Fe] Ratios from Low-resolution Stellar Spectratwo

CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 43 (2019) 237–251

A Method of Estimating the [α/Fe] Ratios from Low-resolution Stellar Spectra†  LU Yu

LI Xiang-ru

LIN Yang-tao

QIU Kai-bin

School of Mathematical Sciences, South China Normal University, Guangzhou 510631

Abstract A method for the determination of [α/Fe] from low-resolution stellar spectra is presented. The proposed scheme includes the following three steps: firstly, the spectrum is decomposed by the multi-scale Haar wavelet, and the high-frequency components are removed to suppress the high-frequency noise; then, based on the correlation of the spectral data component with [α/Fe], the spectral features are selected by the LASSO (Least Absolute Shrinkage and Selection Operator) algorithm; finally, [α/Fe] is measured by the multiple linear regression method based on the MARCS stellar spectrum library. The effectiveness of the method is verified with the low-resolution stellar spectra of ELODIE, SDSS (Sloan Digital Sky Survey), LAMOST (Large Sky Area Multi-Object Fibre Spectroscopic Telescope), and four star clusters. The systematic deviations and accuracies are as follows: (0.04 dex, 0.064 dex) for the 317 ELODIE spectra; (0.16 dex, 0.065 dex) for the 412 SDSS spectra; (0.05 dex, 0.062 dex) for the 1276 LAMOST spectra (with the signal-noise ratio in the g band (SNRG) greater than 20). The averages of [α/Fe] obtained for the likely members of the globular star clusters (M13, M15) and open star clusters (NGC2420, M67) are in agreement with the literature values. Key words stars: fundamental parameters—stars: abundances—methods: data analysis—methods: statistical 1.

INTRODUCTION

The chemical elements O, Mg, Si, Ca, and Ti etc. are called as α-elements. The ratio between the α-element abundance and the Fe-element abundance is called as [α/Fe]. In the †

Supported by National Natural Science Foundation (61273248, 61075033), NNSF-CAS Association

Foundation (U1531242), and Guangdong Natural Science Foundation (2014A030313425, S2011010003348) Received 2018–01–30; revised version 2018–03–28  

A translation of Acta Astron. Sin. Vol. 59, No. 4, pp. 35.1–35.13, 2018 [email protected]



[email protected]

c Elsevier 0275-1062/01/$-see front matter © 2019 B. V. AllScience rights reserved. 0275-1062/01/$-see front matter  2019 Elsevier B. V. All rights reserved. doi:10.1016/j.chinastron.2019.04.006 PII:

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Galaxy, the stars in different stellar populations have different values of [α/Fe], it can be taken as an observational probe to trace the evolutionary process of a star[1] . Hence, [α/Fe] has important significance for exploring the composition, formation, and evolution of the Galaxy. Along with the implementations of successive survey projects, such as Gaia (Gaia astronomical satellite), SDSS (Sloan Digital Sky Survey), especially LAMOST (Large Sky Area Multi-Object Fibre Spectroscopic Telescope), the number of stellar spectra that we have is tremendously increasing[2] . For example, SDSS has released 1075113 stellar spectra (DR14)[3] , and LAMOST has released 8171443 stellar spectra (DR5)[4] . These huge number of spectral data not only make the large-scale study of stellar properties become possible, but also bring us a new challenge. How to make the automatic measurement of [α/Fe] for the stellar spectra of medium-and-low resolution is the problem necessary to be solved by the current survey projects. At present, the methods to estimate [α/Fe] for the medium-and-low resolution stellar spectra include mainly the module matching method, linear regression method, etc. For the LAMOST low-resolution spectra, Li et al.[1] adopted the LSP3 (LAMOST Stellar Parameter Measurement Software Package of Beijing University) method to study the estimation of [α/Fe], the basic principle of this method is: to build first a stellar spectrum module library (such as the KURUCZ module library), the [α/Fe] parameter of the every spectrum in the library is known; then to match the measured spectrum with the spectrum in the module library, and to determine the estimated [α/Fe] of the measured spectrum by minimizing χ2 . When the signal-noise ratio (SNR) of the spectrum is greater than 20, the accuracy is better than 0.1 dex. Lee et al.[5] adopted the module matching method to make the measurement of [α/Fe] for a sample consisted of 425 ELODIE spectra and 91 SDSS spectra, the accuracies are respectively 0.062 dex and 0.069 dex. Bu Y. D. et al.[6] proposed a method for measuring [α/Fe] based on the LASSO algorithm (Least Absolute Shrinkage and Selection Operator), when this method is applied to the measurement of [α/Fe] for a sample of ELODIE spectra (with a resolution of R =2100), the accuracy is 0.067 dex, and the accuracy for a sample of SDSS spectra is 0.097 dex. Xiang et al.[7] discussed the applications of the multiple linear regression and KPCA (Kernel Principal Component Analysis) methods in the [α/Fe] measurement, at first the KPCA nonlinear method is adopted to make the feature extraction from the high-dimensional LAMOST stellar spectrum, then the multiple linear regression method is used to make the estimations of stellar atmosphere parameters (including the surface temperature Teff , surface gravitational acceleration lg g, and chemical abundance [Fe/H]), and of [α/Fe]. When the SNR of the spectrum is greater than 50, the accuracy of [α/Fe] is better than 0.05 dex. This paper has proposed a multiple linear regression model based on the Haar wavelet and LASSO algorithm (briefly called as HLM model). The basic idea of this model is: first of all, the Haar wavelet is used to make the 4-grade wavelet decomposition on the original spectrum, to remove the high-frequency components and to suppress the high-

