The RAVE-on Catalog of Stellar Atmospheric Parameters and Chemical Abundances for Chemo-dynamic Studies in the Gaia Era

Given our assumptions, the model we adopt is

$$\begin{eqnarray}&&{y}_{{jn}}={\boldsymbol{v}}({{\ell }}_{n})\cdot {{\boldsymbol{\theta }}}_{j}+{e}_{{jn}},\end{eqnarray} \tag{ 1 }$$

where y_jn is the pseudo-continuum-normalized flux for star n at wavelength pixel j, ${\boldsymbol{v}}({{\ell }}_{n})$ is the vectorizing function that takes as input the K labels ℓ_n for star n and outputs functions of those labels as a vector of length D > K, ${{\boldsymbol{\theta }}}_{j}$ is a vector of length D of parameters influencing the model at wavelength pixel j, and e_jn is the residual (noise). Here we will only consider vectorizing functions with second-order polynomial expansions (e.g., T_eff², see Sections 3.3–3.5). The noise values e_jn can be considered to be drawn from a Gaussian distribution with zero mean and variance ${\sigma }_{{jn}}^{2}+{s}_{j}^{2}$, where ${\sigma }_{{jn}}^{2}$ is the variance in flux y_jn and ${s}_{j}^{2}$ describes the excess variance at the jth wavelength pixel.

At the training step, we fix the K-lists of labels for the n training set stars. At each wavelength pixel j, we then find the parameters ${{\boldsymbol{\theta }}}_{j}$ and ${s}_{j}^{2}$ by optimizing the penalized likelihood function

$$\begin{eqnarray}{{\boldsymbol{\theta }}}_{j},{s}_{j}^{2} & \leftarrow & \mathop{\mathrm{argmin}}\limits_{{\boldsymbol{\theta }},s}\left[\displaystyle \sum _{n=0}^{N-1}\displaystyle \frac{{[{y}_{{jn}}-{\boldsymbol{v}}({{\ell }}_{n})\cdot {\boldsymbol{\theta }}]}^{2}}{{\sigma }_{{jn}}^{2}+{s}^{2}}\right.\\ & & +\left.\displaystyle \sum _{n=0}^{N-1}\mathrm{ln}({\sigma }_{{jn}}^{2}+{s}^{2})+{\rm{\Lambda }}\,Q({\boldsymbol{\theta }})\right],\end{eqnarray} \tag{ 2 }$$

where Λ is a regularization parameter, which we will set by leave-one-out cross-validation in later sections, and $Q({\boldsymbol{\theta }})$ is a L1 regularizing function (e.g., see LASSO in Tibshirani 1996, and related literature) that encourages ${\boldsymbol{\theta }}$ values to take on zero values without breaking convexity (Casey et al. 2016b):

$$\begin{eqnarray}&&Q({\boldsymbol{\theta }})=\displaystyle \sum _{d=1}^{D-1}| {\theta }_{d}| .\end{eqnarray} \tag{ 3 }$$

Note that the d subscript here is zero-indexed; the function $Q({\boldsymbol{\theta }})$ does not act on the (first) θ₀ coefficient, as this is a "pivot point" (mean flux value) that we do not expect to diminish with increasing regularization (e.g., see Equation (5)). In practice, we first fix ${s}_{j}^{2}=0$ to make Equation (2) a convex optimization problem, then we optimize for ${{\boldsymbol{\theta }}}_{j}$, before solving for ${s}_{j}^{2}$.

The test step is where we fix the parameters ${{\boldsymbol{\theta }}}_{j},{s}_{j}^{2}$ at all wavelength pixels j, and optimize the K-list of labels ℓ_m for the mth test-set star. Here the objective function is

$$\begin{eqnarray}&&{{\ell }}_{m}\leftarrow \mathop{\mathrm{argmin}}\limits_{{\ell }}\left[\displaystyle \sum _{j=0}^{J-1}\displaystyle \frac{{[{y}_{{jm}}-{\boldsymbol{v}}({\ell })\cdot {{\boldsymbol{\theta }}}_{j}]}^{2}}{{\sigma }_{{jm}}^{2}+{s}_{j}^{2}}\right].\end{eqnarray} \tag{ 4 }$$

After optimizing Equation (4) for the mth star, we store the covariance matrix ${{\boldsymbol{\Sigma }}}_{m}$ for the labels ℓ_m, which provides us with the formal errors on ℓ_m. The formal errors are expected to be underestimated, and in Section 4 we judge the veracity of these errors through validation experiments.

We sought to construct a training set of stars across the main sequence, the sub-giant branch, and the red giant branch. We required stars with precisely measured effective temperature T_eff, surface gravity log g, and elemental abundances of O, Mg, Si, Ca, Al, Fe, and Ni. This proved to be difficult because the magnitude range of RAVE does not overlap substantially with high-resolution spectroscopic surveys. The fourth internal data release of the Gaia-ESO survey includes giant and main sequence stars, but only 142 overlap with RAVE, which is too small to be a useful training set for our purposes. The 13 data release from the Sloan Digital Sky Survey (SDSS Collaboration et al. 2016) includes labels for APOGEE stars on the giant branch and (uncalibrated values for) the main sequence, but our tests indicated that the APOGEE main sequence labels suffered from significant systematic effects. A flat, then "up-turning" main sequence is present, and the metallicity gradient trends in the opposite direction with respect to log g on the main sequence (i.e., metal-poor stars incorrectly sit above an isochrone in a classical Hertzsprung–Russell diagram). If we consider lower-resolution studies as potential training sets, there are 2369 stars that overlap with LAMOST—of which 2213 have positive S/Ns in the g-band (snrg). However, the labels are expectedly less precise given the lower resolution, there are no elemental abundances available for the main sequence stars,²⁶  and the LAMOST lower main sequence suffers from the same systematic effects seen in the APOGEE data.

