Data-driven Spectroscopy of Cool Stars at High Spectral Resolution

To prepare the spectral library for The Cannon, we must ensure that the spectra satisfy certain conditions; the spectra must share a common wavelength grid, be shifted onto the rest wavelength frame, share a common line-spread function, and be continuum-normalized via a method independent of S/N (Ness et al. 2015). The first two conditions are already satisfied for the library spectra, and we can assume that they effectively share a line-spread function, though there may be negligible variation due to variable observation seeing conditions. To carry out normalization, we applied error-weighted, broad Gaussian smoothing with

$$\begin{eqnarray}&&\bar{f}({\lambda }_{0})=\displaystyle \frac{{\displaystyle \sum }_{j}({f}_{j}{\sigma }_{j}^{-2}{w}_{j}({\lambda }_{0}))}{{\displaystyle \sum }_{j}({\sigma }_{j}^{-2}{w}_{j}({\lambda }_{0}))},\end{eqnarray} \tag{ 1 }$$

where f_j is the flux at pixel j of the wavelength range, σ_j is the uncertainty at pixel j, and the weight w_j (λ₀) is drawn from a Gaussian:

$$\begin{eqnarray}&&{w}_{j}({\lambda }_{0})={e}^{-\tfrac{{\left({\lambda }_{0}-{\lambda }_{j}\right)}^{2}}{{L}^{2}}},\end{eqnarray} \tag{ 2 }$$

where L was chosen to be 3 Å. If larger L values are chosen for HIRES spectra, continuum-normalization begins to remove high-resolution features. For reference, Ho et al. (2017) used a width of L = 50 Å to normalize low-resolution Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) spectra (R ≈ 1800). The Gaussian smoothing procedure is illustrated in Figure 2.

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We used The Cannon 2, the second implementation of The Cannon developed by Casey et al. (2016). Hereafter, we will refer to The Cannon 2 simply as The Cannon. This version builds upon the original with additional features that are designed to aid prediction of a larger label set including elemental abundances that go beyond bulk metallicity ([Fe/H]), such as regularization.

As outlined in Section 1, in the training step, The Cannon uses a set of reference objects with well-determined labels to construct a predictive model of the flux at every pixel of the wavelength range that is a function of the stellar labels. Model construction is based on two assumptions: that continuum-normalized spectra with identical labels look identical at every pixel, and that the flux at every pixel in a spectrum changes continuously as a function of the stellar labels. While The Cannon can be trained on any set of empirical spectra and their labels, the resultant model will only be capable of predicting labels for spectra with properties that are represented in the training set. In other words, The Cannon is not able to accurately extrapolate outside the training set parameter space, so the training set spectra must be representative of the test set spectra in order to predict accurate label values. It is also important to note that the Cannon-predicted labels will only be as accurate as those of the training set.

The flux model ${f}_{{jn}}$ for a spectrum n at pixel j can be written as

$$\begin{eqnarray}&&{f}_{{jn}}={\boldsymbol{v}}({l}_{n})\cdot {{\boldsymbol{\theta }}}_{j}+{e}_{{jn}}\hspace{1mm},\end{eqnarray} \tag{ 3 }$$

where ${{\boldsymbol{\theta }}}_{j}$ is the set of spectral model coefficients at each pixel j and ${\boldsymbol{v}}({l}_{n})$ is a function of the label list l_n that is unique for each spectrum n. The function ${\boldsymbol{v}}({l}_{n})$ is referred to as the "vectorizer" which can accommodate functions that are linear in the coefficients ${{\boldsymbol{\theta }}}_{j}$, but not necessarily simple polynomial expansions of the label list l_n. The noise term is described by e_jn and can be taken as sampled from a Gaussian with zero mean and variance ${\sigma }_{{jn}}^{2}+{s}_{j}^{2}$ where ${\sigma }_{{jn}}^{2}$ is the uncertainty reported on the input HIRES spectra (flux variance) and s_j² is the intrinsic scatter of the model at each pixel j. This intrinsic scatter can be likened to the expected deviation of the model from the spectrum at j.