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frequency noise interference; then, based on the correlation of spectral data component with [α/Fe] and the LASSO algorithm to select the optimal spectral features; finally, to make the measurement of [α/Fe] based on the MARCS stellar spectrum library and the multiple linear regression method. 2.

METHOD

2.1 Synthetic Spectrum In order to make the rapid and accurate measurement of [α/Fe] on a stellar spectrum, we employ the existing MARCS module library, which contains about 52000 stellar spectra of the F, G, and K types[8,9] . The MARCS spectrum library gives two sets of module libraries with the resolutions of R =20000 and R =2000[10] . We have selected 4524 spectra from the low-resolution module library of R =2000 to be the training sample. The wavelength range of each spectrum is 3000-10000 ˚ A, with the step length of 1 ˚ A, there are totally 7001 characteristic points. The range of Teff is [2500 K, 8000 K], in which the step length in the [2500 K, 4000 k] region is 100 K, the step length in the [4000 K, 8000 K] region is 250 K. The range of lg g is [–0.5 dex, 3.5 dex], the step length is 0.5 dex. The range of [Fe/H] is [–5.0 dex, 1.0 dex]. The range of [α/Fe] is [0 dex, 0.4 dex], the step length is 0.1 dex. The distribution of [α/Fe] for the training sample is shown in Fig.1 2000

N

1500

1000

500

0 −0.1

0

0.1

0.2

0.3

0.4

0.5

[α/Fe](MARCS)/dex

Fig. 1 The distribution of [α/Fe] for the MARCS training sample. In this data set, [α/Fe] has 5 optional values.

2.2 Spectral Data Pretreatment As the MARCS synthetic spectrum and the measured spectrum are not the same origin, their flux scales are nonidentical, hence we have to make the treatment of continuum normalization on the spectrum. At first, we adopt the segmented linear interpolation method to have the flux points of the MARCS synthetic spectrum interpolated to the wavelengths of the measured spectrum, so that the MARCS synthetic spectrum and the measured spectrum have the same step length and wavelength range; then, to extract the continuum by using the polynomial iteration method and removing the points greater than 3σ; finally, the spectral line spectrum is obtained by dividing the measured spectrum by the continuum. In the

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estimation of [α/Fe] for the ELODIE, SDSS, and LAMOST spectrum samples, we have made respectively the 6th, 13th, and 16th order polynomial fittings on the measured spectrum in order to obtain its continuum. 2.3