These constraints forced us to construct a heterogeneous training set. Given previous successes in transferring high S/N labels from APOGEE (Ness et al. 2015, 2016; Casey et al. 2016b; Ho et al. 2017), we chose to use the 1355 stars in the APOGEE—RAVE overlap sample for giant star labels in the training set. Of these, about 900 are giants according to APOGEE. From this sample, we selected stars to have determinations in all abundances of interest ([X/H] > −5 for O, Mg, Al, Si, Ca, Fe, and Ni); S/N of >200 pixel⁻¹ in APOGEE and >25 pixel⁻¹ in RAVE; and we further required that the ASPCAP did not report any peculiar flags (ASPCAPFLAG = 0). These restrictions left us with 536 stars along the giant branch, with metallicities ranging from [Fe/H] = −1.79 to 0.26. Intermediate tests with globular cluster members showed that the metallicity range of the training set needed to extend at least below [Fe/H] ≲ −2 in order for our catalog to be practically useful. Without additional metal-poor stars, the lowest metallicity labels reported by our model would be around [Fe/H] ≈ −2, even for well studied stars with [Fe/H] ∼ −4 (e.g., CD 38-245). For this reason, we supplemented our sample of APOGEE giant stars with 176 known metal-poor giant stars observed by RAVE. The effective temperature T_eff, surface gravity log g, and iron abundance [Fe/H] labels were adopted from Fulbright et al. (2010) and Ruchti et al. (2011). For this sample of metal-poor stars, we assumed that the elemental abundances of O, Mg, Al, Si, Ca, and Ni followed typical trends of Galactic chemical evolution: we asserted [Mg/Fe] = +0.4, [O/Fe] = +0.4, [Al/Fe] = −0.5, [Ca/Fe] = +0.4, [Si/Fe] = +0.4, and [Ni/Fe] = −0.25. We stress that this decision is made solely to ensure that our overall metallicity scale reflects that of the RAVE survey, down to [Fe/H] ∼ −4. Indeed, it is likely that for most of these elements, these abundances cannot be measured from RAVE spectra for ultra-metal-poor stars: the atomic transitions in the RAVE spectral region are simply too weak to influence the spectrum. For this reason, our adopted abundances for these very metal-poor stars represent an "anchor point" in order to ensure our overall metallicity scale is correct. We do not recommend the use of our individual abundance labels at [Fe/H] ∼ −4. We discuss this issue in more detail in Section 5.

Assembling a suitable training set for the main sequence and sub-giant branch was less trivial. There are no spectroscopic studies that extend the range of stellar types we are interested in (e.g., FGKM-type stars), and which also have a large enough sample size that overlaps with RAVE. Moreover, most of the spectroscopic studies we considered also showed a flat lower main sequence, a systematic consequence of the analysis method adopted (see Bensby et al. 2014 for a discussion on this issue). For these reasons, we chose to make use of the K2/EPIC catalog (Huber et al. 2016) for the training set labels on the main sequence and sub-giant branch. The K2/EPIC catalog follows from the successful Kepler input catalog (Brown et al. 2011), and provides probabilistic stellar classifications for 138,600 stars in the K2 fields based on the astrometric, asteroseismic, photometric, and spectroscopic information available for every star. There are 4611 stars that overlap between K2/EPIC and RAVE.

K2/EPIC differs from the Kepler input catalog because K2/EPIC does not benefit from having narrow-band DDO₅₁ photometry to aid dwarf/giant classification. Despite this limitation, the labels in the K2/EPIC catalog have already been shown to be accurate and trustworthy (Huber et al. 2016). However, when the posteriors are wide (i.e., the quoted confidence intervals are large) due to limited information available, it is possible that a star has been misclassified. This is most prevalent for sub-giants, where Huber et al. (2016) note that ≈55%–70% of sub-giants are misclassified as dwarfs. The probability of misclassification is usually quantified in the uncertainties given for each star; most dwarfs that have a higher possibility of being sub-giants have large confidence intervals. Therefore, requiring low uncertainties will decrease the total sample size, but in practice it removes most misclassifications. The situation is far more favorable for dwarfs and giants. Only 1%–4% of giant stars are misclassified as dwarfs, and about 7% of dwarfs are misclassified as giants. To summarize, the K2/EPIC labels with narrow confidence intervals are usually of high fidelity, and given that we have spectra, we can identify any obvious misclassifications.

