Estimating stellar atmospheric parameters based on LASSO and support-vector regression

Abstract

A scheme for estimating atmospheric parameters T_eff, log g and [Fe/H] is proposed on the basis of the Least Absolute Shrinkage and Selection Operator (LASSO) algorithm and Haar wavelet. The proposed scheme consists of three processes. A spectrum is decomposed using the Haar wavelet transform and low-frequency components at the fourth level are considered as candidate features. Then, spectral features from the candidate features are detected using the LASSO algorithm to estimate the atmospheric parameters. Finally, atmospheric parameters are estimated from the extracted spectral features using the support-vector regression (SVR) method. The proposed scheme was evaluated using three sets of stellar spectra from the Sloan Digital Sky Survey (SDSS), Large Sky Area Multi-object Fibre Spectroscopic Telescope (LAMOST) and Kurucz's model, respectively. The mean absolute errors are as follows: for the 40 000 SDSS spectra, 0.0062 dex for log T_eff (85.83 K for T_eff), 0.2035 dex for log g and 0.1512 dex for [Fe/H]; for the 23 963 LAMOST spectra, 0.0074 dex for log T_eff (95.37 K for T_eff), 0.1528 dex for log g and 0.1146 dex for [Fe/H]; for the 10 469 synthetic spectra, 0.0010 dex for log T_eff (14.42K for T_eff), 0.0123 dex for log g and 0.0125 dex for [Fe/H].

INTRODUCTION

The implementation of large-scale and deep sky survey programmes, such as the Sloan Digital Sky Survey (SDSS: York et al. 2000; Ahn et al. 2012), Gaia–ESO Survey (GES: Gilmore et al. 2012; Randich et al. 2013) and Large Sky Area Multi-object Fibre Spectroscopic Telescope (LAMOST/Guoshoujing Telescope: Zhao et al. 2006; Cui et al. 2012), has resulted in the collection of a large number of stellar spectra. The bulk of this stellar spectral information necessitates the utilization of a fully automated characterization process.

In particular, estimating the effective temperature, surface gravity and metallicity from stellar spectra remains a fundamental problem (Wu et al. 2011; Song et al. 2012). For example, Muirhead et al. (2012) investigated the estimation of the effective temperature T_eff and metallicity [M/H] for late-K and M-type planet-candidate host stars from the K-band spectra released by the the Kepler mission. In this study, a surface fitting method was used along with three spectral indices, namely the equivalent widths of Na i (2.210 μm) and Ca i (2.260 μm) lines and an index describing the flux change between three 0.02 μm wide bands centred at 2.245, 2.370 and 2.080 μm. Koleva et al. (2009) developed a full-spectrum fitting package, ULySS (University of Lyon Spectroscopic analysis Software), and explored its application in spectral parametrization. Manteiga et al. (2010) parametrized stellar spectra by extracting features based on Fourier analysis and wavelet decomposition as well as by constructing a mapping from feature space to parameter space using a forward neural network (FNN) with three layers.

An automated estimation system for atmospheric parameters from a stellar spectrum is also referred to as a spectrum parametrization system (SPS) in related studies. An SPS system consists of two key processes, namely feature extraction and parameter estimation. Feature extraction determines the representation of spectral information. Principal component analysis (PCA: Re Fiorentin et al. 2007; Bu & Pan 2015) and fast Fourier transforms (FFT: Manteiga et al. 2010) are the two most frequently used feature extraction methods. Parameter estimation constructs a mapping from the extracted features of a spectrum to its atmospheric parameters. Some commonly used estimation methods are the FNN (Re Fiorentin et al. 2007), Gaussian process (GP: Bu & Pan 2015) and support-vector regression (SVR: Li et al. 2014).