Feature Extraction

The spectral line spectrum contains rich data components, there are not only the information closely related with the parameter of this paper, but also some redundance and noise, and the rich line features exhibit the significant multi-resolution character[11] . In order to give rise to an effective scheme of parameter estimation in accordance with the above-mentioned characteristics of spectral data, we adopt the wavelet transform method to make the data pretreatment. The wavelet transform posses the ability of multi-resolution analysis, it can decompose the data according to frequency, its ability of temporal resolution in the stellar parameter estimation is represented by the ability to analyze the wavelength positions of spectral features, and the noise effect exhibits generally a high-frequency characteristic. In theory the wavelet transform method is suitable for separating the effective spectral features from the noise, and conforming the different spectral features. Hence, based on the decomposition and extraction of wavelet on the spectral features, it is hopeful to raise the effectiveness of the spectral parameter estimation, and the result of the synthetic experiment in this paper demonstrates also its effectiveness. We make the 4-grade decomposition on the spectral signal by using the Haar wavelet, and extract the 4th-grade low-frequency components to be the candidate features[12] . The study of this paper indicates that in the candidate features extracted in the above manner exist still a large number of redundant features. Hereby, we employ the LASSO algorithm to make the further selection on the candidate features, to remove the redundant i i i i components[13] . Given the training set D = {(xi , y i )}N i=1 , in which x = (x1 , x2 , · · · , xm ) i represents the candidate features of the i-th sample, y is the parameter [α/Fe], then the punishment function of the LASSO algorithm is: ⎡

⎛ ⎞2 ⎤ N m   ⎢ ⎥ ⎝y i − β0 − βˆ = arg min ⎣ βj xij ⎠ ⎦ , β

i=1

j=1

s.t.

m 

|βj |  t ,

(1)

j=1

in which, βj is the regression coefficient of the candidate feature, t is the compromising coefficient, by adjusting t, to make the minor βj become nonzero, and therefore to obtain the optimal feature relevant to [α/Fe][14] , β0 is the constant term. How to derive the regression coefficient βˆ is the key problem of the LASSO algorithm. Here. we select the LARS algorithm to derive βˆ[15] . 2.4

Regression Model

After the optimal features are extracted according to the above procedures, we adopt a multiple linear regression model to estimate [α/Fe]. Given the training set D = {(xi , yi )}N i=1 ,

LU Yu et al. / Chinese Astronomy and Astrophysics 43 (2019) 237–251

then yi =

p 

wj xij + bi ,

241

(2)

j=1

in which, xi = (xi1 , xi2 , · · · , xip ) expresses the optimal features of the i-th training sample, each sample has p features. yi is the parameter [α/Fe], wj is the regression coefficient, bj is the constant term. We search for the optimal value of (w, ˆ ˆb) by minimizing the sum of the squared errors of all training samples, the calculation formula is given as follows: ⎡ ⎢ (w, ˆ ˆb) = arg min ⎣ w,b

3.

N  i=1

⎛ ⎝yi − bi −

p 

⎞2 ⎤ ⎥ wj xij ⎠ ⎦ .

(3)

j=1

EXPERIMENTAL RESULTS

In order to verify the feasibility of the HLM model, we make the estimation of [α/Fe] for the ELODIE, SDSS, and LAMOST stellar spectra. In addition, we have estimated also the averaged [α/Fe] for the members of the M13, M15 two globular star clusters and the NGC2420, M67 two open star clusters. In order to evaluate the performance of the HLM model, we select the systematic deviation μ and accuracy σ as the criterions. The systematic deviation is the mean value of measuring errors. The accuracy is the standard deviation of errors, it represents the dispersion of errors, and it is an important specification to evaluate the stability and robustness of the model[16] . 3.1 ELODIE Spectrum Sample The ELODIE spectrum library indicates the data obtained by the Observatoire de HauteProvence of France through an 1.93 m telescope. This data set contains 1962 spectra of 1388 stars, with the wavelengths in the range of 390–680 nm, and the mean signal-noise ratio of SNR=130.12[17] . Lee et al.[5] have given the values of the parameter [α/Fe] of 425 spectra, from which we have selected 317 spectra with the parameters in the range of [0.04 dex, 0.35 dex]. The data of the spectrum sample are originated from the R =42000 high-resolution spectrum library. In order to verify the effectiveness of the HLM model in the case of low-resolution ELODIE spectrum sample, we reduce the resolution of the ELODIE spectrum sample to be 2100 by means of Gaussian convolution. Fig.2 shows the comparison between the measured [α/Fe]-values of the HLM model and the literature values. In this figure, ELODIE indicates the literature values, which are taken from the estimated values of high-resolution spectra in Reference [5], HLM indicates the measured values of the HLM model, N is the number of spectra. The left panel is the comparison between the measured values of the HLM model and the literature values, in which the dashed line and solid line show the one-by-one correspondence after the correction of the 0.04 dex zero-point deviation. The right panel is the number density histogram of