To determine the optimal model labels (${{\boldsymbol{\theta }}}_{j}$,s_j²), we can relate the flux model to a single-pixel log-likelihood function:

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}p({f}_{{jn}}| {{\boldsymbol{\theta }}}_{j},{\boldsymbol{v}}({l}_{n}),{s}_{j}^{2}) & = & -\displaystyle \frac{{\left[{f}_{{jn}}-{\boldsymbol{v}}({l}_{n})\cdot {\boldsymbol{\theta }}\right]}^{2}}{{\sigma }_{{jn}}^{2}+{s}_{j}^{2}}\\ & & -\,\mathrm{ln}({\sigma }_{{jn}}^{2}+{s}_{j}^{2})-{\rm{\Lambda }}Q({\boldsymbol{\theta }}),\end{array}\end{eqnarray} \tag{ 4 }$$

where Λ is a regularization parameter and $Q({\boldsymbol{\theta }})$ is a regularizing function that encourages the model coefficients ${{\boldsymbol{\theta }}}_{j}$ to take on zero values, resulting in a simpler model that is less prone to overfitting. In the case of L1 regularization implemented within The Cannon, the regularizing function takes the form

$$\begin{eqnarray}&&Q({\boldsymbol{\theta }})=\displaystyle \sum _{j=0}^{J-1}| {\theta }_{j}| .\end{eqnarray} \tag{ 5 }$$

L1 regularization was chosen because The Cannon is designed for predicting large sets of elemental abundances, and it is reasonable to assume that only one or a few elemental abundances will affect the flux at a single pixel of the wavelength range. For more details on regularization or the model itself, see Casey et al. (2016).

In the training step, the log-likelihood is maximized via the Broyden–Fletcher–Goldfarb–Shanno algorithm to derive the best-fit model coefficients ${{\boldsymbol{\theta }}}_{j}$ and intrinsic scatter s_j²:

$$\begin{eqnarray}&&{{\boldsymbol{\theta }}}_{j},{s}_{j}^{2}\leftarrow \mathop{\mathrm{argmax}}\limits_{{\boldsymbol{\theta }},{s}_{j}}\left[\displaystyle \sum _{n=0}^{N-1}\mathrm{ln}p({f}_{{jn}}| {{\boldsymbol{\theta }}}_{j},{\boldsymbol{v}}({l}_{n}),{s}_{j}^{2})\right].\end{eqnarray} \tag{ 6 }$$

Plugging in the explicit form of the log-likelihood (Equation (4)) leads to

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{\theta }}}_{j},{s}_{j}^{2}\leftarrow \mathop{\mathrm{argmax}}\limits_{{\boldsymbol{\theta }},{s}_{j}}\left[\displaystyle \sum _{n=0}^{N-1}-\displaystyle \frac{{\left[{f}_{{jn}}-{\boldsymbol{v}}({l}_{n})\cdot {\boldsymbol{\theta }}\right]}^{2}}{{\sigma }_{{jn}}^{2}+{s}_{j}^{2}}\right.\\ \quad -\left.\,\displaystyle \sum _{n=0}^{N-1}\mathrm{ln}({\sigma }_{{jn}}^{2}+{s}_{j}^{2})-{\rm{\Lambda }}Q({\boldsymbol{\theta }})\right].\end{array}\end{eqnarray} \tag{ 7 }$$

In the test step, we set the model labels (${{\boldsymbol{\theta }}}_{j}$, s_j²) to the optimized values determined during the training step at every pixel j, and fit for the label list l_n for each star n that minimizes the log-likelihood:

$$\begin{eqnarray}&&{l}_{n}\leftarrow \mathop{\mathrm{argmin}}\limits_{{l}_{n}}\left[\displaystyle \sum _{j=0}^{J-1}-\displaystyle \frac{{\left[{f}_{{jn}}-{\boldsymbol{v}}({l}_{n})\cdot {\boldsymbol{\theta }}\right]}^{2}}{{\sigma }_{{jn}}^{2}+{s}_{j}^{2}}\right].\end{eqnarray} \tag{ 8 }$$

Optimization of the log-likelihood in the test step is carried out via weighted least squares.

Before running The Cannon on our cool star sample, we sought to establish a measure of baseline performance. We did this by constructing a sample of synthetic spectra that mimics the cool star sample: 141 "stars" with the same label values as the true cool sample. Because the labels of the synthetic spectra, by definition, lack uncertainty, they can provide a sense of how well The Cannon performs under perfect conditions. The synthetic spectra are generated from the publicly available code SpecMatch-Syn, which fits five regions of optical spectra by interpolating within a grid of model spectra from the library described in Coelho et al. (2005). For more details on SpecMatch-Syn, see Petigura (2015). See Figure 3 for examples of synthetic spectra.

