A Robust Identification Method for Hot Subdwarfs Based on Deep Learning

Based on the LAMOST DR7-V1 catalog, we created a sample set that extracted the data with all spectra classes (O, hot subdwarf, white dwarf, B, A, F, G, K, and M) to meet the labeling requirements of the deep-learning techniques. Referring to the sample selection criteria of Bu et al. (2019), we selected samples with a signal-to-noise ratio (S/N) greater than 10 to construct the sample set. When the S/N was set to be greater than 10, there were sufficient samples for hot subdwarf, white dwarf, B, A, F, G, K, and M stars, while there were only 72 O-type stars.

To ensure the balance of the sample, we have to augment O-type stars according to the characteristics of the existing 72 samples. The spectrum of a star is a blackbody radiation spectrum, which is accompanied by emission and absorption lines. That is to say, the generation of a spectrum needs to satisfy characteristics of both two aspects, such as a blackbody radiation spectrum and emission/absorption lines.

1. Blackbody radiation spectrum fitting

The blackbody radiation spectrum of a star is defined by the Planck formula (Equation (1), a relationship between the flux, the wavelength (λ), and the temperature (t) of the spectrum). To further compensate for the effects of relative flux correction and sky background, we modify the formula to Equation (2), where a compensates for effects of relative flux correction, and b is compensated for by effects of the skylight background.

$$\begin{eqnarray}&&\mathrm{flux}(\lambda ,t)=\displaystyle \frac{8\pi {hc}}{{\lambda }^{5}}\cdot \displaystyle \frac{1}{{e}^{\tfrac{\mathrm{hc}}{\lambda {kt}}}-1}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\mathrm{flux}(\lambda ,t,a,b)=a\cdot \displaystyle \frac{8\pi \mathrm{hc}}{{\lambda }^{5}}\cdot \displaystyle \frac{1}{{e}^{\tfrac{{hc}}{\lambda {kt}}}-1}+b\end{eqnarray} \tag{ 2 }$$

In order to fit the temperature t, a, and b in Equation (2), in practice we obtained 3700 flux values with wavelengths from 3700–8672 Å by truncating each O-type star spectrum. The emission and absorption lines were removed by median filtering, and the corresponding blackbody radiation was obtained. The fitting process can be realized by the scipy.optimize.curve_fit ⁸ function in the Python Scipy package. For each known O-type star spectrum, optimal values were obtained for the parameters (t, a, b) so that the sum of the squared residuals of ${\mathrm{flux}}_{i}-{\mathrm{flux}}_{i}^{{\prime} }$ was minimized (i.e., Equation (3) was minimized). In Equation (3), flux is the original blackbody radiation spectrum and $\mathrm{flux}^{\prime}$ is the fitted flux value obtained by substituting the fitted parameters into Equation (2).

$$\begin{eqnarray}&&\alpha =\sum _{i=1}^{3700}{({\mathrm{flux}}_{i}-{\mathrm{flux}}_{i}^{{\prime} })}^{2}\end{eqnarray} \tag{ 3 }$$

Consequently, 72 sets of parameters (t, a, b) were obtained by fitting 72 O-type samples. Figure 1 shows the real blackbody radiation spectrum and the generated blackbody radiation spectrum, proving consistent in their physical characteristics.

Zoom In Zoom Out Reset image size

2. Obtaining emission and absorption lines

Considering that the emission and absorption lines for a specific class are basically the same, the emission and absorption lines of O-type stars can be obtained by subtracting the blackbody radiation spectrum (obtained by median filtering) from the original spectrum. For the 72 known O-type stars, 72 sets of emission and absorption lines can finally be obtained.

To generate a new blackbody radiation spectrum of the O-type star, we set a random temperature (between 28,000 and 100,000 K) and pick a random set of parameters a, b to obtain the flux calculating by Equation (2). The final generated spectrum was obtained by randomly combining the fitted blackbody radiation spectrum with the emission and absorption lines. Algorithm 1 lists the spectrum generation steps in pseudocode format. From the model training results in Section 3.3, the generated data can effectively solve the overfitting problem caused by the insufficient amount of data and get a good accuracy rate at the same time, which meets the requirements of deep learning.