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residuals (differences between the HLM measurements and the literature values), in which the black curve is the Gauss fitting curve of residuals. From the figure we can find that the systematic deviation of the measured results is 0.04 dex, and the accuracy is 0.064 dex. Lee et al.[5] adopted the module matching method to make the measurement of [α/Fe] for the ELODIE spectrum sample consisted of 425 spectra, the systematic deviation was -0.01 dex, the accuracy was 0.062 dex. Li et al.[1] adopted the LSP3 method to make the measurement for the same sample taken by Lee et al., the systematic deviation was -0.125 dex, the accuracy was 0.071 dex. From the above comparison we can find that the measured result of the HLM model is better than that of the LSP3 method, but a little worse than that of Lee’s module matching method. 50

N =317, μ=0.04 dex, σ=0.064 dex

40

0.4

30 0.2

N

[α/Fe](HLM) /dex

0.6

20 0

−0.2 −0.2

10

0

0.2 0.4 [α/Fe](ELODIE)/dex

0.6

0 −0.2

−0.1 0 0.1 0.2 Δ[α/Fe](HLM−ELODIE)/dex

0.3

Fig. 2 Comparison of [α/Fe] between the measurements of HLM and the values in the literature for the ELODIE sample stars. The left panel plots the measurements of HLM against the values in the literature, while the right panel is a Gaussian fit to the residuals between the measurements of HLM and the values in the literature.

In order to discuss the effects of physical parameters on the measuring accuracy of [α/Fe], Fig.3 shows the variations of the residual of [α/Fe] with [α/Fe], Teff , lg g, and [Fe/H]. In the figure the solid line shows the variation of accuracy, the dashed line shows the variation of systematic deviation. Fig.3(a) displays the relationship between the residual and the value of [α/Fe]. From this figure we can find that when [α/Fe] approaches to the parameter boundary of the MARCS module library, the residual is rather large. In the ELODIE spectrum sample, there are about 80 spectra with the values of parameter [α/Fe] in the range of [0.04 dex, 0.06 dex], they occupy 25% of the total sample. The errors of these sample spectra are rather large, and have an influence on the estimation accuracy. Hence, the next step of our work is to perfect the MARCS module library, and to expand the range of the parameter [α/Fe], so as to improve the measuring accuracy. Fig.3(b) shows the correlation between the residual and the value of Teff , when Teff is greater than 5800 K, the accuracy is 0.051 dex, when Teff is less than 5800 K, the accuracy is 0.076 dex. From Fig.3(c), we can find that there is no significant correlation between the residual and the value of lg g. Fig.3(d) shows the relationship between the residual and the value of [Fe/H], when [Fe/H] is less than –1.5 dex, the accuracy is 0.044 dex, when [Fe/H] is greater than

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–1.0 dex, the accuracy is 0.065 dex. From the trend of variation in the figure we can find that there is a certain correlation of the residual with Teff and [Fe/H], the accuracy for the high-temperature metal-poor spectra is relatively low. (a) Δ[α/Fe](HLM−ELODIE)/dex

0.4 0.2 0 −0.2 −0.4 0

0.05

0.1

0.15 0.2 0.25 [α/Fe](ELODIE)/dex

0.3

0.35

0.4

(b) Δ[α/Fe](HLM−ELODIE)/dex

0.4 0.2 0 −0.2 −0.4 4500

5000

5500 6000 Teff(ELODIE)/K

6500

7000

4.5

5

(c) Δ[α/Fe](HLM−ELODIE)/dex

0.4 0.2 0 −0.2 −0.4 2.5

3

3.5 4 lg g(ELODIE)/dex (d)

Δ[α/Fe](HLM−ELODIE)/dex

0.4 0.2 0 −0.2 −0.4 −3

−2.5

−2

−1.5

−1 −0.5 [Fe/H](ELODIE)/dex

0

0.5

1

Fig. 3 Variations of the [α/Fe] residual for the ELODIE spectra as functions of [α/Fe], Teff , lg g, and [Fe/H] from upper to lower panels, respectively

In order to verify the robustness of the HLM algorithm, we have added respectively the Ganssian noises of different SNRs (50 db, 30 db, 25 db, 20 db, and 15 db) into the ELODIE spectrum sample, the experimental results are given in Table 1. From this table we can find that when the SNR is greater than 20 db, the measured result is rather good, the accuracy is better than 0.1 dex. When the SNR is less than 20 db, the measured result is rather bad.

