SPECTRAL ANALYSIS USING EVOLUTION STRATEGIES

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1 SPECTRAL ANALYSIS USING EVOLUTION STRATEGIES J. FEDERICO RAMÍREZ AND OLAC FUENTES Insttuto Naconal de Astrofísca, Óptca y Electrónca Lus Enrque Erro # 1 Santa María Tonanzntla, Puebla, 784, Méxco framrez@cseg.naoep.mx, fuentes@cseg.naoep.mx ABSTRACT In ths paper we show how evoluton strateges (ES) are successfully appled to the problem of fttng lne profles of stellar spectra, whch provdes a relable decomposton. Usng a stellar spectrum as nput, we mplemented an evoluton strategy to fnd ts components: contnuos spectrum and spectral lnes. In our experments we used Gaussan functons to match spectral lnes and three dfferent equatons to represent the contnuos spectrum. We appled ths method to smulated and real spectra. Keywords: Spectral Analyss, Evoluton Strateges, and Optmzaton 1. INTRODUCTION A task n the analyss of stellar spectra s to dentfy and measure the flux of emsson and absorpton lnes n the spectra. These lne profles can be compared wth the spectral profle of onc and atomc transtons of known wavelengths. Sometmes, the observer bases these assocatons producng an ncorrect nterpretaton of the data. The lne flux n spectra may be measured by multple technques, ncludng fttng ndvdual lnes. Standard spectral decomposton technques usng non-lnear least squares fttng algorthms such as the Levenberg- Marquardt method [1, ] or unconstraned optmzaton methods such as smplex search of Nelder and Mead [3] proved to be unstable wth nosy data and dependent on ntal parameters provded by the user [4]. In ths paper we use a summaton of Gaussan functons for fttng lne profles and an evoluton strategy as optmzaton method to fnd the parameters of the model. ES are stable n the presence of nosy data, and can be appled to non-lnear systems [5]. Another advantage of ths technque s that ES does not requre the ntalzaton of search parameters n contrast to other optmzaton algorthms. The remander of ths paper s organzed as follows: Secton gves a bref overvew of evoluton strateges. Secton 3 presents a bref descrpton of stellar spectra. Secton 4 presents the method used n our experments, Secton 5 presents expermental results and dscusson and Secton 6 presents conclusons and outlnes drectons for future work.. EVOLUTION STRATEGIES Evoluton Strateges (ES) are a class of probablstc search algorthms loosely based on bologcal evoluton. They work on a populaton of ndvduals, where each of them represents a search pont n the space of potental solutons to a gven problem. In evoluton strateges a vector of real numbers represents an ndvdual. Ths s a good representaton when the problem at hand deals wth contnuous parameters. Each ndvdual s formed by a vector of elements called object varables x, and each varable has assocated to t a standard devaton called strategy parameter σ, as shown n fgure 1. x 1 x x 3 x n σ 1 σ σ 3. σ n Fgure 1. Representaton of ndvdual Intally, the algorthm randomly generates a populaton of ndvduals; subsequently ths populaton s updated by means of randomzed processes of recombnaton, mutaton, and selecton. Each ndvdual s evaluated accordng to a ftness functon that depends on the problem to be solved. The selecton process favors the ft ndvduals from the current populaton to reproduce n the next generaton. In evoluton strateges, ths process s completely determnstc. In (µ+λ)-selecton, the µ best ndvduals from the unon of µ parents and λ offsprng are selected to form the next parent generaton, and n (µ, λ)-selecton ths operator selects the µ best ndvduals from the λ offsprng only; for ths µ< λ s requred. The recombnaton process allows to combne nformaton from dfferent members of a populaton, creatng offsprng from them. In [5] Bäck shows a varety of recombnaton mechansms used n evoluton strateges. Typcal examples of them are: dscrete recombnaton, whch s a sexual operaton that creates two offsprng vectors from two parent vectors copyng selected elements from each parent; and ntermedate recombnaton, whch s commonly used as an arthmetc average wth some