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We tested the validity of Cannon-predicted labels through a bootstrap leave-one-out cross-validation scheme where we trained the spectral model on all objects in the synthetic spectral sample but one, and predicted labels for the object that was left out. We carried out this scheme iteratively to pass through the entire sample and predict labels for every object. Following the work of Ness et al. (2015) and Casey et al. (2016), we began with a spectral model in which the label list l_n was quadratic in the labels, resulting in the following label list:

$$\begin{eqnarray}&&\begin{array}{l}{l}_{n}\equiv [1,{T}_{\mathrm{eff}},{R}_{* },[\mathrm{Fe}/{\rm{H}}],{T}_{\mathrm{eff}}^{2},{T}_{\mathrm{eff}}\cdot {R}_{* },\\ {T}_{\mathrm{eff}}\cdot [\mathrm{Fe}/{\rm{H}}],{R}_{* }^{2},{R}_{* }\cdot [\mathrm{Fe}/{\rm{H}}],{\left[\mathrm{Fe}/{\rm{H}}\right]}^{2}].\end{array}\end{eqnarray} \tag{ 9 }$$

Where 1, the first element in the label list, is there to allow for a linear offset in the fitting (Ness et al. 2015). We found that modeling the projected rotational velocity v sin i as a fitted-for parameter in addition to T_eff, R_*, and [Fe/H] resulted in more accurate label predictions; a second-order model without v sin i achieves accuracies of 40 K in T_eff, 13% in R_*, and 0.06 dex in [Fe/H], while a second-order model with v sin i achieves accuracies of 32 K in T_eff, 13% in R_*, and 0.03 dex in [Fe/H].

Using a third-order rather than second-order (quadratic-in-label) model with v sin i further improves label predictions; a third-order model achieves accuracies of 22 K in T_eff, 8% in R_*, and 0.03 dex in [Fe/H]. Thus, these tests with synthetic spectra motivate a third-order Cannon model with v sin i included as a label. The third-order model results in a label list composed of additional third-order cross terms, bringing the total number of terms up to 20.

To run The Cannon on the cool star sample, we employed the same bootstrap leave-one-out cross-validation scheme. As in the case of synthetic spectra, the cool star HIRES spectra are best described by a third-order model, which is unsurprising given their resolution of R ≈ 60,000 (∼3 times the resolution of APOGEE spectra). The more flexible model may also better-describe our more diverse training set, composed of stars with a wider T_eff range (APOGEE stars are confined to T_eff = 3500–5500 K). A third-order model fitting for T_eff, R_*, and [Fe/H] achieves precisions of 80 K in T_eff, 6% in R_*, and 0.1 dex in [Fe/H].

We found that The Cannon predicted anomalously poor label values for one source (GL896A). Upon inspection, we found that the spectrum of GL896A exhibits significantly broader features than any other source in our sample. Because GL896A is not well-represented in the training set, The Cannon is unable to construct a model that well-describes GL896A (Figure 4, top panel). While such fast rotators are rare among K and M dwarfs, the presence of this target indicates that our implementation of The Cannon must still take them into consideration. We modified our implementation by augmenting and diversifying the training set; we created x copies of each spectrum in the cool star sample (exploring different values of x to see which resulted in the best performance), and applied differential values of artificial broadening to simulate faster stellar rotation. Artificial broadening was carried out by convolving the spectra with a rotational-macroturbulent broadening kernel described in Hirano et al. (2011).

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In order for this scheme to work, v sin i must be specified as a fitted-for label as in the tests with synthetic spectra. This is problematic because more than half of the cool stars do not have reported v sin i values. We dealt with this by assigning all sources in the augmented sample a new label that describes general broadening, taken to be the FWHM of a Gaussian fitted to the spectral autocorrelation peaks (Figure 5). This resulted in better flux predictions for the spectrum of GL896A (Figure 4), and better label predictions overall. The most precise labels are achieved when the cool star sample is augmented by x = 5 (five copies generated for each spectrum), and the copies are artificially broadened by 0–5 km s⁻¹ as the cool star sample does not appear to include any significantly rapid rotators (v sin i > 5 km s⁻¹). We ultimately achieved precisions of 68 K in T_eff, 5% in R_*, and 0.08 dex in [Fe/H] (Figure 6, left panel, and verified that these label predictions vary within the reported precisions for different Cannon runs. While it may seem surprising that The Cannon achieves better predictions in R_* for empirical spectra compared to synthetic spectra (5% versus 8% in R_*), it should be noted that the synthetic spectra may not accurately reflect the input R_* values as a conversion to log g was required, which in turn required M_*. We do not have M_* values for the cool star sample, and instead assumed a linear relationship between R_* and M_* (${M}_{* }/{M}_{\odot }={R}_{* }/{R}_{\odot }$) to obtain mass estimates. This is a valid approximation for the main sequence, but is not perfect.

