An upgraded interpolator of the radial basis functions network for spectral calculation based on empirical stellar spectral library
aa r X i v : . [ a s t r o - ph . I M ] J u l Research in Astronomy and Astrophysics manuscript no.(L A TEX: manuscript.tex; printed on July 10, 2020; 0:30)
An upgraded interpolator of the radial basis functions network forspectral calculation based on empirical stellar spectral library
Lian-Tao Cheng , , , , Feng-Hui Zhang , , Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming,650216, P. R. China; c [email protected], [email protected] Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, ChaoyangDistrict, Beijing, 100012, P. R. China Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 396Yangfangwang, Guandu District, Kunming, 650216, P. R. China University of Chinese Academy of Sciences, Beijing, 100049, China
Received 20XX Month Day; accepted 20XX Month Day
Abstract
Stellar population synthesis is an important method in the galaxy and star-clusterstudies. In the stellar population synthesis models, stellar spectral library is necessary forthe integrated spectra of the stellar population. Usually, the stellar spectral library is usedto the transformation between the stellar atmospheric parameters and the stellar spectrum.The empirical stellar spectral library has irreplaceable advantages than the theoretical library.However, for the empirical spectral library, the distribution of stars is irregularly in the stellaratmospheric parameter space, this makes the traditional interpolator difficult to get the accu-rate results. In this work, we will provide an improved radial basis function interpolator whichis used to obtain the interpolated stellar spectra based on the empirical stellar spectral library.For this interpolator, we use the relation between the standard variance σ in the Gaussianradial basis function and the density distribution of stars in the stellar atmospheric parame-ter space to give the prior constraint on this σ . Moreover, we also consider the anisotropicradius basis function by the advantage of the local dispersion of stars in the stellar atmo-spheric parameter space. Furthermore, we use the empirical stellar spectral library MILES totest this interpolator. On the whole, the interpolator has a good performance except for theedge of the low-temperature region. At last, we compare this interpolator with the work in L.-T. Cheng et al.
Cheng et al. (2018), the interpolation result shows an obvious improvement. Users can usethis interpolator to get the interpolated spectra based on the stellar spectral library quicklyand easily.
Key words: stars: fundamental parameters—stars: atmospheres—(Galaxy:) globular clus-ters: general—methods: numerical
Stellar population synthesis model is important in astronomical research. Most of the observation data arefrom star, and these data can be used in the study of the system of stars (galaxy, the star-cluster, etc.).Stellar population synthesis model is a widely used tool in this kind of study, this is because that the stellarpopulation is the basic ingredient in the system of stars. Moreover, the integrated spectrum of stellar pop-ulation contains large effective information, so the calculation of integrated spectra is necessary for stellarpopulation synthesis model.In the widely used evolutionary population synthesis models (Bruzual & Charlot 2003; Moll´a et al.2009; Zhang et al. 2013, etc.), the below three key components are used for the integrated spectra ofthe stellar populations. 1. initial mass function (IMF), which gives the relative number for stars with ainitial mass M ; 2. isochrone library, which is derived from the stellar evolution model and is used to givethe stellar parameters (include the atmospheric parameters) for stars in a stellar population; 3. the stellarspectral library, which is used to convert the stellar atmospheric parameters to the stellar spectra. At last,the integrated spectrum of stellar population is the sum of the stellar spectrum.The stellar spectral library gives a correspondence between the stellar atmospheric parameters and thespectrum. Stellar spectral library can be divided to two kinds: theoretical and empirical libraries. The the-oretical spectra are calculated from the stellar atmosphere model (Kurucz 1992; Gonz´alez Delgado et al.2005, etc.), the empirical spectra are from observations (Prugniel & Soubiran 2001; Le Borgne et al. 2003;S´anchez-Bl´azquez et al. 2006; Chen et al. 2014, etc.).For the empirical and theoretical spectral libraries, each one has its advantages and disadvantages.Because the theoretical spectra are calculated from the atmospheric model, they have larger coverage inthe wavelength, spectral resolution and the stellar atmospheric parameters range. However, they are lim-ited by the incomplete atomic- and molecular-line lists, the uncertain abundance pattern, the assumptionand idealized treatment in the model calculation, and so on (Kurucz 2014). Unlike the theoretical spectrallibrary, the empirical spectra are from observations, they have the limited wavelength, resolution, noise anderror that is raised from observation and the data process (e.g. flux calibration, stellar atmospheric parameter n upgraded spectral interpolator base on RBF 3 extraction). They avoid many disadvantages of the theoretical spectra. The theoretical and empirical stellarspectral libraries are complementary.Because the stars in the spectral library are discrete, we need an interpolator to get the spectra of anyset of possible stellar atmospheric parameters. The distribution of stars in the theoretical spectral librarieshave less atmospheric parameter limitations and usually is dense and regular. Under this situation, classi-cal linear interpolation method can give a reliable interpolation result. However, for the empirical stellarspectral library, the stars are discrete in the atmospheric parameter space, this irregular distribution makesthe traditional interpolator difficult to get the expected interpolation results. Therefor, in the work of Chenget al. (2018) we constructed an interpolator based on the radial basis function (RBF) network to get the stel-lar spectra in the stellar population synthesis model, the algorithm is different from those using polynomialform (e.g. Wu et al. 2011 and Prugniel et al. 2011).The interpolator based on the radial basis function network is called RBF interpolator in this text. Thecomputational formula of the RBF interpolator (Eq. 3) is similar to the formula of field in the smooth particlehydrodynamics simulation (SPHs) and the formula of expected number in the likelihood estimation. All ofthem are based on the calculations of the kernel functions. In this work, Gaussian function is used as kernelfunction as shown in Eq. 5, the σ is the standard deviation used to