We sought to have a small overlap between our giant and main sequence star training sets. Most of our giant training set is encapsulated within 0 < log g < 3.5; however, there is a sparse sampling of stars reaching to log g ≈ 4. We required log g > 3.5 for the K2/EPIC main sequence/sub-giant star training set, allowing for ≈0.5 dex of overlap between the two training sets. We further employed the following quality constraints on the K2/EPIC catalog: the upper and lower confidence intervals in T_eff must be below 150 K; the upper and lower confidence intervals in log g must be less than 0.15 dex; the S/N of the RAVE spectra must exceed 30 pixel⁻¹; and T_eff ≤ 6750 K. Unfortunately, these strict constraints removed most metal-poor stars, which we later found to cause the test labels to have under-predicted abundances for dwarfs of low metallicity. For this reason, we relaxed (ignored) those quality constraints for stars with [Fe/H] < −1, and included an additional 12 turn-off stars with −1.6 ≳ [Fe/H] ≳ −2.1 from Ruchti et al. (2011). After training a model based on main sequence and giant stars (Section 3.1), we found we could identify misclassifications by leave-one-out cross-validation. However, we chose not to do this because the number of likely misclassifications in the training set was negligible (≈1%), and the improvement in main sequence test-set labels was minimal. The distilled sample of the RAVE–K2/EPIC overlap catalog contains 595 stars (583 of 4611 from K2/EPIC). The full training set for each model (see the next sections) is shown in Figure 1.

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We have constructed a justified training set for stars across the main sequence, sub-giant, and red giant branch. However, the lack of overlap between RAVE and other works have resulted in a somewhat peculiar situation. Detailed abundances are available from APOGEE for all giant stars in our sample; however, only imprecise (but accurate on expectation) metallicities are available from K2/EPIC for stars on the main sequence and the sub-giant branch. Here we will construct a simple model for all stars that only makes use of three labels (T_eff, log g, [Fe/H]), before we outline how we derive abundances for giant branch stars. The complexity for this model will be quadratic (T_eff² is the highest term), where the vectorizer ${\boldsymbol{v}}({{\ell }}_{n})$ expands as

$$\begin{eqnarray}&&{\boldsymbol{v}}({{\ell }}_{n})\to [1,{T}_{\mathrm{eff},n},\mathrm{log}\,{\text{}}{g}_{n},{[\mathrm{Fe}/{\rm{H}}]}_{n},{T}_{\mathrm{eff},n}^{2},\mathrm{log}\,{\text{}}{g}_{n}\,{T}_{\mathrm{eff},n},\\ &&\,\,{[\mathrm{Fe}/{\rm{H}}]}_{n}\,{T}_{\mathrm{eff},n},\mathrm{log}\,{\text{}}{g}_{n}^{2},{[\mathrm{Fe}/{\rm{H}}]}_{n}\,\mathrm{log}\,{\text{}}{g}_{n},[\mathrm{Fe}/{\rm{H}}{]}_{n}^{2}]\end{eqnarray} \tag{ 5 }$$

such that ${\boldsymbol{v}}({\ell })$ produces the design matrix:

$$\begin{eqnarray}&&{\boldsymbol{v}}({\ell })\to \left[\begin{array}{c}{\boldsymbol{v}}({{\ell }}_{0})\\ \vdots \\ {\boldsymbol{v}}({{\ell }}_{N-1})\end{array}\right].\end{eqnarray} \tag{ 6 }$$

We used no regularization (Λ = 0) for this model. After training the model, we treated all 520,781 spectra as test-set objects. In the left-hand panel of Figure 2, we show the effective temperature T_eff and surface gravity log g for all spectra. The main sequence and red giant branch are clearly visible. However, the details of stellar evolution are no longer present: the sub-giant branch is not discernible, and there are a number of systematic artifacts (over-densities) present in label space. These artifacts disappear when we require additional quality constraints (e.g., no peculiar morphological classifications), but the complexity of the Hertzsprung–Russell diagram is still not present. Thus, we concluded that while this model could be useful for deriving stellar classifications (e.g., F2-type giant), the labels are too imprecise.

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We chose to adopt separate models for the main sequence and the red giant branch rather than switch to a single model with higher complexity. This choice allowed us to derive stellar parameters for stars on the main sequence and sub-giant branch, as well as detailed elemental abundances for red giant branch stars. However, adopting two separate models introduces the challenge of how to combine the results from two models, or how to assign one star as "belonging" to a single model. In Section 3.6, we describe how we will use the simple model introduced in this section to discriminate between results from a three-label main sequence model in Section 3.4 and a nine-label giant star model in Section 3.5.

We constructed a three-label quadratic model using only main sequence and sub-giant stars. In order to set the regularization hyperparameter Λ for this model, we trained 30 models with different regularization strengths, spaced evenly in logarithmic steps between Λ = 10⁻³ to Λ = 10³. We then performed leave-one-out cross-validation for each model. Specifically, for each star in the training set: we removed the star; trained the model; and then inferred labels from the removed star as if it was a test object. We also performed leave-one-out cross-validation on an unregularized (Λ = 0) model, which we will use as the basis for comparison. For the unregularized case, we calculated the bias and root-mean-square (rms) deviation between: the training set labels, and the labels we derived by cross-validation, where one star was removed at a time and the model was re-trained. We repeated this calculation of bias and rms deviation for all 30 models with different regularization strengths Λ.

We show the percentage difference in the rms deviation of the labels with respect to the unregularized model in Figure 3. The upper and lower envelope represent the boundaries across all labels, showing that with increasing regularization, the rms decreased in all labels. We found similar improvements in the biases; however, these were already minimal in the unregularized case. The improvement in rms reaches a minimum value near Λ = 35.6 (≈10^1.5), where we achieve rms deviations that are about 10% better than the unregularized case. Based on this improvement, we set Λ = 35.6 for this model. At this regularization strength, the bias and rms values found by leave-one-out cross-validation are, respectively: 38 K and 256 K for T_eff, 0.05 dex and 0.29 dex for log g, with 0.03 dex and 0.17 dex for [Fe/H].