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Because of the large number of terms in our model, we considered overfitting to be a potential issue. That is, overly precise modeling of the training set flux may lead to less accurate label predictions. To address this, we added regularization to our Cannon model and assessed whether label prediction improved. We explored a grid of regularization strengths from Λ = 10⁻⁶ to Λ = 10² uniform in log space. We found that no matter the regularization strength, adding regularization to the model always resulted in less precise label predictions. It is possible that regularization does not lead to better predictions for the three labels (T_eff, R_*, and [Fe/H]) we are considering because all of these labels affect the flux at each wavelength point. Thus, we do not benefit from regularization that encourages sparsity (L1, encourages the model coefficients to go to zero). L1 regularization may lead to better label predictions if we expand our label set to include elemental abundances, but that is beyond the scope of this study.

To investigate the effect of photon shot noise on the precision of label predictions made with The Cannon, we carried out the same procedure employed by Yee et al. (2017) for the empirical spectroscopic tool SpecMatch-Emp; we isolated a subset of 20 stars from the cool star sample with S/N > 160/pixel and degraded their spectra by injecting Gaussian noise to simulate target S/N values of 120, 100, 80, 60, 40, 20, and 10 per pixel. We generated 20 S/N-degraded spectra for each spectrum in the subset and S/N target value, then compared the precision of the Cannon label predictions for the degraded spectra with those of the original S/N > 160/pixel spectra, which we took as ground truth. The results are summarized in Figure 7.

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As expected, lower S/N leads to larger median scatter in label predictions made with The Cannon. However, the median scatter at S/N = 10/pixel is still low, with 3.5 K in T_eff, 0.4% in R_*, and 0.006 dex in [Fe/H]. This demonstrates that The Cannon is quite robust, even at low S/N values. This performance is better than that achieved by SpecMatch-Emp, which has median scatter values at S/N = 10/pixel of 10.4 K in T_eff, 1.7 % in R_*, and 0.008 dex in [Fe/H] (Yee et al. 2017), though it should be noted that SpecMatch-Emp conducted this test with stars spanning the HR diagram while our sample is T_eff-limited.

Motivated by the small observed scatter in [Fe/H], we attempted to estimate the minimum change in [Fe/H] that is theoretically detectable. To do so, we considered the difference between two spectra corresponding to stars with slightly different metallicities (Δ[Fe/H]). We defined a quantity ${ \mathcal S }$ that relates three quantities: Δ[Fe/H]; the derivative of the spectrum with changing metallicity, $\delta f/\delta [\mathrm{Fe}/{\rm{H}}];$ and the flux uncertainty σ_f. ${ \mathcal S }$ can be thought of as analogous to S/N. For the jth pixel, the relation is

$$\begin{eqnarray}&&{{ \mathcal S }}_{j}=\displaystyle \frac{{\left(\tfrac{\delta f}{\delta [\mathrm{Fe}/{\rm{H}}]}\right)}_{j}{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}{{\sigma }_{f,j}}.\end{eqnarray} \tag{ 10 }$$

This equation can be rewritten as

$$\begin{eqnarray}&&{{ \mathcal S }}_{j}=\displaystyle \frac{{\left(\tfrac{\delta f}{\delta [\mathrm{Fe}/{\rm{H}}]}\right)}_{j}{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}{{c}_{j}\langle {\sigma }_{f}\rangle },\end{eqnarray} \tag{ 11 }$$

where $\langle {\sigma }_{f}\rangle$ is the average flux uncertainty and c_j is directly related to the blaze function. The total ${ \mathcal S }$ (summing over pixels) of the metallicity measurement can be written as

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal S } & = & \sqrt{\displaystyle \sum _{j}{\left({{ \mathcal S }}_{j}\right)}^{2}}\hspace{3mm}\\ & = & \displaystyle \frac{{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]}{\langle {\sigma }_{f}\rangle }\sqrt{\displaystyle \sum _{j}{\left[\left(\displaystyle \frac{\delta f}{\delta [\mathrm{Fe}/{\rm{H}}]}\right)/{c}_{j}\right]}^{2}}.\end{array}\end{eqnarray} \tag{ 12 }$$