This work describes a scheme for estimating atmospheric parameters from the stellar spectrum. The three processes of the proposed scheme are performed as follows. Several candidate features are initially obtained by transforming a spectrum into a low-frequency space using the Haar wavelet transform. Then, some representative candidate features are selected as spectral features using the Least Absolute Shrinkage and Selection Operator (LASSO) algorithm (Tibshirani 1996). Finally, atmospheric parameters are estimated from extracted features using a regression method.

High-frequency components are removed in the first process. These components are usually more affected by noise than low-frequency components, which will be discussed further in Section 5.3. The second process reduces the number of spectral features efficiently, a feature closely related to the estimation efficiency and exploration of representative features (Section 5.3). In the third process, the spectrum parametrization problem is investigated using four typical estimation methods: the radial basis function neural network (RBFNN: Schwenker, Kestler & Palm 2001), SVR (Chang & Lin 2001; Schökopf & Smola 2002; Smola & Schölkopf 2004), K-nearest neighbour regression (KNNR: Altman et al. 1992) and least-squares regression (LSR: James et al. 2013). Experimental results indicate that SVR is superior to the other three methods.

This article is organized as follows. Section 2 describes some applied experimental data. After introducing four regression methods in Section 3, Section 4 describes a scheme for extracting spectral features. In Section 5, the proposed scheme is evaluated experimentally. Finally, Section 6 concludes this work.

DATA

The proposed scheme was evaluated by performing experiments using three data sets: 33 963 actual spectra from LAMOST, 50 000 actual spectra from SDSS and 18 969 synthetic spectra computed from Kurucz's model. Actual spectra usually carry some disturbances from noise and pre-processing imperfections.

The proposed scheme is a statistical learning method. It basically involves the discovery of a mapping from stellar spectra to their atmospheric parameters using empirical data, which is referred to as a training set in machine learning. At the same time, the performance of the mapping discovered should be evaluated objectively. Therefore, an independent set of stellar spectra is needed for this evaluation. This independent set is usually referred to as a test set. Thus, in each experiment, stellar spectra are split into two subsets, namely the training and test sets.

Actual spectra from SDSS

A set of 50 000 stellar spectra and their physical parameters collected from SDSS are selected (Abazajian et al. 2009; Yanny et al. 2009). The selected spectra span the ranges [4088, 9740] K for T_eff, [1.015, 4.998] dex for surface gravity and [−3.497, 0.268] dex for [Fe/H]. All stellar spectra are shifted to their rest frames (zero radial velocity) using the radial velocity provided by the SDSS/the Sloan Extension for Galactic Understanding and Exploration (SEGUE) Spectroscopic Parameter Pipeline (SSPP: Beers et al. 2006; Allende Prieto et al. 2008; Lee et al. 2008a,b, 2011; Smolinski et al. 2011) and rebinned to a maximal common log(wavelength) range [3.581862, 3.963961] with a sampling step of 0.0001. The common wavelength range is approximately [3818.23, 9203.67] Å. The SDSS training and test sets are labelled as |$S_{\rm tr}^{\rm SD}$| and |$S_{\rm te}^{\rm SD}$|⁠, respectively. Without special instructions, the sizes of |$S_{\rm tr}^{\rm SD}$| and |$S_{\rm te}^{\rm SD}$| are 10 000 and 40 000, respectively.

Actual spectra from LAMOST

A set of 33 963 LAMOST stellar spectra (Luo et al. 2012) and their physical parameters from the LAMOST pipeline are selected based on two signal-to-noise ratio (SNR) constraints: namely SNR_g ≥ 20 and SNR_r ≥ 20 in the g and r bands (Luo et al. 2012).¹ This constraint is proposed by considering the current quality stability of the LAMOST spectra. The LAMOST spectra are also shifted to their rest frames using the radial velocity provided by LAMOST SSPP and rebinned to their maximal common log(wavelength) range [3.5845, 3.9567] with a sampling step of 0.0001. The common wavelength range is approximately [3841.49, 9051.07] Å. All LAMOST spectra span the ranges [3853.2, 9927] K for T_eff, [0.8920, 4.9959] dex for log g and [−2.3280, 0.9360] dex for [Fe/H]. The LAMOST training and test sets are labeled as |$S_{\rm tr}^{\rm LA}$| and |$S_{\rm te}^{\rm LA}$|⁠, respectively. |$S_{\rm tr}^{\rm LA}$| consists of 10 000 stellar spectra, while |$S_{\rm te}^{\rm LA}$| consists of 23 963 spectra.

Synthetic spectra

A set of 18 969 synthetic spectra are calculated from Kurucz's NEWODF (new opacity distribution function) models (Castelli & Kurucz 2003) using the spectrum (v2.76) package (Gray & Corbally 1994) as well as 830 828 atomic and molecular lines (contained in two files, luke.lst and luke.nir.lst). These atomic and molecular data are stored in the file stdatom.dat, which includes solar atomic abundances from Grevesse & Sauval (1998). The spectrum package and the three data files can be downloaded from the website.²

The grids of the synthetic stellar spectra span parameter ranges [4000, 9750] K for T_eff (45 values, step sizes of 100 K between 4000 and 7500 K and 250 K between 7750 and 9750 K), [1, 5] dex for log g (17 values, step size of 0.25 dex) and [−3.6, 0.3] dex for [Fe/H] (27 values, step sizes of 0.2 dex between −3.6 and −1 dex and 0.1 dex between −1 and 0.3 dex). The synthetic stellar spectra are also split into two subsets: a training set |$S_{\rm tr}^{\rm SY}$| and a test set |$S_{\rm te}^{\rm SY}$| consisting of 10 469 and 8500 spectra, respectively.

ESTIMATION MODELS AND EVALUATION METHODS

Here, x represents a description of a stellar spectrum, while y is the atmospheric parameter T_eff, log g or [Fe/H] of x.

Estimation models

In this work, f is estimated using four typical regression methods: RBFNN (Schwenker et al. 2001), SVR (Chang & Lin 2001; Schökopf & Smola 2002; Smola & Schölkopf 2004), KNNR (Altman et al. 1992) and LSR (James et al. 2013). RBFNN and KNNR are non-linear regression methods, while LSR is a linear regression method. SVR can be implemented with a Gaussian kernel and a linear kernel. These two implementation cases are labelled as SVR_G and SVR_l, respectively. Therefore, SVR_G is a non-linear method whereas SVR_l is a linear method.

Evaluation criteria

Suppose that |$\skew4\hat{f}$| is an estimate of the spectrum parametrization mapping f and S = {(x, y)} is a data set, where x is a representation of a stellar spectrum and y is an atmospheric parameter of the corresponding star. The data set S can be a training or test set.

For convenient comparison with related reports, this study evaluates the performance of an estimate |$\skew4\hat{f}$| using three methods: mean error (ME), mean of absolute error (MAE) and standard deviation (SD). These three evaluation methods are widely used in related studies (Re Fiorentin et al. 2007; Jofré et al. 2010; Tan et al. 2013).

FEATURE EXTRACTION

Extract candidate features

Atmospheric parameters T_eff, log g and [Fe/H] show an evident non-linear dependence on stellar spectra (tables 6, 10 and 11 in Li et al. 2014). Therefore, the spectrum parametrization problem is investigated by non-linearly transforming a spectrum before estimating atmospheric parameters. In this work, stellar spectra are transformed using a Haar wavelet (Mallat 2009) and decomposed into a series of components with different wavelengths and frequencies (time–frequency localization).

High-frequency components are usually more affected by noise than low-frequency components. Thus, this work obtains candidate features by removing high-frequency components. This process will be discussed further in Section 5.3.

In addition to the Haar wavelet, multiple alternatives for a non-linear transformation (Mallat 2009; Daubechies 1992), such as the Coiflets, Daubechies, Symmlet and biorthogonal wavelets, are available based on the wavelet transform.