characterize the effect region of the kernelfunction.For the RBF interpolator, the kernel function does not have any strict constraint, the different settingsof the kernel functions will influence the interpolation results. If all the Gaussian kernel functions have thesame σ , the relatively small σ in the sparse area will make the interpolation results be discrete, however,the relatively large σ in the dense area will make the interpolation results be oversmoothed (lack of detailedinformation). In the work of Cheng et al. (2018), the local average distance is used to give the σ for theGaussian kernel function. This is because the distribution of stars in the spectral library is non-homogeneousin the stellar atmospheric parameter space.In this work, we compare the interpolation calculation of the RBF network with the calculation of thedensity field in SPHs. Under this comparison, we include a constraint about the σ in the Gaussian kernelfunction of RBF network from the relation between the smooth length and the density in SPHs. Underthis constraint, the size of σ for each kernel function is related to the local density of sample points in theparameter space. We use this constraint to replace the coarse determination about σ in the work of Chenget al. (2018) for the RBF network Gaussian kernel function. Moreover, same as in SPHs, the sphericallysymmetric kernel function usually is not a better selection (Bicknell & Gingold 1983; Shapiro et al. 1996;Owen et al. 1998). We will refer to the process of adaptive smooth particle hydrodynamics simulation(ASHPs) and take anisotropic kernel function in the RBF interpolation calculations. As a result, We will L.-T. Cheng et al. present an upgraded RBF interpolator which can be used for the spectral calculation based on the empiricalstellar spectral library.The outline of this paper is as follows. In Section 2, we briefly introduce the RBF network and its struc-ture, then explain the constraint on the kernel function used in this work and the constructing anisotropickernel function in the RBF network. In Section 3, we use the Beetle Antennae Search algorithm (Jiang& Li 2017, hereafter BAS) to search for the best kernel function parameters. In Section 4, we presentthe interpolation spectra and test this upgraded RBF interpolator by using the Medium-resolution IsaacNewton Telescope library of empirical spectra (hereafter MILES, S´anchez-Bl´azquez et al. 2006; Cenarroet al. 2007), and compare it with our previous work in Cheng et al. (2018). At last, in Section 5, we give theconclusion of this work.
In the stellar population synthesis model, the stellar spectral library provides a fast way to get the spectraof any star compared with the direct calculation by the stellar atmospheric model. This process is a fittingor an interpolation process of the spectra in the stellar atmospheric parameter space (usually it includesthree parameters: effective temperature T eff , logarithmic surface gravity acceleration lg g and metallicity [ F e/H ] ). The stellar spectral library gives a correspondence between stellar atmospheric parameters andstellar spectra.In this section, we will give a detailed introduction of the RBF network and the upgraded RBF interpo-lator in our work. In Section 2.1, we describe the RBF network and its calculated process as an interpolator(RBF interpolator). In Section 2.2, we describe the kernel function of upgraded RBF interpolator in ourwork. In Section 2.3, we give a summary of the upgraded RBF interpolator in our work. Early introduction of RBF interpolator can be found in Powell (1987). Broomhead & Lowe (1988) broughtthe RBF into the artificial neural network (ANNs). Up to now, the RBF interpolation and fitting method hasbeen applied widely in many fields (such as, mineral analysis, aircraft design, image processing and patternclassification).RBF interpolator can be thought as an application of RBF network which is a kind of kernel methods.RBF also is called kernel function in this paper. The construction of RBF network is shown in Fig. 1.Moreovemsxxr, the sample in this work is a set of points with the coordinate set { x , x , ..., x i , ..., x N } and the values { y , y , ..., y i , ..., y N } ( x i ( x , x , ..., x D ) i is the i th sample point coordinate in the D -dimensional space. In this paper, we use {} to represent set in mathematics). n upgraded spectral interpolator base on RBF 5 Fig. 1
The three-layer structure of RBF network is shown. The left is the input layer and x ( x , x , ..., x D ) is the input sample coordinate in the D -dimensional space. The middle is a hidden layer that is constitutedby RBF functions K i ( x − µ i ) , i = 1 , , , ..., m , µ i ( µ , µ , µ , ..., µ D ) i is the central coordinate of K i ( x − µ i ) in the D -dimensional space and m is the number of RBF functions in the hidden layer. Theright is the output layer and y is the prediction value which is the sum of RBF function multiplied by thecorresponding weight factor c ( c , c , c , ..., c m ) .From Fig. 1, we can find the RBF network consists of three layers. The left is the input layer and x ( x , x , ..., x D ) is the input point coordinate. The middle is the hidden layer which is constituted byKernel functions K i ( x − µ i ) ( i = 1 , , ..., m ) , µ i ( µ , µ , µ , ..., µ D ) i is the central coordinate of the i th kernel function in D -dimensional space. The right is the output layer which is the sum of the kernelfunction multiplied by the weight factor ( y = P mi =1 c i · K i ( x − µ i ) ) and the size of the output layer hasno any limitation. In the spectral interpolation, y is the flux within a given wavelength interval, it is onedimensional scalar, the interpolation spectrum consist of the interpolated fluxes at different wavelengths.For a sample with a huge size ( N ), a fast RBF network can be constituted by a much less number ofkernel functions in the hidden layer ( m ≪ N ). Usually, K-means clustering method (MacQueen 1967;Ding & He 2004) is used to search for the kernel central coordinates µ i ( i = 1 , , , ..., m ) , then linearregression method is used to get the weight factor array c ( c , c , c , ..., c m ) for the sample set { ( x , y ) i , i =1 , , , ..., N } .The empirical stellar spectral library usually comprise several hundred or thousand spectra. So in ourwork, the number of kernel functions m is set to be same as the sample number N . Moreover, we do notneed the calculations of the K-means and the linear regression. We take the sample points as the centersof the kernel functions directly in the spectral RBF interpolator ( µ i = x i , i = 1 , , , ..., N ). The weightfactor array c of kernel function in the hidden layer is obtained by solving the system of linear equations N X j =1 K j ( x i − x j ) · c j = y i ( i = 1 , , , ..., N ) , (1) L.-T. Cheng et al.
