The completeness and contamination fraction of LAMOST giants are shown in Figure 16. Nearly all the misclassified giants visible in Figure 15 have [Fe/H] > −1.0, and contamination starts to increase at [Fe/H] > −1.3, demonstrating consistency with the SDSS/SEGUE sample that was used to train the algorithm.

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The right panel of Figure 14 shows that the distribution of the predicted absolute magnitude for the LAMOST stars with ϖ/δϖ ≥ 5 is similar to the corresponding distribution for the SDSS/SEGUE stars (left panel). We note that the relative fraction of actual dwarfs misidentified as giants (lower right corner) is smaller than for the SDSS/SEGUE data set. This is because the LAMOST data set is intrinsically brighter than the SDSS/SEGUE data set. Indeed, since LAMOST stars are ∼2 mag brighter than SDSS/SEGUE, the giants are on average closer. Thus, the number of giants that have a relative parallax precision better than 20% is higher in the LAMOST data set.

The number of stars present in the horizontal feature between $0\leqslant {M}_{G,{Gaia}}\leqslant 4$ is higher in the LAMOST data set than for SDSS/SEGUE. These stars correspond to actual giants misclassified as dwarfs. The higher number of these misclassified stars is higher in the LAMOST data set than in SDSS/SEGUE because the number of metal-rich giants ([FeH] > −1.0) is higher in this first one and, as mentioned earlier, the dwarf/giant classification does not work well for these stars.

Figure 17 shows that the predicted metallicities for LAMOST stars are generally consistent with the spectroscopic metallicities obtained by the LAMOST pipeline. However, the predicted metallicity seems to be slightly underestimated for stars with [Fe/H] ≥ −0.5. Interestingly, we trace this to a systematic difference between the spectroscopic metallicity for the ∼4500 stars in common between the SDSS/SEGUE and LAMOST data sets, as shown by the red dots in the bottom panel of Figure 17. Since the metallicity is calibrated on the FeHadop metallicity from SDSS/SEGUE, it is not surprising to see a similar trend in the predicted photometric metallicity.

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Based on this comparison, we conclude that our method, applied to the LAMOST data set, demonstrates the same behaviors, agreements, and biases as found using the SDSS/SEGUE data set.

Several Galactic satellites with known distances are included in the CFIS data set and present another opportunity to independently test our algorithm. We first examine the four closest GCs present in the CFIS footprint, NGC 6205, NGC 6341, NGC 5272, and NGC 5466, where both dwarfs and the giants are detected. These are located between 7 and 16 kpc from us (see Table 2).

For this analysis, we only consider stars at more than 3, 5, 2.5, and 1 half-light radius for NGC 6205, NGC 6341, NGC 5272, and NGC 5466, respectively. This removes the (significant) effect of crowding on the input photometry. We remove obvious foreground contamination by selecting stars using their Gaia proper motions. The absolute color–magnitude diagrams (CMDs) of these GCs are shown in Figure 18. The left panels show the CMDs using the distances found in the literature (fourth column of Table 2). Each star is color-coded by the probability that it is a dwarf according to our algorithm.

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The color-coding in the left panels of Figure 18 reveals that our algorithm generally identifies the giants correctly. A noticeable exception is for horizontal branch stars, which are not present in significant numbers in our training/test sets. In addition, the large majority of the dwarfs in each GC are also correctly identified. The exception to this is for the faintest stars in NGC 6205, where the fraction of dwarfs misclassified as giants increases at the faintest magnitudes. This is likely a direct consequence of the misidentification of metal-poor dwarfs in the temperature range occupied by giants, and it is discussed at length in Section 3.2.

The middle panels of Figure 18 show the CMDs of the clusters where the absolute G-band magnitude is computed directly from the Gaia parallax using Equation (2). It is clear that this method is inadequate for these objects, since the uncertainties on the Gaia parallaxes for stars more distant than a few kiloparsecs are generally prohibitively large. We stress that it is this fact that motivated the development of this algorithm in the first place. The right panels of Figure 18 show the CMDs where the absolute magnitude of the stars is computed via our algorithm. It is clear that the derived CMDs are much better than for the middle panels and are reasonably close to the reference CMDs for each cluster. We note that the absence of any stars on the subgiant branch in the CMDs in the right panel is a direct consequence of the preference of our algorithm to define those stars as dwarfs.

To verify that our algorithm predicts correct metallicities and distances for dwarfs and giants, we compare the mean distance and metallicity of the four GCs according to our algorithm to the value found in the literature. We select stars that have a probability to be a dwarf or a giant of more than 0.7. For the dwarf samples, we require stars to have an intrinsic absolute magnitude larger than 3.7, to remove the impact of the subgiants. The mean distances and metallicities for dwarfs and giants for each cluster are listed in Table 2. For all the parameters, the derived values are within 1σ of the literature values for the clusters.

It is important to note that the lower precision on the distance estimated for the four GCs by our method is 26% using the giants and 18% using the dwarfs, validating our method to estimate the absolute magnitude of those stars. One could notice that the mean metallicities obtained for the dwarfs for NGC 6205 and NGC 5272 are ≃0.2–0.25 dex lower than listed in the literature, though they are both in the 1σ of the estimated metallicity. However, the few dwarfs of NGC 6205 present in the SDSS/SEGUE catalog have metallicities between −2 < [Fe/H] < −1.5 with a peak around 1.7. Therefore, it is more likely that the underestimation of the metallicity is related to the SDSS/SEGUE metallicity than directly related to our algorithm.