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Hence, the HLM model is suitable for the estimation of the abundance ratio [α/Fe] of the low-resolution stellar spectrum with an SNR greater than 20 db. Table 1 Measured Results (systematic deviation, accuracy) of the spectra with different SNRs SNR/db

50

30

25

20

15

HLM

0.042, 0.065

Match template[5]

–0.001, 0.067

0.038, 0.069

0.033, 0.077

0.027, 0.109

0.033, 0.125

–

–0.004, 0.076

–

–

Method

3.2

SDSS Spectrum Sample

From the low-resolution SDSS-DR13 spectrum library we have selected a sample consisted of 412 stellar spectra originated from the same sources in the APOGEE (Apache Point Observatory Galactic Evolution Experiment) high-resolution spectrum library[18] . The SNR distribution of the spectrum sample is shown as Fig.4(a). Because of the Doppler effect, the actually measured SDSS/SEGUE (Sloan Extension for Galactic Understanding and Exploration) spectra have redshifts[19,20] . According to the radial velocities provided by the SSPP (SEGUE Stellar Parameter Pipeline), we have removed the spectrum redshifts[21,22] . The common rest logarithmic wavelength range of the total sample is [3.6176, 3.9539], with the step length of 0.0001, there are totally 3364 sampling points. The [α/Fe] parameter values are taken from the APOGEE spectrum library. As APOGEE provides only the single αelement abundance, and the line intensity for each α-element abundance differs, hence we set a different weight for the every α-element abundance, then we obtain [α/Fe] by calculating the weighted average, the calculation formula is given as follows:

n x i Wi x = i=1 , n i=1 Wi

(4)

in which, xi represents respectively [Mg/Fe], [Ti/Fe], [Ca/Fe], and [Si/Fe], the corresponding Wi is respectively 5, 3, 1, and 1[5] . The [α/Fe] parameter range of the 412 sample spectra is [0.05 dex, 0.35 dex]. Fig.5 shows the comparison between the measured values of the HLM model for the SDSS spectrum sample and the measured values taken from the APOGEE spectrum library. In the figure, HR indicates the measured values taken from the APOGEE spectrum library. The systematic deviation of the experimental results is 0.16 dex, and the accuracy is 0.065 dex. Lee et al[5] . estimated the [α/Fe] ratios of 91 SDSS sample spectra, they found also that there exists a systematic deviation of 0.13 dex between the estimated values and the high-resolution results in the literature, and that the accuracy is 0.069 dex. Hence, the estimated results of the HLM algorithm on the SDSS spectrum sample are believable, and the accuracy is better than the module matching algorithm.

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(b) LAMOST

60

300

50

250

40

200

30

150

N

N

(a) SDSS

20

100

10

50

0

20

40

60 80 SNR/db

245

100

0

120

20

40

60 80 SNRG/db

100

120

Fig. 4 Signal-to-noise ratio distributions of the SDSS and LAMOST samples 80

0.6

N =412, μ=0.16 dex, σ=0.065 dex 60

0.4 0.3

N

[α/Fe](HLM) /dex

0.5

40

0.2 0.1

20

0 −0.1

0

0.2 0.4 [α/Fe](HR) /dex

0.6

0 −0.1

0

0.1 0.2 0.3 Δ[α/Fe](HLM−HR)/dex

0.4

Fig. 5 Comparison of [α/Fe] between the measurements of HLM and the values taken from the APOGEE for the SDSS sample stars. The left panel plots the measurements of HLM against the values from the APOGEE, while the right panel is a Gaussian fitting to the residuals between the measurements of HLM and the values from the APOGEE.

Fig.6 displays the residual distribution, the relationships of the residual with [α/Fe], Teff , lg g, and [Fe/H] are same as the conclusion in Sec.3.1. As the SDSS spectrum sample has only few spectra with the [α/Fe] values close to the parameter boundary of the MARCS module library, so the experimental result of the SDSS spectrum sample is better. 3.3 Globular and Open Star Clusters In the SDSS/SEGUE surveys, multiple globular and open star clusters have been discovered. In which, we have selected the M13, M15 two globular star clusters, and the M67, NGC2420 two open star clusters. The spectral data of the member stars of the four clusters are taken from the SDSS-DR13. The SNR distribution of the sample spectra is shown in Fig.7. In their papers, Smolinski et al.[23] and Lee et al.[24] have introduced the method for the selection of cluster members, and given the averaged [α/Fe] values of these clusters.