Fig. 2
A example of the 2-D space σ distribution for the kernel function. Black points are the centers ofthe kernel functions ( { µ i , i = 1 , , , ..., m } ). From the left to the right and from the bottom to the top, thedensity of points decrease. The radius of black rings is used to characterize the σ influence range for theclassical RBF network. The red ellipse is used to characterize a more reasonable influence range of kernelfunctions for anisotropic distribution.In our works the Gaussian kernel function e − P Dd =1 ( x d − µ d ) / (2 σ ) is used, K j ( x i − x j ) = e − P Dd =1( xd,i − xd,j )22 σ j , (2)where x d,i and x d,j are the coordinates of the i th and j th sample point in the D -dimensional space ( d =1 , , , ..., D ). The central coordinate of the j th kernel function K j is x j , σ j is the standard deviation ofthe j th kernel function K j and can be used to characterize the influence range of the kernel function. c j isthe weight factor of the j th Gaussian kernel function. Solving the linear Eq. 1, we can get the weight factorarray c , if sample coordinate in set { x , x , x , ..., x N } is different from each other . After obtaining c ,we can get a simple formula for the interpolation calculation, y ( x ) = N X j =1 K j ( x − x j ) · c j , (3)where x is the input coordinate, and y ( x ) is the interpolation result. For spectrum interpolation calculation, x and y correspond to the stellar atmospheric parameter and the interpolated flux within a wavelengthinterval, respectively.The RBF network is base on the kernel functions. However, a const σ usually is used for all the Gaussiankernel functions (Lowe 1989). For the irregular distribution of stars in the stellar spectral library, all thekernel functions with the same σ is not a better selection. Next, we will introduce the kernel function in ourspectral RBF interpolator. For Gaussian kernel function, this conclusion can get a fixed factor array c from the Micchelli theorem (Micchelli 1986) n upgraded spectral interpolator base on RBF 7 The const σ in the Gaussian kernel functions has a disadvantage: large σ of the kernel function makes theinterpolation results oversmoothed in the dense area, the relative small σ makes the interpolation resultsdiscrete in the sparse area. In Fig. 2, we give a example of the 2-D σ distribution, black point is the centersof the kernel functions { µ i , i = 1 , , , ... } , and the space density decreases from the left to the right andthe bottom to the top, and the anisotropic distribution exists in top-left and bottom-right parts. The blackrings are used to characterize the influence range of kernel functions in the traditional RBF network and thered ellipses show a better choice for the anisotropic kernel functions.Stars in the empirical spectral library face a more complex situation than Fig. 2. The distribution isnonuniform and its density varies significantly in the stellar atmospheric parameter space (a typical situationcan be found in Fig. 3). This situation is resulted from the observational and theoretical limitations . In ourwork, we set a constraint by including a relation between the σ and the spatial density of sample pointsinto the RBF network, this relation can be used to determine the σ value. Moreover, we also considerthe anisotropy of the kernel function in the RBF network by referring to the adaptive smoothed particlehydrodynamics simulation (hereafter ASPHs).In Section 2.2.1, we give an introduction of the constraint on σ . In Section 2.2.2, we show genericGaussian kernel function for the anisotropic situation in our works. σ and its size calculation In this part, we introduce the smooth length constraint of SPHs into the RBF network. Under this constraint,we show the computing method of σ for the kernel function in the RBF network.In the SPHs, the sample consists of particles with the coordinate set { x i , i = 1 , , , ..., N } and massset { m i , i = 1 , , , ..., N } , x i and m i are the position and mass of the i th particle. The fluid density ρ inthe position x i is ρ i ≈ N X j =1 m j · W ij ( i = 1 , , , ..., N ) , (4)where N is the number of particles, W ij is a · K ( x i − x j ) and a is the normalized coefficient. We use theGaussian kernel function W ij = 1( √ πσ j ) D · e − P Dd =1( xd,i − xd,j )22 · σ j , (5) For spectroscopic observations, only solar neighbor stars can be observed with high quality. In stellar theory, the bright starsallows have a short evolutionary time scale. Both of them make the observed spectral sample have obvious selection effect
L.-T. Cheng et al. where D is the space dimension and σ j is the smooth length of the j th kernel function. Replacing m j with ρ j V j , where V j is the volume of j th particle in SPHs. We have ρ i ≈ N X j =1 ρ j · V j · W ij = N X j =1 ρ j · V j · √ πσ j ) D · e − P Dd =1( xd,i − xd,j )22 · σ j . (6)The fluid density varies with time. A better smooth length should vary dynamicly. Usually σ = σ · ( ρ /ρ ) /D is used to give the current smooth length ( σ and ρ are initial value of SPHs). This relationgives a constraint about the σ by spatial density of sample points. Here, we use this relation in the kernelfunction of the RBF network at different positions (in the stellar atmospheric parameter space). We give theconstraint about σ , σ ∝ ρ − D ∝ V D , (7)where V is used to characterize the particle volume which is related to the influence range of the kernelfunction. Usually, the dimension number D is 3 for the stellar spectral library.Now, including the constraint (formula 7) in Eq. 6, we have ρ i ≈ N X j =1 ρ j · c ( √ π ) D · e − P Dd =1( xd,i − xd,j )22 · σ j ( i = 1 , , , ..., N ) , (8)where c ≡ V / ( σ D ) . Eq. 8 gives a set of nonlinear equations of σ . However, it is difficult to be solved.To simplify the Eq. 8, we approximate Eq. 8 to the following expression by replacing the j with i forthe subscripts of σ and ρ , ≈ N X j =1 c ( √ π ) D · e − P Dd =1( xd,i − xd,j )22 · σ i ( i = 1 , , , ..., N ) , (9)this is because ρ and σ are continuous and the adapted kernel function (here, it is Gaussian function) is forlocal region. The equation becomes independent of each other. From Eq. 9, we can use the bisection methodto calculate σ i ( i = 1 , , , ..., N ) quickly. In this process the σ can be adjusted by the control parameter c . From Eq. 9, we can know c is within the interval of ((2 π ) D/ /N, (2 π ) D/ ] , its upper and lower limitscorrespond to ∞ and for σ .In Fig. 3, we show the resulted σ i ( i = 1 , , , ..., N ) for the dimensionless stellar atmospheric pa-rameter of MILES library. In this work, the coordinates ( T eff , lg g, [ F e/H ]) are dimensionless which areexpressed in the units of mean square error (hereafter MSE ). The size of σ is relatively large in the sparesarea and small in the dense area.We should also notice that the σ i in Eq. 9 is scalar, which corresponds to the isotropic kernel function. Infact, the anisotropic distribution of stars in the stellar atmospheric parameter space is common for empiricalspectral library (shown in Fig. 3). n upgraded spectral interpolator base on RBF 9 Fig. 3
The size of σ in the kernel function for the MILES stellar spectral library. The coordinatesof black points are the dimensionless stellar atmospheric parameters of MILES library ( T eff / MSE T eff , (lg g ) / MSE lg g and [ F e/H ] / MSE [ F e/H ] , the MSE is the mean square error function). The radius of semi-transparent sphere around the black point characterizes the corresponding σ . This part gives the introduction of the anisotropic kernel function by generalizing the Gaussian kernelfunction.In SPHs, the fluid varies with time and the deformation usually is not isotropy. For the anisotropy de-formation, the isotropy kernel function usually is not able to give a better smoothing effect as shown in Fig.2. Similarly, Bicknell & Gingold (1983) has even considered the kernel function with different smoothinglength in y and z axis direction in the tidal destroy simulation. Moreover, the generalized anisotropy kernelfunction has been discussed by Shapiro et al. (1996) and Owen et al. (1998).The anisotropic kernel function is the generalization of the isotropic Gaussian kernel function. If we let M be P d =1 ( x d − µ d ) / (2 σ ) , which is the exponential part of the isotropic Gaussian kernel function, then the corresponding vector M aniso of anisotropic kernel function can be described by the following M aniso = T · ( x − µ ) = X k =1 T dk · ( x k − µ k ) (10) = T T T T T T T T T x − µ x − µ x − µ = Z Z
00 0 Z R R R R R R R R R x − µ x − µ x − µ ( d = 1 , , , where d and k is the dimensional subscript, the matrix T is a linear translation (from x − µ to M aniso ).The generalized Gaussian kernel function result can be written as e − M .Matrix T can be separated into two parts T = Z · R as shown in Eq. 10. The diagonal matrix Z is usedto give a scale transform along with axis, the matrix R is constituted by three orthogonal basis and givesa rotational transform. Both Z and R change with the position of kernel function center µ in the stellaratmospheric parameter space. The isotropy kernel function can be recovered by setting I = R = √ σ · Z ,here the I is the unit matrix.In the stellar spectral library, most of stars distribute along the main sequence and in the red giantregion, these two parts are distributed almost along the T eff and lg g axis direction, the distribution of starsat different [ F e/H ] range only have a slight bias. Therefore, in most cases, the anisotropy is in the axisdirection. To simplify the calculation process, in this work we ignore the rotational matrix R by setting R = I and only consider the Z , Z and Z in the generic Gaussian kernel function.The degree of anisotropy of kernel function is dependent of the ratio of Z , Z and Z . In our work,the axial local dispersion D is used to give the e Z for any kernel function with central coordinate µ , Z dd ∝ e Z dd = c + (1 − c ) D d ( P d =1 D d ) / ( d = 1 , , ,D d = vuut N X i =1 ( x d,i − µ d ) · e − P d =1( xd,i − µd )22 σ (11)where c is a control parameter which is used to adjust the degree of anisotropy, the Gaussian function isused to limit the calculation in the local region. So after the calculation of Eq. 11, we can know the ratio of Z , Z and Z in the matrix Z for all kernel functions and get the corresponding normalized matrix e Z .The last Z is given by solving the Eq. 9 again with the anisotropic kernel function e − [ e Z ( x − µ )] . n upgraded spectral interpolator base on RBF 11 Here, we give a summary of the mathematical process of the RBF interpolator in this work. In general, asshown in Fig. 1, there are two key parts for the RBF network need to be given, one is the kernel functions { K , K , K , ..., K N } , another is the coefficients array c . For the latter, c can be obtained by solving thesystem of linear equations 1 for the given kernel functions and the sample { ( x , y ) i , i = 1 , , , ..., N } .Next, we will list the determination process of the RBF kernel functions { K , K , K , ..., K N } .First of all, we use the stellar spectral library as the sample of RBF network, { x , x , x , ..., x N , } corresponds to the stellar atmospheric parameter and { y , y , y , ..., y N } corresponds to the flux. We selectthe Gaussian kernel function and set the central coordinates of all kernel functions are the coordinates ofthe sample ( µ i = x i , i = 1 , , , ..., N ). Next, for the i th kernel function ( i = 1 , , , ..., N ), three steps tobe executed.1. Solving equations 9 to get σ iso ,i of the isotropic Gaussian kernel function. In this step, we includeparameter c for all points.2. Inputting σ iso ,i in Eq. 11 and calculating the matrix e Z i . In this step, c is used to control the degree ofanisotropy of kernel functions.3. Rewriting the exponential part of the equations 9 by e − [ e Z ( x − µ )] and solving it, we can get the last Z i and T i (in this step, a new parameter c is included which corresponds to c in the first step).After those three steps, we get the last kernel function of RBF network, then solve the system of equations1 to get the weight factor array c . At last, we get the last RBF network, and the calculation of spectral RBFinterpolator corresponds to Eq. 3.The three control parameters c , c , c are included, the optimization calculation of these parametersis shown below. For distinguishing the RBF interpolator in Cheng et al. (2018) from that in this work, wename there two spectral RBF interpolators RBF18 and