Rearranging terms to solve for the minimum theoretically detectable change in metallicity yields

$$\begin{eqnarray}&&{\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]={ \mathcal S }\langle {\sigma }_{f}\rangle /\sqrt{\displaystyle \sum _{j}{\left[\left(\displaystyle \frac{\delta f}{\delta [\mathrm{Fe}/{\rm{H}}]}\right)/{c}_{j}\right]}^{2}}.\end{eqnarray} \tag{ 13 }$$

A metallicity change is detectable at 1σ if ${ \mathcal S }=1$. For an S/N = 10/pixel as considered above, i.e., $\langle {\sigma }_{f}\rangle =0.1$, we find Δ[Fe/H] = 0.001 dex. This is much smaller than the median scatter in [Fe/H] predictions made with The Cannon at S/N = 10/pixel (0.006 dex). Therefore, the sensitivity of The Cannon lies within theoretical bounds.

While HIRES spectra are observed at R ≈ 60,000, many large spectroscopic surveys are observed at lower spectral resolutions. Thus, it is valuable to quantify how spectral resolution affects the accuracies of label predictions with The Cannon. We expected performance to decrease as spectral resolution decreases because lines will blend together, resulting in less spectral information for The Cannon to work with.

To investigate spectral resolution dependence, we followed the same procedure used for the S/N degradation test; we used the same subset of 20 stars with S/N > 160 and simulated lower resolution by convolving their spectra with a Gaussian kernel. We again treated the label predictions of the original high-resolution (R ≈ 60,000) spectra as ground truth. We simulated spectra with target resolution values of R = 50,000, 40,000, 30,000, 20,000, 10,000, 7500, and 5000. The results are summarized in Figure 7, which also illustrates how the precisions of label predictions are affected when both S/N and resolution are degraded.

As in the case of degraded S/N, the accuracy of Cannon label predictions decrease with spectral resolution. At R = 30,000, median scatter in the labels is 6.7 K in ${T}_{\mathrm{eff}}$, 0.2% in R_*, and 0.009 dex in [Fe/H]. This performance is better than that of SpecMatch-Emp's at equivalent resolution, with median scatter values of 10.1 K in T_eff, 1.3% in R_*, and 0.014 dex in [Fe/H] (Yee et al. 2017).

The Cannon also exhibits a much slower reduction in label accuracy as resolution continues to decrease; at R = 5000, the median scatter in Cannon predictions is 24.1 K in T_eff, 4.7% in R_*, and 0.026 dex in [Fe/H], while SpecMatch-Emp's is 962 K in T_eff, 228% in R_*, and 0.094 dex in [Fe/H] (Yee et al. 2017). This suggests that The Cannon would be a favorable method for predicting labels for spectra from many large, lower resolution spectroscopic surveys (e.g., SEGUE (Beers et al. 2006), R ≈ 2000, RAVE (Steinmetz et al. 2006), R ≈ 7000, LAMOST (Newberg et al. 2012), R ≈ 1800).

To investigate the effect of errors in the library labels on predictions made with The Cannon, we followed the same procedure used for the S/N and resolution degradation tests; we used the same subset of 20 stars with S/N > 160 and injected Gaussian noise into the labels to simulate additional uncertainty up to 1× the achievable precisions (68 K in T_eff, 5% in stellar radius R_*, and 0.08 dex in [Fe/H]). We found that the labels are quite robust to realistic random noise in the library labels; adding 1× uncertainty leads to an increase in label prediction uncertainties of 22 K in T_eff, 4% in R_*, and 0.06 dex in [Fe/H]. The results are summarized in Table 1.

Table 1.  Median rms Scatter in All Cannon-derived Labels after Adding 1× the Amount of Uncertainty to All Labels

Added Uncertainty	σ(Teff)	σ(Δ ${R}_{* }/{R}_{* }$)	σ([Fe/H])
(Teff, R*, [Fe/H])	K	%	dex
68 K	22	2	0.02
5% R*	17	4	0.02
0.08 dex	10	1	0.06

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It is worth noting that the scatter in Cannon-predicted values of T_eff is lower than the original label uncertainty by more than 50%, suggesting that in the limit of a very large library with labels containing a certain amount of random noise, The Cannon can derive a model that yields a higher T_eff precision compared to that of the library spectra. We note that this result is insensitive to zero-point offsets; it is not possible to bootstrap to higher label precisions using The Cannon.