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Δ[α/Fe](HLM−HR)/dex

(a) 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0

0.05

0.1

0.15

0.2 0.25 [α/Fe](HR) /dex

0.3

0.35

0.4

Δ[α/Fe](HLM−HR)/dex

(b) 0.4 0.3 0.2 0.1 0 −0.1 −0.2 3500

4000

4500

5000 Teff(HR) /K

5500

6000

6500

Δ[α/Fe](HLM−HR)/dex

(c) 0.4 0.3 0.2 0.1 0 −0.1 −0.2 0

0.5

1

1.5

2

2.5 3 lg g(HR) /dex

3.5

4

4.5

5

Δ[α/Fe](HLM−HR)/dex

(d) 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −3

−2.5

−2

−1.5

−1 −0.5 [Fe/H](HR) /dex

0

0.5

1

Fig. 6 Variations of [α/Fe] for the SDSS spectra as functions of [α/Fe], Teff , lg g, and [Fe/H] from upper to lower panels

Table 2 lists the estimated results of the four clusters. The second column “Number” indicates the number of cluster members, the third column “Literature” means the estimated result of high-resolution spectra, the fourth column is the measured result of SSPP, both of them are taken from Reference [5], the fifth column gives the averaged [α/Fe] value of the star cluster estimated by the method of this paper and its standard deviation, in which the averaged value is the estimated result after the systematic deviation of 0.16 dex has been removed. From Table 2 we can find that for the star clusters M13, M67 and NGC2420 the estimated result of the HLM algorithm has not a large difference from the value in the

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literature, being better than the SSPP measurement. However, the estimated result of the M15 star cluster has a rather large difference from the literature value, being a little worse than the SSPP measurement. Besides, the accuracies of the measured results of the four star clusters are all better than 0.055 dex, and the dispersions are relatively small. This experiment has further verified the feasibility of the HLM model for the measurements of the [α/Fe] ratios of low-resolution spectra. 70

30

20

50

M13 60

NGC2420

M67

M15

25

40

15

50

20 30 10

N

15

N

N

N

40 30

20 10

20

5

10

5

10 0 0

20

40 SNR/db

60

80

0 0

20

40 SNR/db

60

80

0 30

35

40 SNR/db

45

50

0 0

20

40 60 SNR/db

80

Fig. 7 Signal-to-noise ratio distributions of the member stars of M13, M15, M67, and NGC2420

Table 2 Average estimated values of [α/Fe] for clusters Cluster

Number

Literature

SSPP

HLM

M13

290

0.192 ± 0.017

0.235 ± 0.092

0.174 ± 0.042

M15

98

0.397 ± 0.149

0.238 ± 0.099

0.216 ± 0.042

M67

52

0.075 ± 0.012

0.032 ± 0.027

0.068 ± 0.020

NGC2420

162

0.053 ± 0.029

0.084 ± 0.044

0.069 ± 0.053

3.4 LAMOST Spectrum Sample By the cross-matching of the APOGEE spectrum library with the low-resolution LAMOSTDR3 spectrum library, we have obtained the [α/Fe] parameter values of more than 6000 LAMOST spectra. From which we have selected 1276 low-noise (the signal-noise ratio in the 403–533 nm band (SNRG) is greater than 20) sample spectra. The SNRG distribution of these sample spectra is shown in Fig.4(b). After the redshifts are removed according to the radial velocities given by the LAMOST, the common rest logarithmic wavelength range of the total sample is [3.5843, 3.9556], with the step length of 0.0001, there are totally 3714 sampling points. The ranges of the four stellar atmosphere parameters are: Teff ∈[3553.7 K, 6355.3 K], lg g ∈[0.4481 dex, 3.9840 dex], [Fe/H]∈[-2.1727 dex, 0.4206 dex], and [α/Fe]∈[0.06 dex, 0.34 dex]. Fig.8 shows the comparison between the measured values of the LAMOST spectrum sample and the values taken from the APOGEE spectrum library. The systematic deviation of the experimental result is 0.05 dex, and the accuracy is 0.062 dex. Adopting the LSP3 method Lee et al.[1] made the measurement of [α/Fe] for the 98 LAMOST sample stellar spectra (SNRG > 30), the systematic deviation was -0.120, and the accuracy was 0.090 dex.