RBF update . We can control the RBF network by adjusting the control parameters ( c , c , c ) (Section 2.3). In thissection, we will introduce the optimization process of these three control parameters.In this work, MILES library is used to build the spectral RBF interpolator RBF update . The MILESempirical stellar spectral library includes ∼ stars obtained on the . Isaac Newton Telescope.The wavelength ranges from . to . ˚ A and the spectral resolution is ∼ . ˚ A (S´anchez-Bl´azquezet al. 2006, FWHM). The coverage of the stellar atmospheric parameters are: < T eff < , . < lg g < . and − . < [ F e/H ] < +1 . . The MILES spectral library has a larger coverage in c −5 0 5 10 15 c c I t e r a t i o n s A v e r a g e v a l u e o f M S E RBF upate
RBF18
Fig. 4
The output of the optimization process by the BAS search algorithm. The left panel shows thevariation of object function as a function of the iteration times. Red point is the objective function valueof the
RBF update interpolator. Black point is the objective function value of
RBF18 interpolator which isused as a comparison. The right panel shows the corresponding track of the point ( c , c , c ) . The cyandash line is displacement, and nodes are the coordinates of ( c , c , c ) in different iterations. The nodecolor corresponds to the number of iterations, 2000 iterations are shown here.the parameter spaces than the other empirical stellar spectral libraries used in the stellar population synthesismodels (Cenarro et al. 2007).In this work, we use the semi-empirical BaSeL-3.1 stellar spectral library (Lejeune et al. 1997, 1998;Westera et al. 2002) as the reference library to find the best control parameters of RBF update interpolator.BaSeL-3.1 is one of the widely used spectral libraries, it provides an extensive and homogeneous grid oflow-resolution spectra in the range of − ˚ A for a large range of stellar parameters: < T eff < ; − . < lg g < . and − . < [ F e/H ] < . .For avoiding extrapolation of the spectral RBF interpolator, the input parameter should be within thecoverage area of stars in the MILES library. So in the BaSeL-3.1 library only those models within thecoverage area of MILES library are used here. Gaussian smooth algorithm is used to degrade the resolutionof the output spectra to ˚ A (the resolution of BaSeL-3.1 in visible wavelength). Iterative calculation isused in the optimization, in each iteration, hundreds of stars in BaSeL-3.1 are selected as the input sample,the output interpolation spectra are used to compare with the original spectra in BaSeL-3.1. The averagevalue of the mean square error between interpolation spectra and original spectra in BaSeL-3.1 are usedas objective function f ( c , c , c ) in the optimizing process, and the best ( c , c , c ) corresponds to theminimum of f ( c , c , c ) . n upgraded spectral interpolator base on RBF 13 T eff =20000K, lgg=4.2, [Fe/H]=0.5 (1, 1) T eff =20000K, lgg=4.2, [Fe/H]=0.0 (1, 2) T eff =20000K, lgg=4.2, [Fe/H]= )0.5 (1, 3)12 T eff =10000K, lgg=4.3, [Fe/H]=0.5 (2, 1) T eff =10000K, lgg=4.3, [Fe/H]=0.0 (2, 2) T eff =10000K, lgg=4.3, [Fe/H]= )0.5 (2, 3)0.51.0 T eff =5000K, lgg=4.5, [Fe/H]=0.5 (3, 1) T eff =5000K, lgg=4.5, [Fe/H]=0.0 (3, 2) T eff =5000K, lgg=4.5, [Fe/H]= )0.5 (3, 3)4000 5000 6000 7000Wavele gth/(0.00.51.0 T eff =4360K, lgg=2.0, [Fe/H]=0.5 (4, 2) F l u x ( o r m a li z e d t o V - b a n d ) T eff =4360K, lgg=2.0, [Fe/H]=0.0 (4, 2) 4000 5000 6000 7000Wavele gth/( T eff =4360K, lgg=2.0, [Fe/H]= )0.5 (4, 3) Fig. 5
The twelve interpolation spectra based on the MILES spectral library by the
RBF update interpolator.Input stellar parameters are shown in each panel. Here the spectra are dimensionless. The first two linesshow the interpolation spectra of six main sequence stars with T eff = 20000 K and T eff = 10000 K , Thethird line shows the interpolation spectra of three main sequence stars with T eff = 5000 K . The forth lineshows the interpolation spectra of three red giant stars with T eff = 4360 K . In each line, the input stellarmetallicity is [Fe / H] = 0 . , . and − . .The widely used optimization algorithm needs massive computation to obtain the best ( c , c , c ) . BASalgorithm is used (Jiang & Li 2017), which is a new and light algorithm by simulating the beetle behavior.In this work, the process of the BAS search is in 3-D space of ( c , c , c ) and comprises four steps.Step1. Setting the initial position of the ’beetle’ P , the initial distance of two antennas A and the initialstep length S . For the i th iteration, it is referred to as P i , A i , S i ;Step2. Generating a unit vector d with random direction, d is used to give the relative position of twoantennas. Then, left antenna position has P l , i = P i − . · A i · d , right antenna position has P r ,i = P i + 0 . · A i · d ;Step3. Calculating the objective function on two antennas f ( P l ,i ) , f ( P r ,i ) . The new position is P new = P i + [ f ( P r ,i ) − f ( P l ,i )] / abs( f ( P r ,i ) − f ( P l ,i )) · d · S i ;Step4. If iteration meets the critical condition, the process jump out of the iterations. If not, i + = 1 , P i = P new and the loop go back to Step2.4 L.-T. Cheng et al.