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We note that while leave-one-out cross-validation has been used to choose a justified regularization strength, it cannot be used to find the optimal regularization strength across all pixels. In the same sense, the regularization strength Λ could—in principle—differ for each pixel and each label. A penalized likelihood function of that description could qualitatively be similar to a Bayesian hierarchical model with strong priors on the spectral derivatives ${\boldsymbol{\theta }}$ being zero. Rather, in this case, we have performed 30 iterations of leave-one-out cross-validation and we have fixed one global Λ value based on the rms improvement in our 30 pre-selected regularization strengths.

We inferred labels for all 520,781 RAVE spectra using this regularized main sequence/sub-giant-star model; we made no a priori decisions as to whether a star was likely a main sequence/sub-giant star or not. The results for the entire survey sample are shown in the center panel of Figure 2. The increased density of solar-type stars is consistent with RAVE observing stars in the local neighborhood, and the high number of turn-off and main sequence stars relative to the sub-giant branch is expected from the relative lifetimes of these evolutionary phases. An over-density of stars near the base of the giant branch is also present. This artifact is due to having giant stars in the test set, but not in the training set, and the model is (poorly) extrapolating outside the convex hull of the training set.

The red giant branch stars in our training set have stellar parameters (T_eff, log g) and up to 15 elemental abundances from the ASPCAP (García Pérez et al. 2016). A subset of these elements have atomic transitions in the RAVE wavelength region: O i, Mg i, Al i, Si i, Ca ii, Ti i, Fe i, and Ni i. However, we excluded [Ti/H] from our abundance list because of systematics in the ASPCAP [Ti/H] abundances (Holtzman et al. 2015; Hawkins et al. 2016). Therefore, we are left with nine labels in our giant star model: T_eff, log g, and seven elemental abundances.

Similar to Sections 3.3 and 3.4, we used a quadratic vectorizer for the giant star model. Here the terms are expanded in the same way as Equation (5), only with nine labels instead of three. We set the regularization hyperparameter Λ in the same way described in Section 3.4, using the same 30 trials of Λ. The results are shown in Figure 3, where again the enveloped region represents the minimum and maximum change in rms label deviation with respect to the unregularized case. At the point of maximum improvement near Λ = 0.13 (≈10^−0.9), the rms in all nine labels has decreased by up to 30%, with all labels showing an improvement >5%, and the mean improvement over all labels is about 10%. Near Λ ≈ 10^−0.9 to 10^−0.3, the regularization also produces a sparser matrix of ${\boldsymbol{\theta }}$, with ≈20% more terms (mostly cross-terms) having zero-valued entries. Based on the increased model sparsity and decreasing rms deviation in the labels, we adopt Λ = 0.57 (10^−0.25) for the giant star model. The bias in labels from a regularized model with Λ = 0.57 is negligible: −0.3 K in T_eff, and <0.007 dex in magnitude for log g and all seven elemental abundances. The rms at this regularization strength is 69 K in T_eff, 0.18 dex in log g, and varies between 0.07–0.09 dex depending on the elemental abundance.

We inferred labels for all 520,781 RAVE spectra using this model, again without regard for whether a star was likely a giant or not. The results for all survey stars are summarized in the right panel of Figure 2. The red clump is clearly visible and in the expected location, without requiring any post-analysis calibration. However, artifacts due to dwarf stars being present in the test set, and not in the training set, are also present.

We have derived labels for all 520,781 RAVE spectra using the three models described in previous sections. The results from our first model (Section 3.3)—which includes the main sequence, sub-giant, and red giant branch—shows that a single three-label quadratic model is too simple for the RAVE spectral range. The other models have problems too: unrealistic over-densities in label space show that the main sequence model and the giant model make very poor extrapolations for stars outside their respective training sets. For these reasons, we were forced to exclude or severely penalize incorrect results from both models. We emphasize that the choices here are entirely heuristic and depart from interpreting The Cannon output labels as the maxima of individual likelihood functions. Each model produces estimates of the labels for a given star, and we use those estimates to produce a unified estimate, but this joint estimate is calculated by disregarding the probabilistic attributes of individual estimates.

Before attempting to join the results from the models in Sections 3.4 and 3.5, we excluded results in either model that had a reduced ${\chi }_{r}^{2}\gt 3$. We further discarded stars with labels that are outside the extent of the training set. Specifically for the results from the giant model, we (conservatively) excluded stars with derived log g > 3.5, and for the results from the main sequence model, we excluded sub-giant stars (log g < 4 and T_eff < 5000 K) that were outside the two-dimensional (T_eff, log g) convex hull of the training set used for the main sequence model. Unfortunately, these restrictions did not remove all spurious results. The reason for this can be explained with an example: consider that our giant star model was trained with only giant stars but tested with both giant stars and dwarf stars. Some classes of stars (e.g., metal-poor dwarfs) can project into a region of label space that would suggest it is a giant (e.g., a clump star). These objects could have relatively low ${\chi }_{r}^{2}$ values (e.g., ${\chi }_{r}^{2}\lt 3$) and, in this example, they would appear as bonafide red clump stars. These incorrect projections are extrapolation errors in high dimensions that project to "normal" parts of the label space in two dimensions. For these reasons, we also made use of the simple model in Section 3.3 to inform whether we should adopt results from the red giant branch model, the main sequence/sub-giant model, or a linear combination of the two.