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From this we can find that the HLM algorithm is more accurate than the LSP3 algorithm. 400

0.5

N =1276, μ=0.05 dex, σ=0.062 dex

300 0.3

N

[α/Fe](HLM) /dex

0.4

0.2

200

0.1

100 0 −0.1 −0.1

0

0.1 0.2 0.3 [α/Fe](HR) /dex

0.4

0 −0.2

−0.1

0 0.1 0.2 Δ[α/Fe](HLM−HR)/dex

0.3

Fig. 8 Comparison of [α/Fe] between the measurements of HLM and values from the APOGEE for the LAMOST sample stars. The left panel plots the measurements of HLM against the values from the APOGEE, while the right panel is a Gaussian fit to the residuals between the measurements of HLM and the values from the APOGEE

The systematic deviation of [α/Fe] for the LAMOST spectrum sample is much less than that of the SDSS spectrum sample. This is mainly because of the difference in the wavelength ranges of the two samples. The logarithmic wavelength range of the LAMOST spectra is [3.5843, 3.9556], while the logarithmic wavelength range of the SDSS spectra is [3.6176, 3.9539]. This leads to the differences in the finally obtained spectral features and the sensitivities to the noise and other factors. Therefore, the systematic deviations of the two samples have a significant difference. If we have the wavelength range of the LAMOST sample truncated to be [3.6176, 3.9539] of the SDSS spectra, then the systematic deviation of the experimental result will be 0.128 dex, the difference between the two is obviously reduced. Hence, the wavelength range of the sample has a rather large influence on the systematic deviation. Fig.9 displays the variations of the [α/Fe] residual with Teff , lg g, [Fe/H], and SNRG. The correlations of the [α/Fe] residual with Teff and [Fe/H] are opposite to the case of the ELODIE spectra mentioned in Sec.3.1. This is caused by the different feature distributions extracted from the two. The feature distributions extracted from the ELODIE, SDSS, and LAMOST spectrum samples are shown in Fig.10. In this figure, the black curves indicate the stellar spectra with different parameter values, the black marks indicate the positions of the extracted features. The extracted features of the ELODIE and SDSS spectrum samples are distributed in different wavebands, while the extracted features of the LAMOST spectrum sample are mainly distributed in the wavelength range of 380–530 nm. The variations of Teff and [Fe/H] have rather large influences on the linear peak values of spectral lines, and therefore affect the measuring accuracy of [α/Fe]. As the feature distributions of the LAMOST and ELODIE spectrum samples have a rather large difference, so the spectral lines obtained from the two also very different, hence the influences of Teff and [Fe/H] on

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the measuring accuracy of [α/Fe] are also different. The relationship of the residual and SNRG is shown as Fig.9(d). When the SNRG of the spectrum sample attains 30 db, the accuracy is 0.071 dex; when the SNRG attains 50 db, the accuracy is 0.054 dex; when the SNRG attains 70 db, the accuracy is 0.047 dex. From this figure, we can find that with the increase of the SNRG, the accuracy becomes high. The experimental result indicates that the noise of the spectral signal has a certain influence on the measuring accuracy. (a) Δ[α/Fe](HLM−HR)/dex

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4.

SUMMARY

We have developed a multiple linear regression model based on the Haar wavelet and LASSO algorithm. We have applied this model to measuring the [α/Fe] ratios of the ELODIE, SDSS, and LAMOST low-resolution stellar spectra, and obtained a better result for all of them. A character of the HLM model is the sparsity. For example, in the LAMOST original spectral signals there are 3714 features, we adopt the Haar wavelet and LASSO algorithm to select the features relevant to [α/Fe], the dimension of the extracted features is 15, occupying only 0.4% the original features. Each feature extracted by us corresponds to a corresponding waveband of the original spectrum, and has a very strong physical meaning. From these features we can acquire the important spectral lines that have an influence on [α/Fe]. Another character of the HLM model is the rapid calculation and high accuracy. This method occupies less computer memory, and has a rapid calculation speed, it suits the treatment of the huge number of spectral data. Compared with the experimental results in the literature, the accuracy of the HLM model is higher. For example, when the HLM algorithm is adopted to make the estimation of [α/Fe] for the 1276 LAMOST sample spectra, the measuring accuracy is 0.062 dex; while Li et al.[1] adopted the LSP3 method to make

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