RBF update interpolator.Input stellar parameters are shown in each panel. Here the spectra are dimensionless. The first two linesshow the interpolation spectra of six main sequence stars with T eff = 20000 K and T eff = 10000 K , Thethird line shows the interpolation spectra of three main sequence stars with T eff = 5000 K . The forth lineshows the interpolation spectra of three red giant stars with T eff = 4360 K . In each line, the input stellarmetallicity is [Fe / H] = 0 . , . and − . .The widely used optimization algorithm needs massive computation to obtain the best ( c , c , c ) . BASalgorithm is used (Jiang & Li 2017), which is a new and light algorithm by simulating the beetle behavior.In this work, the process of the BAS search is in 3-D space of ( c , c , c ) and comprises four steps.Step1. Setting the initial position of the ’beetle’ P , the initial distance of two antennas A and the initialstep length S . For the i th iteration, it is referred to as P i , A i , S i ;Step2. Generating a unit vector d with random direction, d is used to give the relative position of twoantennas. Then, left antenna position has P l , i = P i − . · A i · d , right antenna position has P r ,i = P i + 0 . · A i · d ;Step3. Calculating the objective function on two antennas f ( P l ,i ) , f ( P r ,i ) . The new position is P new = P i + [ f ( P r ,i ) − f ( P l ,i )] / abs( f ( P r ,i ) − f ( P l ,i )) · d · S i ;Step4. If iteration meets the critical condition, the process jump out of the iterations. If not, i + = 1 , P i = P new and the loop go back to Step2.4 L.-T. Cheng et al. eff /K−10123456 l g ( g / ( c m ⋅ s − )) MS − − TP−AGBlg(age/yr) = 7, 7.5, 8, 8.5, 9, 9.5, 10[Fe/H]= −0.5, 0.0, 0.5
Fig. 6
The positions (green points) of the stars in Fig. 5. The isochrones are used to give the relative position.For each isochrone, stars from the zero-age main sequence to the end of TP-AGB phases are displayed. Thered dot lines have [ Fe/H ] = − . , black lines have [ Fe/H ] = 0 and yellow dot dash lines have [ Fe/H ] = 0 . .From left to right lg ( age / yr) are , . , , . , , . and 10.During iteration, the distance of two antennas A i and the step length S i change slowly. In our calculation, S i +1 = b · S i and A i = c · S i , b is a constant coefficient close to 1 but less than 1 and c also is a constantcoefficient.Fig. 4 gives the iteration process of the ( c , c , c ) in the optimization. In this figure, 2000 iterationsare shown. The left panel shows the objective function output, red points are the objective function valuesof RBF update interpolator. As a comparison, the black points are the objective function value of
RBF18 interpolator with the same calculation. In the right panel, the cyan dash line shows the movement trailof ’beetle’, and the color nodes are the ’beetle’ positions in iteration. The node color corresponds to thenumber of iterations. We can find the result of ( c , c , c ) converges to a fixed value at last. The objectivefunction of RBF update interpolator is lower than the result of
RBF18 interpolator, this means that spectralinterpolator
RBF update has a better performance than
RBF18 in spectral interpolation calculation of thestars in BaSeL-3.1 library.
In this section, we show the interpolation results and the analysis of
RBF update interpolator based onthe empirical stellar spectral library MILES. In Section 4.1, we present the interpolated stellar spectra fordifferent spectral types. In Section 4.2, we give a test of the
RBF update interpolator and give a comparisonbetween the interpolation spectra by
RBF update and
RBF18 interpolators based on the MILES stellarspectral library. In Section 4.3, we give an analysis and a discussion for the behave of the
RBF update interpolator in our test. n upgraded spectral interpolator base on RBF 15
For the sake of clarity, we only show the interpolation spectra of twelve stars with typical stellar atmosphericparameters in Fig. 5. All of the interpolation spectra are normalized to its V-band flux. The twelve panels aredivided in to four rows, and each row corresponds to three input stars with different metallicity [ F e/H ] = − . , and . (the corresponding atmospheric parameters is shown in each panel). In the first row, theinput stars with T eff = 20000 K and lg g = 4 . correspond to the massive main-sequence star and havea great effect on the U-band flux of the stellar population integrated spectra with stellar population ageless than . yr . In the second row, the input stars with T eff = 10000 K and lg g = 4 . correspond tomedium mass main-sequence star and have a great effect on the B-band flux of integrated spectra withstellar population age less than . yr . In the third row, the input stars with T eff = 5000 K and lg g = 4 . correspond to the low-mass main-sequence star which have a large number in the stellar population, theyhave a significant effect on the V-band of the stellar populations integrated spectra. In the last row, the inputstars with T eff = 4360 K and lg g = 2 . correspond to red giant stars. It is very bright and have a greateffect on the infrared band flux of the older stellar population integrated spectra.In Fig. 6, we give the positions of these stars (in Fig. 5) on the T eff and lg g plane (the green points).Moreover, we present three sets of the isochrones at metallicity [ F e/H ] = − . , and . to give thelocations of the stellar populations. For each set of isochrones, from left to right the stellar age lg (age / yr) =7 , . , , . , , . and , The isochrones only show the stars from the zero-age main sequence phase tothe end of thermal pulsating asymptotic giant branching (TP-AGB) phase. The relative positions betweenstars and the isochrones correspond to the above analysis of Fig. 5. In this figure, the isochrones are theresults of MIST (Dotter 2016; Choi et al. 2016), they are calculated by using the stellar evolution codeMESA (Paxton et al. 2011, 2013, 2015, 2018). In this part, we test
RBF update interpolator based the empirical spectral library MILES in Section 4.2.1,and we give a comparison between the
RBF update and
RBF18 interpolators in Section 4.2.2.