Refining the candidate features

Experiments indicate the presence of many redundancies in the extracted candidate features (Section 5.3). Therefore, this work proposes a scheme for detecting spectral features from the extracted candidate features using the LASSO algorithm (Tibshirani 1996).

Based on the training set from SDSS (Section 2.1), 17 spectral features are detected for T_eff, 24 for log g and 25 for [Fe/H] (Table 1).

EXPERIMENTS AND DISCUSSION

Performance for SDSS spectra

From the detected features in Table 1, a spectrum parametrization model can be learned from the training set |$S_{\rm tr}^{\rm SD}$| (Section 2.1). The performance obtained using the test set |$S_{\rm te}^{\rm SD}$| is presented in Table 2 .

For the SDSS test set, MAE errors are 0.0062, 0.2035 and 0.1512 dex for log T_eff, log g and [Fe/H], respectively. To compare the proposed scheme with those in previous related reports, the performance of the proposed scheme was also evaluated using measures ME and SD. More experimental results are presented in Table 2, as well as in Figs 1 and 2. Direct comparisons with reports are shown in Section 6 and more discussion on the dispersion in Fig. 1 is presented in Section 5.4.

Performance for LAMOST spectra and synthetic spectra

The proposed scheme is also tested on actual spectra from LAMOST (Section 2.2) and synthetic spectra (Section 2.3).

The performances of the scheme on LAMOST spectra is presented in Table 3, as well as in Figs 3 and 4. The test results using synthetic spectra are presented in Table 4, as well as in Figs 5 and 6. The results in Tables 2, 3 and 4 show that SVR_G and RBFNN are more suitable than KNNR, LSR and SVR_l for estimating atmospheric parameters.

Filtering and selection: positive or negative?

The proposed scheme extracts spectral features by removing high-frequency components from the Haar wavelet transform and rejecting most low-frequency components by the LASSO algorithm. In this study, we determine whether these processes eliminate important spectral information, such as weak lines.

Four experiments are conducted and the results are listed in Table 5. The results show the possibility of eliminating useful spectral information. However, in application, the observed spectrum is inevitably contaminated with noise. In theory, weak lines should be more sensitive to noise.

Therefore, the loss from elimination is trivial. The wavelet components with the lowest frequency are traditional choices for spectral features for estimating atmospheric parameters (Lu et al. 2013). In the experiments for T_eff, when all low-frequency wavelet components are used, the number of features will increase from 17 to 239 (an increase of (239 − 17)/17 = 1305.88 per cent), but the MAE can only decrease by 0.0007 dex (11.29 per cent: Experiments 1 and 2). When no component is eliminated while estimating T_eff, the number of features will increase from 17 to 3823 (an increase of 22 388.23 per cent), but the MAE error increases by 0.0398 dex (641.93 per cent). A small number of detected features indicates an efficient process for estimating atmospheric parameters from spectral features. The above results suggest that the model developed in this work can estimate stellar atmospheric parameters with high accuracy.

Knowledge base, dispersion and performance

The proposed scheme is a statistical learning method. Its primary principle is to discover automatically the mapping from a stellar spectrum to its atmospheric parameters from a training set. The training set is the carrier of knowledge and affects the accuracy of the scheme.

Therefore, the size of the training set affects the performance of the proposed scheme. For example, if the size of the training set is increased from 10 000 to 15 000, 20 000 and 25 000 in the experiments on SDSS spectra,³ the test dispersion can clearly be improved (Fig. 7). Similar experiments are conducted on synthetic spectra and the corresponding results are presented in Fig. 8.

Actual data usually present some disturbances arising from noise and pre-processing imperfections (e.g. sky lines and/or cosmic-ray removal residuals, residual calibration defects and interstellar extinction instability⁴). The negative effect from these factors can be reduced to a certain extent by enriching the knowledge carrier, i.e. the training set (Figs 7 and 8).