In Figure 4, we show the differences in effective temperature T_eff and surface gravity log g between the main sequence model (Section 3.4) and the simple model (Section 3.3), and the differences between the red giant branch model (Section 3.5) and the simple model (Section 3.3). We have scaled the differences in T_eff and log g to make the central peak near (0, 0) to be approximately isotropic by setting ${\delta }_{{T}_{\mathrm{eff}}}=90\,{\rm{K}}$ for the main sequence model, ${\delta }_{{T}_{\mathrm{eff}}}=50\,{\rm{K}}$ for the giant model, and ${\delta }_{\mathrm{log}g}=0.15\,\mathrm{dex}$ for both models. It is important to note that these δ values do not represent any kind of intrinsic uncertainty or precision: they are merely normalization factors. Empirically, we found that adopting substantially different scaling factors (e.g., a relative factor of two change) would produce clear inconsistencies in our results (e.g., sub-giants being misclassified as giants). Thus while the normalization factors are likely sensitive at the 100% level, our tests suggested that they were not sensitive within the ≈30% level (e.g., 120 K and 65 K, respectively). Therefore, we chose these factors empirically to make the distributions in Figure 4 approximately isotropic, and to some extent, comparable. We also note that these normalization factors are comparable to the rms scatter in the training sets of the giant model and the main sequence model, which qualitatively describes why differences with respect to the simple model become approximately isotropic when scaled with these normalization factors. In Figure 4, the stars within the peak at (0, 0) represent objects where the simple model and the comparison model both report similar labels. The artifacts seen in the Hertzsprung–Russell diagrams in Figure 2 are also present in Figure 4 as over-densities far away from the central peak. Therefore, we can adopt the scaled distance in labels T_eff and log g from the simple model to the main sequence model d_ms,

$$\begin{eqnarray}&&{d}_{\mathrm{ms}}={\left(\displaystyle \frac{{T}_{\mathrm{eff},\mathrm{ms}}-{T}_{\mathrm{eff},\mathrm{simple}}}{{\delta }_{{T}_{\mathrm{eff},\mathrm{ms}}}}\right)}^{2}+{\left(\displaystyle \frac{\mathrm{log}{g}_{\mathrm{ms}}-\mathrm{log}{g}_{\mathrm{simple}}}{{\delta }_{\mathrm{log}g,\mathrm{ms}}}\right)}^{2},\end{eqnarray} \tag{ 7 }$$

and the scaled distance from the simple model to the giant model d_giant,

$$\begin{eqnarray}&&{d}_{\mathrm{giant}}={\left(\displaystyle \frac{{T}_{\mathrm{eff},\mathrm{giant}}-{T}_{\mathrm{eff},\mathrm{simple}}}{{\delta }_{{T}_{\mathrm{eff},\mathrm{giant}}}}\right)}^{2}+{\left(\displaystyle \frac{\mathrm{log}{g}_{\mathrm{giant}}-\mathrm{log}{g}_{\mathrm{simple}}}{{\delta }_{\mathrm{log}g,\mathrm{giant}}}\right)}^{2},\end{eqnarray} \tag{ 8 }$$

to derive the weights,

$$\begin{eqnarray}&&{w}_{\mathrm{ms}}=\displaystyle \frac{1}{{{d}_{\mathrm{ms}}}^{2}}\,\mathrm{and}\,{w}_{\mathrm{giant}}=\displaystyle \frac{1}{{{d}_{\mathrm{giant}}}^{2}},\end{eqnarray} \tag{ 9 }$$

and produce the weighted labels $\hat{{\ell }}$:

$$\begin{eqnarray}&&\hat{{\ell }}=\displaystyle \frac{{w}_{\mathrm{ms}}\,{{\ell }}_{\mathrm{ms}}+{w}_{\mathrm{giant}}\,{{\ell }}_{\mathrm{giant}}}{{w}_{\mathrm{ms}}+{w}_{\mathrm{giant}}}.\end{eqnarray} \tag{ 10 }$$

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We calculate weighted errors of $\hat{{\ell }}$ in the same manner. In Figure 5, we show the mean relative weight w_ms/(w_ms + w_giant) within each two-dimensional hexagonal bin of ${\hat{T}}_{\mathrm{eff}}$ and $\hat{\mathrm{log}g}$. Hereafter, when we refer to labels (e.g., T_eff), we refer to those from the joint estimate $\hat{{\ell }}$, not individual estimates from separate models. For giant stars the relative weight of the main sequence model is zero, and vice-versa for main sequence stars. The relative weights smoothly transition from 0 to 1 on the sub-giant branch near $\mathrm{log}g\approx 3.5$, in the training set overlap region of both models. For abundance labels in the giant model that are not in the main sequence model (e.g., [O/H], [Mg/H]), we only report abundances for objects if w_ms < 0.05.