Algorithm 1. O-type spectrum generation

Input:

• Sepcoriginal: 72 O-type samples, each spectrum contains 3700 flux values at wavelength from 3700 to 8672 Å

• Totype: Temperature of O-type star, $T\_{\rm{otype}}\in [28,000{\rm{K}},100,000{\rm{K}}]$

• Modified Planck formula (Equation (2)): ${\rm{flux}}(\lambda ,t,a,b)=a\cdot \displaystyle \frac{8\pi {hc}}{{\lambda }^{5}}\cdot \displaystyle \frac{1}{{e}^{\tfrac{{hc}}{\lambda {kt}}}-1}+b$

Output:

• Generated O-type spectrum, Sepcgeneration

1: for each spectrum (${\rm{Sepc}}\_{{\rm{original}}}_{i}$), $i\in [1,72]$ do

2: /*Step 1: blackbody radiation fitting*/

3: 1) Perform median filtering on the spectrum and obtain the blackbody radiation spectrum, ${{\rm{flux}}}_{i};$

4: 2) According to the modified Planck formula, fit and minimize the value of the objective function ${\sum }_{{i}=1}^{3700}{({{\rm{flux}}}_{{i}}-{{\rm{flux}}}_{{i}}^{{\prime} })}^{2};$

5: 3) Obtain the corresponding value of the parameters, temperature (ti ), ai , and bi .

6:

7: /*Step 2: acquisition of absorption and emission lines*/

8: 4) Obtain the emission and absorption line: ${\rm{em}}\_{\rm{ab}}\_{{\rm{line}}}_{i}={\rm{sepc}}\_{{\rm{original}}}_{i}-{{\rm{flux}}}_{i}$.

9: end for

10:

11: for $i\in [1,N({\rm{number}}\_{\rm{of}}\_{\rm{spectrum}}\_{\rm{to}}\_{\rm{be}}\_{\rm{generated}})]$ do

12: /*Step 3: O-type spectrum generation*/

13: 1) Randomly choose a temperature ($T\_{{\rm{otype}}}_{k}$) between 28,000 K and 100,000 K;

14: 2) Randomly choose a set of a and b obtained in Step 1, and feed a, b and $T\_{{\rm{otype}}}_{k}$ to Equation (2) (where t = $T\_{{\rm{otype}}}_{k}$) to generate the blackbody radiation spectrum, ${\rm{spec}}\_{\rm{blackbody}}\_{{\rm{gen}}}_{k};$

15: 3) Randomly choose a set of emission and absorption line obtained (${\rm{em}}\_{\rm{ab}}\_{{\rm{line}}}_{i}$) in Step 2;

16: 4) Generate O-type spectrum: Sepcgeneration=${\rm{spec}}\_{\rm{blackbody}}\_{{\rm{gen}}}_{k}+{\rm{em}}\_{\rm{ab}}\_{{\rm{line}}}_{i}$

17: end for

Download table as:  ASCIITypeset image

In this section, a CNN-based model for classifying the spectra is proposed. A straightforward idea of model construction is to classify all the spectra into nine classes and get different types of candidates. However, the preliminary experiments proved that the probability of confusing hot subdwarfs and white dwarfs in the nine-class classification model is high due to their similar blackbody radiation characteristics. Therefore, a more refined hybrid model with an eight-class classification model and a binary classification model is constructed. The eight-class classification model disregards the effect of white dwarfs, and the binary classification model further separates hot subdwarfs from white dwarfs.

The network consists of six convolutional layers (Figure 2). A max-pooling layer follows each convolutional layer. After the convolutional and max-pooling layers, two fully connected layers are added. In the convolutional layer, we process the output data by the ReLU function (Nair & Hinton 2010). The cross entropy function (Hinton & Salakhutdinov 2006) is used as the loss function, and the stochastic gradient descent function (Hardt et al. 2016) is used as the optimizer. During training, we start with small convolutional kernels and gradually increase the size of the kernels, and the final parameters of each layer are given in Table 1. The Softmax function (Ian et al. 2016) gives the final classification results, which are probability values of the samples being classified into different classes.

After sample augmentation of the O-type star spectrum, we randomly selected data and performed manual validation until there were 2000 samples for each class (O, hot subdwarf, white dwarf, B, A, F, G, K, and M stars; a total of 18,000 samples). 70% of the samples were randomly selected as the training set, and 30% were used for testing. Considering that our goal was to construct a model for searching hot subdwarfs, we did not include white dwarfs in the training and testing set of the eight-class classification model. A total of 3700 flux values in a wavelength range of 3700–8672 Å were obtained by truncating each sample. Specifically, LAMOST DR7-V1 provides spectral data with a wavelength increment of lg(λ_i+1) − lg(λ_i ) = 0.0001, therefore there are 3700 data points sampled in the wavelength range of 3700–8672 Å. We did not resample the original spectral data in our analysis.