Compactness

For simplicity, this section considers the features of log g as an example to discuss compactness. Other features of T_eff and [Fe/H] can be discussed similarly.

The original SDSS spectra are described by 3821 fluxes. To estimate log g, 24 features are detected and data reduction is (3821 − 24)/(3821) ≈ 99.37 per cent. This result indicates that log g can be estimated from a spectral description with a dimension of 24 instead of 3821. The small number of features also implies high efficiency in estimating the parameter from spectral information.

Compactness indicates the study of whether the number of features can be reduced further. Experimental results show that if seven features, namely L1, L8, L13, L19, L20, L21 and L23, are rejected, the feature number decreases by 7/24 ≈ 29 per cent and the MAE error increases by 0.0027 dex (approximately 0.0027/0.2035 ≈ 1.32 per cent).⁵ Therefore, the feature number can be refined further if a slight decrease in accuracy is accepted.

CONCLUSION AND FUTURE WORK

This work investigated the estimation of effective temperature (T_eff), surface gravity (log g) and metallicity ([Fe/H]) from stellar spectra based on the Haar wavelet transform and LASSO algorithm. The proposed scheme is evaluated using actual spectra from SDSS and LAMOST as well as synthetic spectra computed from Kurucz's model. Favourable results are achieved in all cases.

The proposed scheme exhibits excellent robustness and sparseness. The features are extracted using two steps. From the SDSS data, the original spectra, described by 3821 fluxes, are initially decomposed using the Haar wavelet transform and low-frequency coefficients (239 features) are selected as candidate features. In this step, some noise and redundancies with high-frequency components are removed. Then a small subset of candidate features are chosen as spectral features using the LASSO algorithm. The second step is a supervised learning process that selects features according to their correlation with the parameter to be estimated. The number of selected features is 17 for T_eff, 24 for log g and 25 for [Fe/H]. A representative work is Re Fiorentin et al. (2007), in which 50 features are extracted for estimating atmospheric parameters.

Another advantage of the proposed scheme is its high accuracy. Using the SVR_G method and 40 000 stellar spectra from SDSS, the MAEs are 0.0062 dex for log T_eff (85.83 K for T_eff), 0.2035 dex for log g and 0.1512 dex for [Fe/H]. Further details are shown in Table 2. In previous reports, Re Fiorentin et al. (2007) estimated the parameters of 19 000 spectra from SDSS with MAEs of 0.0126 dex for log T_eff, 0.3644 dex for log g and 0.1949 dex for [Fe/H]. Jofré et al. (2010) first highly compressed the data using a likelihood method and then estimated the parameters T_eff, log g and [Fe/H] from low-resolution stellar spectra measured by SEGUE; the standard deviations (SDs) of the errors were 130 K for T_eff, 0.5 dex for log g and 0.25 dex for [Fe/H]. Therefore, the results estimated using the proposed scheme exhibit higher accuracy compared with those reported in the literature.

In this work, the proposed scheme is evaluated using three different data sets (SDSS, LAMOST and synthetic spectra). For each kind of data, the proposed model is learned and tested independently. An interesting problem is the estimation of atmospheric parameters of one data set (e.g. LAMOST) using a model learned from the other data sets (e.g. SDSS or synthetic spectra). For example, Re Fiorentin et al. (2007) investigated how to estimate the atmospheric parameters of SDSS data from synthetic spectra and vice versa. In this process, residual calibration defects should be considered. This work focuses on sparse feature extraction and the abovementioned problem will be investigated in future work.

The authors thank the reviewer and editor for their instructive comments and extend their thanks to Professor Ali Luo and Fang Zuo for their support and discussions. This work is supported by the National Natural Science Foundation of China (grant No: 61273248, 61075033, 61202315), the Natural Science Foundation of Guangdong Province (2014A030313425,S2011010003348), the Open Project Program of the National Laboratory of Pattern Recognition (NLPR) (201001060) and the high-performance computing platform of South China Normal University.

REFERENCES