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The weighted labels for all stars are shown as normalized histograms in Figure 6. The normalization in each axis is arbitrary, since main sequence stars do not have detailed abundances here, there are fewer stars with—for example, [Ni/H] labels—than there are stars with T_eff labels. The [X/H] abundance distributions peak near solar values, and the peaks in the T_eff and log g histograms are consistent with our expectations from astrophysics: the bulk of clump stars is visible for cool giant stars (log g ≈ 2.5 and T_eff ≈ 4750 K), and the increase in turn-off stars at T_eff ≈ 6000 K is expected given the longer timescales for the turn-off.

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The weighted T_eff and log g values for stars meeting different S/N constraints are shown in Figure 7, both in logarithmic density and mean metallicity. The artifacts from individual models are no longer apparent, and the complete structure of the Hertzsprung–Russell diagram is visible. However, there are a number of caveats introduced by the decisions we have made on how to combine estimates from multiple models. We discuss these issues in detail in Section 5.

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4.1.1. Repeat Observations

The RAVE survey performed repeat observations for 43,918 stars with time intervals ranging from a few hours to up to four years. This timing was constructed to be quasi-logarithmic such that spectroscopic binaries could be optimally identified. Most of the stars that were observed multiple times were only observed twice, with 13 visits being the maximum number of observations for any target. These repeat observations allow us to quantify the level of (in)correctness in our formal errors.

We calculated all possible pair-wise differences between the labels we derived from multiple visits. If RAVE observed a star H times, there are $H!/2(H-2)!$ possible a-to-b pair-wise combinations where we can calculate the difference between the derived label (for example, log g) over the quadrature sum of their formal errors $(\mathrm{log}\,{\text{}}{g}_{a}-\mathrm{log}\,{\text{}}{g}_{b})/\sqrt{{{\sigma }_{\mathrm{log}{\text{}}g,a}}^{2}+{{\sigma }_{\mathrm{log}{\text{}}g,b}}^{2}}$. If our derived labels were unbiased and our formal errors were correct, the distribution of these pair-wise comparisons would be well-represented by a Gaussian distribution with zero mean and variance of unity. However, our formal errors are likely to be underestimated, and therefore we introduce a systematic error floor for each label, which is added in quadrature to every observation, such that (for example, log g),

$$\begin{eqnarray}&&{\eta }_{\mathrm{log}{\text{}}g}=\displaystyle \frac{\mathrm{log}\,{\text{}}{g}_{a}-\mathrm{log}\,{\text{}}{g}_{b}}{\sqrt{{{\sigma }_{\mathrm{log}{\text{}}g,a}}^{2}+{{\sigma }_{\mathrm{log}{\text{}}g,b}}^{2}+2{{\sigma }_{\mathrm{log}{\text{}}g,\mathrm{floor}}}^{2}}}.\end{eqnarray} \tag{ 11 }$$

We increased the minimum label error until the variance of the η distribution approximately reached unity. We found the minimum error in T_eff to be 70 K, 0.12 dex in log g, and varied between 0.06 and 0.08 dex for individual elements. The minimum errors are given with the distributions of η for each label in Figure 8. These minimum values form part of our error model, such that they have been added in quadrature with the formal errors; the quoted label errors in our catalog include these minimum errors.

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4.1.2. Precision as a Function of S/N

We further used the repeat observations in RAVE to build intuition for the label precision that was achievable as a function of S/N. Specifically, we stacked all spectra for a given star by summing the fluxes weighted by the inverse variances, then treated the stacked spectra as normal survey stars. We inferred labels for the stacked spectra using all three models, and derived a joint estimate as per Section 3.6. The labels we inferred from each stacked spectrum then served as a basis of comparison for the labels we derived from the individual visit spectra of the same star, which are of lower S/Ns.

In Figure 9, we show the rms difference in labels between the stacked spectra and single visit, binned by the S/N of the individual visit spectrum. Here we only show stars where the stacked spectrum had S/N > 100 pixel⁻¹ to ensure that our baseline comparisons were in a region where we are dominated by systematic uncertainties. The precision in all labels tends to flatten out past S/N > 40 pixel⁻¹, and the precision at high S/N is comparable to the minimum error floors we adopted in Section 4.1.1. The median S/N of RAVE spectra is 50 pixel⁻¹, at which point our abundance precision is about 0.07 dex, varying a few tenths of a dex between different elements.

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4.2.1. Comparison with RAVE DR4

We cross-matched our results against the official fourth RAVE data release (Kordopatis et al. 2013) as an initial point of external comparison (Figure 10). In order to provide a fair comparison, we only show stars that meet a number of quality flags in both samples. Our constraints require that the S/N exceeds 10 pixel⁻¹, and ${\chi }_{r}^{2}\lt 3$. For this comparison, we further required that the QK flag from Kordopatis et al. (2013) is zero, indicating that no problems were reported by the pipeline; ${T}_{\mathrm{eff},{DR}4}\gt 4000\,{\rm{K}}$; the error in radial velocity e_HRV is <8 km s⁻¹; and the three principal morphological flags c1, c2, c3, from Matijevič et al. (2012) all indicate "n" for a normal FGK-type star. There is good agreement in T_eff, with a bias and rms of just 4 K and 240 K, respectively. The offset in log g on the giant branch between this study and Kordopatis et al. (2013) has been noted in other studies (e.g., APOGEE), and this issue has been minimized in the fifth RAVE data release by correcting log g values with a calibration sample consisting of asteroseismic targets and the Gaia benchmark stars. There is also a slight discrepancy in the log g values along the main sequence, where our work tends to taper down toward higher log g values at cooler temperatures, and the RAVE DR4 sample tends to have a slightly flatter lower main sequence. This difference is not likely to have a very significant effect on the detailed abundance or spectrophotometric distance determinations between these studies (Binney et al. 2014).