The binary classification model was constructed to accurately distinguish hot subdwarfs from white dwarfs. Since the blackbody radiation spectrum of hot subdwarfs and white dwarfs are very similar, it is difficult for the model to distinguish them when trained with the original spectra. Therefore, for the binary classification model, the emission and absorption lines of the spectra were used for training and testing. Experiments showed that the model fitted best when trained with the 1400 flux values (the emission and absorption lines) corresponding to wavelengths between 3700 and 5106 Å for each spectrum. A total of 835 hot subdwarfs with S/Ns greater than 10 and not involved in the training process were used to further validate the model. The eight-class classification model and binary classification model were trained using the same hot subdwarf training set to obtain a more robust model. All of the samples were normalized by Equation (4) to prevent gradient problems during deep-learning model training, where x_min was the minimum value of the flux and x_max was the maximum value of the flux:

$$\begin{eqnarray}&&x^{\prime} =\displaystyle \frac{x-{x}_{\min }}{{x}_{\max }-{x}_{\min }}.\end{eqnarray} \tag{ 4 }$$

The performance of a deep-learning model is usually evaluated by accuracy, precision, recall, and F1 score (Forman et al. 2003), which are parameters calculated from the confusion matrix. A confusion matrix includes four parts: true positive (TP), false positive (FP), true negative (TN), and false negative (FN). TP and TN are the observations that are correctly predicted. FP and FN are cases of misclassification. Accordingly, the definitions of accuracy, precision, recall, and F1 score are in Equations (5), (6), (7), and (8), respectively.

Zoom In Zoom Out Reset image size

For the eight-class classification model, we set the batch size to 100 and the number of iterations to 2500. After training, we found that the model gradually fitted after about 2200 iterations. For the binary classification model, we set the batch size to 10 and the number of iterations to 2000. The model gradually fitted after about 1750 iterations. Through repeated training and testing, we finally got an accuracy of 94.21% for the eight-class classification model and accuracy of 96.17% for the binary classification model. The confusion matrices are shown in Figure 3. The precision, recall, and F1 score for each class are presented in Tables 2 and 3. We also output the mean and variance μ of the difference between the predicted and true labels of the two models. For the eight-class classification model, the mean is 0.0166 and the variance μ is 0.67. For the binary classification model, the mean is 0.023 and the variance μ is 0.46. The results show that the model performs well, with a tiny difference between the predicted and true labels.

$$\begin{eqnarray}&&\mathrm{Accuracy}=\displaystyle \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{FP}+\mathrm{FN}+\mathrm{TN}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\mathrm{Precision}=\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\mathrm{Recall}=\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&F1=\displaystyle \frac{2\ast \mathrm{Precision}\ast \mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}\end{eqnarray} \tag{ 8 }$$

Zoom In Zoom Out Reset image size

To further test the ability of our model at searching for hot subdwarfs, we used the 835 identified hot subdwarfs that were not involved in the training process for validation. The classification results showed that 802 hot subdwarfs (96.05% of the total) were correctly identified. An analysis and discussion of the misclassified samples are given in Section 5.

Compared with the previous binary classification model (hot subdwarfs and other types), a more accurate hybrid deep-learning model is constructed in this paper. It achieved an accuracy rate of 96.17% on the testing set and 802 of the 835 hot subdwarfs (96.05%) in the validation sets were correctly identified. From Table 3, we can see that the F1 value for searching hot subdwarfs is 96.13%, which is much higher than the 76.98% in Bu et al. (2019).

Comparative analysis shows that the overfitting problem caused by insufficient data can be solved by data augmentation. The improvement of training samples also enables the model to learn more complete features, resulting in better results. Overall, the model structure that combined eight-class classification with the binary classification model obtains better performance than the binary classification model alone.

By analyzing the classification results in Section 4, it can be found that for 2428 hot subdwarfs in the LAMOST catalog, the model eventually found 2067. Therefore, we analyzed the remaining 361 hot subdwarfs. Through sample analysis, it can be concluded that 89 hot subdwarfs were misclassified by our model, and 272 hot subdwarfs were predicted as hot subdwarfs by our model with a low probability and filtered out during candidate screening (in Section 4).

The spectral characteristics of most correctly classified hot subdwarfs are consistent with Figure 4(a). We performed a detailed analysis of the misclassified spectra and the low-probability spectra, and it was found that there are mainly three conditions: (1) low S/N data (Figure 4(b)). It is difficult to identify the absorption and emission lines. For the effect of noise, our experiments find that the prediction probability gradually decreases as the noise increases. (2) A spectrum with abnormal value, that is, a spectrum dominated by sharp spikes either from cosmic rays or instrumental artifacts (Figure 4(c)). We found through experiments that artificially setting abnormal values for hot subdwarfs filtered out by our models can also lead to misclassification and a low prediction probability. (3) A spectrum exhibits the features of a binary star, as seen in Figure 4(d), which is probably a binary star (possibly a hot subdwarf and an M star). To improve the classification accuracy, we need more labeled training data. However, there are very few samples of these types in the LAMOST catalog. These three factors lead to a sample being misclassified or classified with a very low probability as a hot subdwarf.

Zoom In Zoom Out Reset image size

For a spectrum with a low S/N and a category that cannot be manually confirmed (Table 4, type 4), repeat observations of the LAMOST can be continuously followed at a later stage.