RBF update interpolator
We test the
RBF update interpolator based on the MILES stellar spectral library. In the test, we delete onemember star from MILES stellar spectral library and use the remained spectra as a library to calculate thespectrum of the deleted star. Every star in the MILES library has been tested by the above process. Andthe comparison between interpolated and original spectra is used to show the interpolation performance of
RBF update interpolator, (the same test is also done for the spectral interpolator
RBF18 ). Fig. 7
The
MSE
RBF update based on MILES spectral library. The top-left panel gives the result in stellaratmospheric parameter space. The points give the positions of test stars, the grey level and point size areused to characterize the test result
MSE of RBF update interpolator. Here,
MSE is the mean square errorbetween the interpolated and the original spectra of test star. The top-right panel gives the projection onthe lg T eff and lg g plane, the bottom-left panel gives the projection on the lg g and [ F e/H ] plane and thebottom right panel gives the projection on the lg T eff and [ F e/H ] plane. The red ”+” and the correspondingletter are used to give the position of the stars for which their spectra are shown in Fig. 8.The MSE between interpolated and original spectra is used to characterize the difference,
MSE = pP λ ( f int , λ − f ori , λ ) / len( f ori , λ ) , where f int , λ and f ori ,λ are the normalized flux on the λ th wavelengthinterval of the interpolated and original spectra, respectively, len( f ori , λ ) is the array length of the spectrum.Every star of MILES library has a corresponding MSE value, we present it in Fig. 7.Fig. 7 plots the overall result of the
MSE distribution. The top-left panel shows the result in the lg T eff , lg g, [ F e/H ] space. For the sake of clarity, the top-right panel shows the projection of MSE onthe lg T eff and lg g plane, the bottom-left panel shows the projection on the lg g and [ F e/H ] plane and thebottom-right panel shows the projection on the lg T eff and [ F e/H ] plane. For each panel, the MSE value is n upgraded spectral interpolator base on RBF 17
Fig. 8
The spectra of eight test stars in the MILES library represented by the red ”+” in Fig. 7. The stellaratmospheric parameters and the test result
MSE of RBF update and
RBF18 interpolators also are listed. Forevery panel, three spectra are included. The black line is the original spectrum of test stars in MILES library.The green and red translucent lines are the interpolation spectra of test star by
RBF update and
RBF18 interpolators, respectively. For the reason of clarity, the interpolation spectra by
RBF update interpolator isshifted upwards, and the corresponding original spectrum also has a copy spectrum moved upwards by thesame distance. Moreover, for avoiding overlap, the red and green lines are translucence. First 4 panels (a-d)list four representative test spectra in different areas of stellar atmospheric space. Panels e and f list therepresentative bad interpolation spectra in the low-mass main sequence region. The last two panels g and hlist the representative bad interpolation spectra in red giant branch region.characterized by the gray level and point size. We can find that most stars have relatively small
MSE values.The obvious difference exists in parts of the lower-temperature region, especially for stars in the edge of thelow-metallicity red giant region.For a more detailed analysis of the
RBF update interpolator, eight representative stellar spectra are shownin Fig. 8 (the position of those eight test stars are marked by red ”+” in Fig. 7). In each panel of Fig. 8, blacklines are the original spectrum of the test star, the green and red translucent lines are the interpolation
Fig. 9
The discrepancy in the
MSE between the
RBF update and
RBF18 interpolators based on the MILESlibrary. The points give the positions of test stars. The color of points is used to characterize the value
MSE
RBF update − MSE
RBF18 , the blue points mean negative value, the red points mean positive value. Thepoint size is used to characterize the absolute value of
MSE
RBF update − MSE
RBF18 . The top-left panel listthe result in 3D stellar atmospheric parameters, the top-right panel gives the projection on the lg T eff and lg g plane, the bottom-left panel gives the projection on the lg g and [ F e/H ] plane, the bottom-right panelshows the projection on the lg T eff and [ F e/H ] plane.spectra of the test star by RBF update and
RBF18 interpolators, respectively. The interpolation spectra ofthe
RBF update interpolator and a copy of the original spectrum are moved upwards for the reason of clarity.The stellar atmospheric parameter and the
MSE value also are given in each panel. Here the combination ofgreen and black spectra gives a direct spectrum performance of the
RBF update interpolator, the combinationof red and black spectra gives a direct spectrum performance of the
RBF18 interpolator. Panels a-d list therepresentative spectra of massive main sequence, medium mass main sequence, low-mass main sequenceand red giant test stars, we can find the interpolation spectra have a good match with the original spectra.Most of the test stars have similar results in our test, but there are still a few test stars have bad test results. n upgraded spectral interpolator base on RBF 19
Panels e-h show four typically spectra of bad performance by
RBF update and
RBF18 interpolators, theyare in the red giant (g-h) and low-mass main sequence (e-f) regions. This bad performance will be discussedin Section 4.3.
RBF18 interpolator
In this section, we give a comparison between the
RBF update and
RBF18 interpolators by the test inSection 4.2.1. For any test star in MILES library, we use the
MSE
RBF update − MSE
RBF18 to character thediscrepancy. The mean square error
MSE is large than zero, the smaller value of
MSE means the bettermatch between the interpolated and the original spectra of test star in MILES library. So, the negativevalue of
MSE
RBF update − MSE
RBF18 means that the
RBF update interpolator has a better performance, thepositive value means that the
RBF18 interpolator has a better performance.In Fig. 9,
MSE
RBF update − MSE
RBF18 is shown in the stellar atmospheric parameter space. The top-left panel gives the result in T eff , lg g and [ F e/H ] space. The other three panels list the correspondingthree projections that are same as those in Fig. 7. The point size is used to characterize the absolute value of MSE
RBF update − MSE
RBF18 , the point color is used to characterize the value
MSE
RBF update − MSE
RBF18 .Here, blue and red points mean negative and positive value of
MSE
RBF update − MSE
RBF18 , respectively.On the whole, the test results of
RBF update interpolator are better than those of the
RBF18 interpolator.In the high temperature main sequence region, it does not have obvious difference for two interpolators.Panels a and b in Fig. 8 give two typical test results, and both two interpolators have smaller
MSE . Indense part of low-temperature main sequence and red giant regions, the
RBF update interpolator has betterperformance than
RBF18 interpolator with a lower
MSE value. Two typical test spectra are shown in panelsc and d in Fig. 8, the green interpolation spectra have a better match with the original spectra than the redone.But for some test stars on the edge of the low-temperature region, both
RBF update and
RBF18 inter-polators have bad performance, the spectra of four typical test stars in those regions are shown in panels e-hof Fig. 8, the detail analysis of the reason will be given in Section 4.3.