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4.2.2. Comparisons with Reddy, Bensby, and Valenti & Fischer

We searched the literature for studies that overlap with RAVE, and which base their analysis on high-resolution, high S/N spectra. We found four notable studies with a sufficient level of overlap: the Milky Way disk studies by Reddy et al. (2003, 2006) and Bensby et al. (2014), as well as the Valenti & Fischer (2005) work on exoplanet host star candidates. These studies perform a careful (manual; expert) analysis using extremely high-resolution, high-S/N spectra, and make use of Hipparcos parallaxes where possible. Most of the stars in these samples are main sequence or sub-giant stars. Therefore, these works constitute an excellent comparison to evaluate the accuracy of our results on the main sequence and sub-giant branch.

In Figure 11, we show Hertzsprung–Russell diagrams for the RAVE stars that overlap with these studies. We only include stars with ${\chi }_{r}^{2}\lt 3$ and S/N > 10 pixel⁻¹, though the latter cut removed only a few stars because the average S/N in the RAVE spectra for these stars is relatively high (≳50 pixel⁻¹). The literature data points in Figure 11 are linked to our derived labels for the same stars, illustrating good qualitative agreement across the turn-off and sub-giant branch in all studies. If we treat all three studies as a single point of comparison, the bias between our work and these studies is −89 K in T_eff, just −0.06 dex in log g, and −0.03 dex in [Fe/H] (see Figure 12). The rms deviations in labels are 237 K, 0.30 dex, and 0.15 dex, respectively. When considering the relative information content available in RAVE (945 pixels in the near-infrared with ${ \mathcal R }\,\approx$ 7500) compared to these literature studies that use Hipparcos parallaxes where possible, and base their inferences on spectra with resolving power ${ \mathcal R }$ between 40,000 to 110,000, and S/N exceeding 150, we consider the agreement to be very satisfactory. Indeed, given the metallicity precision available in the RAVE-on catalog, these results will likely be useful for future studies based on exoplanet host star properties (e.g., the Transiting Exoplanet Survey Satellite, TESS²⁷ ).

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4.2.3. Comparison with the Gaia-ESO Survey

There are 142 stars that overlap between RAVE and the fourth internal data release of the Gaia-ESO survey. These are a mix of main sequence, sub-giant, and red giant branch stars. About half (67) of the sample were acquired with the UVES instrument—the other with the GIRAFFE spectrograph—and the S/N of the Gaia-ESO spectra peaks at ≈140 pixel⁻¹. Despite most of these stars having relatively low S/N in RAVE (≈25 pixel⁻¹), there is good agreement in with Gaia-ESO and the RAVE-on stellar parameters (Figure 13). The rms in effective temperature, surface gravity, and metallicity is 233 K, 0.37 dex, and 0.17 dex, respectively.

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Based on this comparison, we find no evidence for a systematic offset in metallicities between stars on the main sequence and those on the giant branch. This is a crucial observation, as the metallicities for stars in our main sequence training set have a principally different source than those on the giant branch. We cannot make these same inferences based on other surveys, like APOGEE, because (1) APOGEE stars formed part of the training set, and (2) they do not include main sequence stars. Even if we found good agreement between K2/EPIC and APOGEE metallicities, this would not be informative, because APOGEE is the source of metallicity for many stars on the giant branch in the K2/EPIC sample. Therefore, although this is a qualitative comparison only, it is reassuring that there is no obvious systematic difference between the metallicities of main sequence and giant branch stars.

The metallicity agreement between this work and Gaia-ESO extends down to low metallicity, near [Fe/H] ≈ −1.5. The scatter increases for the few stars in the overlap sample with [Fe/H] < −1, in the regime where the influence from atomic transitions of these elements becomes very small in RAVE spectra. Moreover, these particular stars have lower S/Ns, which is reflected in the larger uncertainties reported for these metallicities.

The fourth internal data release of the Gaia-ESO includes detailed chemical abundances of up to 45 species (≈32 elements at different ionization stages). This provides us with an independent validation for our detailed abundances on the giant branch. These comparisons are shown in Figure 14, where markers are colored by the S/N of the RAVE spectrum. The number of stars available in each abundance comparison varies due to what is available in the Gaia-ESO data release, which is itself a function of the instrument used, the spectral type, and other factors. The absolute bias for individual elements varies from as low as 0.06 dex ([Al, Mg/H]) to as high as 0.26 dex ([Si/H]), where we overestimate [Si/H] abundances relative to the Gaia-ESO survey. The large bias in [Si/H] is likely a consequence of an offset between [Si/H] abundances in Gaia-ESO and APOGEE, the source of our training set for giant star abundances. The rms deviation in each label is small for stars with [X/H] > −0.5, before increasing at lower metallicities. If we consider all stars, the smallest abundance rms we see with respect to Gaia-ESO is 0.16 dex for [Ca/H] and [Al/H]. The increasing rms at low metallicity is likely a consequence of multiple factors, namely, inaccurate abundance labels for metal-poor stars (Section 3.2); only weak, blended lines being available in RAVE, which cease to be visible in hot and/or metal-poor stars; and to a lesser extent, low S/N for those particular stars being compared. Unfortunately, not all of these factors are represented by the quoted errors in each label. For these reasons, although it affects only a small number of stars, we recommend caution when using individual abundances for very metal-poor giant stars in our sample.