When we applied the algorithm to any stars in the field, it was not possible to remove the subgiant contamination. The distances predicted by the algorithm for dwarfs in the four GCs without the constraint on the intrinsic absolute magnitude of the dwarf to be larger than 3.7 are shown in Table 3. Releasing this constraint reduces systematically the predicted distance of the GCs. However, these values are still consistent with the distance found in the literature, with the exception of NGC 5466. Due to its distance, the fraction of subgiants/giants is larger than in the other GCs, leading to a more significant impact of these stars on the distance estimation. This bias should be considered when working with the dwarf stars at large distance (typically >10 kpc). Interestingly, taking into account the subgiant contamination has very little impact on the estimated metallicity, the difference with the metallicity found previously being much smaller than the scatter found with the spectroscopic measurement.

In addition to these relatively nearby clusters, NGC 2419 and the Draco dwarf spheroidal (dSph) are also present in the CFIS footprint. These systems are much more distant, at 79.7 ± 0.3 kpc and 74.26 ± 0.18 kpc,¹⁵ respectively (Hernitschek et al. 2019). As such, only the upper portions of the giant branch in these two satellites are visible in our data, and most of these stars have relatively poor photometric measurements. This leads to large uncertainties on the distances predicted by our algorithm. In what follows, we only use stars whose intrinsic absolute magnitudes are known to better than δM_G ≤ 0.5. This corresponds to a precision on the distance of at least 23%. The left panels of Figure 19 show the CMDs for these two satellites, adopting the literature distance for the absolute magnitude on the y-axis. The middle panels only keep stars that also have δM_G ≤ 0.5.

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We remind the reader that these two objects currently provide the best opportunity to test the validity of our technique at large distances and were used to determine the global offset of the parallax for the giants, ϖ₀ = 0.033 mas, that we adopted in Section 3.3.2. Using this calibration, for NGC 2419, the mean distance of the giants according to our algorithm is 77.6 ± 15.2 kpc, in agreement with the distance found by Hernitschek et al. (2019) of 79.7 ± 0.3 kpc. For Draco, we find a mean distance of 73.4 ± 9.9 kpc for the giants, consistent with the 74.26 ± 0.18 kpc found by Hernitschek et al. (2019). We note that a smaller offset of ϖ₀ = 0.029 mas (as used for the dwarfs; Lindegren et al. 2018) would lead to a distance for these two satellites closer to ∼50 kpc.

Finally, the CFIS footprint contains a large portion of the GD-1 stream (Grillmair & Dionatos 2006). Applying a similar proper-motion selection to that of Price-Whelan & Bonaca (2018), we measure a heliocentric distance for the stream of 8.0 ± 1.5 kpc and a mean metallicity of [Fe/H] = −1.9 ± 0.4, consistent with the recent measurement of Malhan & Ibata (2019).

In this section we apply the algorithm to the ∼12.8 million stars present in the current version of the merged CFIS-PS1-Gaia catalog. The catalog contains 12.2 million stars identified as dwarfs (P_Dwarf > 0.5) and 600,000 stars (∼5%) identified as giants (P_Giant > 0.5). The distribution of the systematic errors (δ[Fe/H]_sys and $\delta {M}_{G,\mathrm{sys}}$) and of the uncertainties due to the photometry (which we will call photometric uncertainties for simplicity; δ[Fe/H]_photo and $\delta {M}_{G,\mathrm{photo}}$) on the metallicity and the absolute magnitude as a function of the apparent magnitude in the G band are presented in Figure 20. The systematic errors and photometric uncertainties for the dwarfs are lower than for the giants, and it is interesting to see that the systematic errors are generally higher than the photometric uncertainties, especially for stars fainter than G = 17.5. This indicates that the relations found by each independent ANN (which give similar results for the training and testing sample) predict different values at fainter magnitudes. The differences between these values are more important than the uncertainties on these parameters due to photometric uncertainties. These differences are likely a consequence of the low number of stars in the training and testing samples below G = 18 (<10%), which has caused the ANNs to determine relationships based mainly on the brighter stars. We expect that a training set based on the next generation of spectroscopic surveys, such as WEAVE, 4-MOST, or SDSS-V, will be able to significantly reduce the systematic errors at faint magnitude, since these catalogs will contain a higher number of stars at fainter magnitudes than SDSS/SEGUE.

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We recommend to only use stars classified as dwarfs or giants with high confidence (i.e., P_Dwarf ≥ 0.7 or P_Giant ≥ 0.7) and that have uncertainties on their estimated absolute magnitude $\delta {M}_{G}=\sqrt{(\delta {M}_{G,\mathrm{sys}}^{2}+\delta {M}_{G,\mathrm{photo}}^{2})}\leqslant 0.5$. This selection removes most of the contamination from the misclassified dwarfs/giants while keeping a large number of stars whose distances and metallicities are "correct" (with a relative uncertainty on the individual distances of less than 26% and on the individual metallicities of less than 0.3 dex).