Here, we will give an analysis of the test result in Section 4.2. On the whole, the
RBF update interpolatorhas good performance than the
RBF18 interpolator in the spectral interpolation calculation. However, ithas bad performance in the edge of the low-temperature region. A typical example of bad performance isshown in the last four panels of Fig. 8. Panels e and f show the test result of two test stars in the edge of thelow-temperature main sequence region, panels g and h show the test result of two test stars in the edge ofthe red giant region. The positions of these four test stars are shown in Fig. 7, we can find the test stars of
Fig. 10
The
MSE distribution of the test for the
RBF update interpolator on the lg T eff and lg g plane. Thetop panel is the same as the top-right panel of Fig. 7 and gives the result MSE distribution of the test whichis based on the MILES library with its original stellar atmospheric parameters (Cenarro et al. 2007). In thebottom panel, we show the
MSE distribution of test based on the MILES library with the corrected coolstellar atmospheric parameters (Sharma et al. 2016). For every panel, the grey level and the size of pointsare used to characterize the
MSE values.panels e and f are adjacent and those of panels g and h are adjacent. From the Gaussian kernel function (Eq.3), we know that the spectrum of the adjacent star has a bigger effect than the distant one in the interpolationcalculation. Therefore, in the test, the calculation of the deleted spectrum depends largely on the adjacentspectra in the stellar atmospheric parameter space. For the test star in panel e, the star f has a big effect,that is the reason the interpolation spectrum in panel e is similar to the original spectrum in panel f. For theinterpolation spectrum of panels f, g and h, the situations are similar.For the empirical stellar spectral library, we list three possible reasons for the bad test results.1. The finite spectra faces its complex change in some stellar atmospheric parameter region. It means thatthe library is incomplete and has not include enough typical spectra. n upgraded spectral interpolator base on RBF 21
2. Three stellar atmospheric parameters can not determine solely the spectra. It means that one set of stellaratmospheric parameters in 3-D space corresponds to several potential spectra with obvious difference (similar to the description in Arentsen et al. 2019.)3. The stellar atmospheric parameters are not self-consistent. The stellar spectra do not vary with the stellaratmospheric parameters by a one-by-one relation .For the first reason, more targeted observational data are needed. For the second reason, more potentialparameters of spectra should be given for a more strict constraint on the spectra. For the third reason, thespectra in the library need a more detailed derivation of atmospheric parameters, we give a test by usingthose self-consistent stellar atmospheric parameters. Here we use a relatively new result of Sharma et al.(2016) to test this idea. In Sharma et al. (2016), ∼ cool stars in MILES library were refined. Wereplace the corresponding parameter of MILES library by the results of Sharma et al. (2016), and use themto test RBF update interpolator as did in Section 4.2.1. In Fig. 10, we show the test results. The top panelshows the
MSE distribution of test stars based on the MILES library with the original stellar atmosphericparameters (Cenarro et al. 2007), it is same as the top-right panel in Fig. 7, the bottom panel shows the MESdistribution of the test based on the MILES library with replaced stellar atmospheric parameters of the coolstars (Sharma et al. 2016). An obvious improvement is appeared in the bottom panel, this result prove theself-consistent parameters of library is important for the spectral interpolation calculation.
In stellar population synthesis models, the empirical stellar spectral library is necessary for the integratedspectra of the stellar populations. In this work, we improve the RBF network by comparing with the otherkernel methods (SPHs and likelihood approximation) and give an upgraded spectral RBF interpolator. Weinclude a constraint about the kernel function (Eq. 7) in the RBF network, this constraint gives the relationbetween the σ of Gaussian kernel function and the sample spatial density in the parameter space.Moreover, we also consider the anisotropic kernel function by relating it to the inhomogeneous distri-bution of stars in the stellar atmospheric parameter space. We use the local axial direction dispersion todetermine the anisotropic kernel function (Eq. 11). By including three control parameters c , c , c , we canget a RBF network for spectral interpolation calculation, here we call it RBF update interpolator. The BASsearch algorithm is used to search the best control parameters c , c , c by matching with the semi-empiricalBaSeL-3.1 stellar spectral library. An example is that [ F e/H ] can not describe the ratio of the different elements in the stellar atmosphere, this problem has not anobvious effect in the high-temperature region, but can not be ignored in the low-temperature region. A simple example is that a smooth change of the spectra in the stellar atmospheric parameter space can be broken and becomesmessy by adding a set of random biases on the stellar atmospheric parameters in the library.
We also use a test to analyze the performance of
RBF update interpolator based on the MILES stellarspectral library. In the test, we select any star in the MILES library as the test object and compare theoriginal with the interpolation spectra which is calculated by the
RBF update interpolator based on theremained stellar spectra in MILES library. We find that
RBF update interpolator has a good performance ingeneral except for some test stars in the edge of the red giant and low-temperature main sequence regions(Fig. 7).Three possible reasons can cause these bad performance for empirical stellar spectral library, the firstis the incomplete spectral coverage in the stellar atmospheric parameter space, the second is the existenceof potential atmospheric parameters, the third is the inconsistent atmospheric parameters. For the first tworeasons, more observations are needed and the additional atmospheric parameters should be included in thestellar spectral library. For the last reason, the modified stellar atmospheric parameters of the stellar spectrallibrary are needed. Moreover, we also give a comparison between the
RBF update interpolator and our earlywork in Cheng et al. (2018), the results show that the
RBF update interpolator has an obvious improvement,except for the edge of the low-temperature region. (Fig. 9), the same reasons make both interpolators havenot a good performance in these regions.At last, the code of
RBF update interpolator is written by Python and you can find it in . The code can be used for different libraries and user can use it with themodified stellar spectral library by adding additional spectra or updating the stellar atmospheric parametersof the library.
Acknowledgements
This project was partly supported by the Chinese Natural Science Foundation (No.11973081 and 11521303), the Yunnan Foundation (grant No. 2011CI053 and 2019FB006) and the YouthProject of Western Light of Chinese Academy of Sciences. We also thank the referee for suggestions thathave improved the quality of this manuscript.
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