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4.2.4. Comparison with the RAVE DR4 Calibration Sample

The fourth RAVE data release made use of a number of high-resolution studies to verify the accuracy of their derived stellar atmospheric parameters. These samples include main sequence stars and giant stars, with a particular focus to include metal-poor stars to identify (and correct) any deviations at low metallicities. We refer the reader to Kordopatis et al. (2013) for the full compilation of literature sources. Although the stellar atmospheric parameters in this compilation come from multiple (heterogeneous) sources, we find generally good agreement with these works (Figure 15). However, we note that some reservation is warranted when evaluating this comparison, as some of the metal-poor stars in this calibration sample formed part of our training set.

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4.3.1. Globular Clusters

After verifying that our atmospheric parameters and abundances are comparable with high-resolution studies, here we verify that our results are consistent with expectations from astrophysics. In the RAVE survey, Anguiano et al. (2015) identified 70 stars with positions and radial velocities that are consistent with being members of globular clusters: 49 stars belonging to NGC 5139 (ω Centauri), 11 members of the retrograde globular cluster NGC 3201, and 10 members of NGC 362. In addition, Kunder et al. (2014) compiled 12 stars thought to belong to NGC 1851, and a further 10 stars in NGC 6752. We refer the reader to those studies for details regarding the membership selection.

In Figure 16, we show our effective temperature T_eff and surface gravity log g for these prescribed globular cluster members. The right-hand panels indicate measurements made in this work, and for comparison purposes we have included the results from the fourth RAVE data release in the left-hand panels. We show representative PARSEC isochrones (Bressan et al. 2012) in all panels, where the isochrone ages and metallicities are adopted from Kunder et al. (2017), Marín-Franch et al. (2009), and the Harris (1996, accessed 6 September 2016) catalog of globular cluster properties. The globular cluster with the highest number of members is NGC 5139 (ω-Centauri), where we find a significant metallicity spread that is consistent with high-resolution studies (Carretta et al. 2009, 2013; Marino et al. 2011). Based on the pre-defined membership criteria, we find the mean metallicity of ω-Centauri to be [Fe/H] = −0.85. However, it is clear that the membership criteria could be improved with our revised metallicities and detailed chemical abundances. Indeed, our individual abundance labels could be further used to identify globular cluster members—at least, of relatively metal-rich clusters—that are now tidally disrupted (Anguiano et al. 2016; Kuzma et al. 2016; Navin et al. 2016), even for stars with low S/Ns.

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4.3.2. Open Clusters

Using positions, proper motions, and metallicities from the RAVE survey (i.e., not ours, such that they can be used as comparison), we identify ∼160 probable members of four open clusters that were observed by RAVE. Specifically, we identify 78 potential Pleiades members, 26 candidates in the Hyades, another 13 in IC4561, and 30 stars in the solar-metallicity open cluster M67. We show the effective temperature T_eff and surface gravity log g for these cluster candidates in Figure 17. The isochrones are sourced from Bressan et al. (2012), with cluster properties adopted from Kharchenko et al. (2013).

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We find good agreement between our atmospheric parameters (right-hand panels) and the isochrones shown. The position of the red clump in IC4651 and M67 are perfectly matched to the isochrone, and the Hyades main sequence is in good agreement down to ${T}_{\mathrm{eff}}\approx 4000\,{\rm{K}}$. Similarly, we find consistency with the literature and our metallicity scale. We find the mean metallicity of M67 stars to be [Fe/H] = −0.02 ± 0.03 dex, in excellent agreement with the expected [Fe/H] = 0.00 value (not accounting for atomic diffusion). We further find the Hyades mean metallicity to be [Fe/H] = 0.07 ± 0.09 dex, consistent with Paulson et al. (2003): [Fe/H] ≈ 0.13. For IC4651 from 13 stars we find a mean [Fe/H] = 0.15 ± 0.03 dex, matching the high-resolution, high-S/N study of Pasquini et al. (2004), where they find [Fe/H] = 0.10 ± 0.03 dex. Finally, from 78 stars, we find the mean metallicity of the Pleiades to be [Fe/H] = −0.02 ± 0.01 dex, in very good agreement with the [Fe/H] = −0.034 ± 0.024 dex measurement reported by Friel & Boesgaard (1990).

Despite the discrepancies between the isochrone and our derived labels for stars in the Pleiades, we have made no attempt to refine the membership selection in any of the aforementioned clusters. We note, however, that the same discrepancy with the isochrone appears present in the fourth RAVE data release Kordopatis et al. (2013).

Source code for this project is available at https://www.github.com/AnnieJumpCannon/rave, and this document was compiled from revision hash 7cafdd5 in that repository. Derived labels, associated errors, and relevant metadata are available electronically from Zenodo (Casey et al. 2016c) at 10.5281/zenodo.154381, or from the RAVE online database. Please note that it is a condition of using these results that the RAVE data release by Kunder et al. (2017) must also be cited, as the work presented here would not have been possible without the tireless efforts of the entire RAVE collaboration, past and present.