2 varants. These operators can be used n sexual or panmctc form. In the sexual form, every element of an offsprng s the result of recombnaton between two ndvduals randomly chosen from the parent populaton. In panmctc form, each element of an offsprng may be the result of recombnaton among one ndvdual and several other ndvduals randomly chosen from the parent populaton. Mutaton s an asexual operator that generates random changes to an ndvdual and often provdes new relevant nformaton. The mutaton operator s appled ndependently to each object varable of an ndvdual. It s carred out as shown n equaton 1. The strategy parameters may be mutated usng a multplcatve, logarthmc normally dstrbuted process as shown n equaton. x = x + σ N(,1) (1) σ = σ exp( N (,1) + N (,1)) () Where N (,1) s a normally dstrbuted random varable havng an expectaton of zero and a standard devaton of one, N (,1) ndcates that the random varable s sampled anew every tme the ndex changes. Rechenberg proposed a determnstc adjustment of strategy parameters durng evoluton, called the 1/5- success rule [6], whch reflects that, on average, one out of fve mutatons should cause an mprovement n the objectve functon values to acheve best convergence rates. If more than 1/5 of the mutatons are successful, σ s ncreased, otherwse t s decreased. narrow dscontnutes supermposed. These dscontnutes are called absorpton lnes when the total flux s less than the contnuum or emsson lnes when the total flux s greater than the contnuum, and are caused by the presence of certan atoms n the star's atmosphere. Each absorpton lne appears as a valley, whle each emsson lne appears as a peak n a stellar spectrum. The depth or heght of each lne ndcates ts strength. The wdth of each lne ndcates the range of wavelengths. Fnally, each lne has a specfc shape. All of these characterstcs convey nformaton about the star. An expert astronomer can analyze these lnes and estmate wth good accuracy several of the most mportant propertes of the star. In ths paper we used the spectrum of a star of type A8V obtaned from a dgtal optcal stellar lbrary [7]. It conssts of fluxes coverng 351. to 893. nm n wavelength wth a resoluton of.5 nm. 4. THE METHODS We can defne a Gaussan functon wth three parameters: a center pont (λ ο ), a varance (σ) and an ampltude (A), as shown n fgure 3. Ths functon s defned by equaton 3. The varance determnes the shape of the Gaussan functon. Two or more Gaussan functons can be combned to ft data as shown n fgure 4. Ampltude A λ ο λ σ S t Fgure 3. Standard Gaussan Functon ( x λ ) G ( x) = Aexp (3) σ A Fgure. Sample stellar spectrum 3. THE STELLAR SPECTRA Astronomers can determne the chemcal composton and physcal nature of a star by analyzng ts spectrum, whch s a plot of flux densty as a functon of wavelength, as shown n fgure. Stellar spectra consst of a contnuous spectrum (background or contnuum), wth λ A λ Fgure 4. Combnaton of Gaussan Functons

3 We can represent a stellar spectrum wth a functon defnng the background and a set of Gaussan functons representng the stellar lnes. The frst functon may be a quadratc equaton, another Gaussan functon coverng the entre spectrum or the Planck functon, whch approxmates a blackbody emsson [8]. Equaton 4 shows the functon used for ths purpose. N P( λ ) = F( λ) + = G 1 ( λ) (4) F(λ) represents the background or contnuum, and G (λ) s are the Gaussan functons supermposed on t. These functons must be spread over the range of the evaluaton data and each one must be assgned wth an approprate varance and central pont n order to cause the requred overlap among Gaussan functons. F(λ) can have one of the followng forms: quadratc aλ + bλ + c1 equaton ( λ λ ) Gaussan Aexp F( λ) = w Functon hc 1 Planck Cy 5 λ hc Functon exp 1 λkt where h = Planck Constant k = Boltzmann Constant c = Speed of Lght T = Temperature of the Star Cy = Adjustment constante An evoluton strategy s used to fnd the free parameters of P(λ). The representaton of the object varables nto an ndvdual for ths problem has the form shown n fgure 5. n g Free Parameters of F(λ) λ o1 A 1 W 1... λ on A N W M Fgure 5. Representaton of the object varables nto an Indvdual Where n g corresponds to the number of Gaussan functons. It means that ndvduals may have dfferent szes. The free parameters of F(λ) are a, b, and c 1 for the quadratc equaton, Cy and T for Planck s functon, and λ o, A, w for the Gaussan functon. Also, the free parameters of G (λ) are λ o, A, W. In our evoluton strategy mplementaton (hereafter GaES), we added an operator called ntellgent-mutaton. Ths procedure checks for the poston of the greatest dfference between data generated by the fttng model and the orgnal data, and adds a Gaussan functon n ths poston wth the same ampltude as the error sze and a random varance value, as shown n fgure 6. Also, we added another procedure called elmnatonmutaton, whch carres out the opposte functon. It suppresses the Gaussan functon that produces the smallest reducton n the error. Ampltude Ampltude Ampltude Max Error Orgnal Data - Predcted o Corrected + Added Functon X Fgure 6 Added functon by Intellgent Mutaton ( P( λ ) f ( λ )) nd = 1 _ = + α n (5) g nd Ftness GaES The ftness functon s shown n equaton 5, where n d s the number of ponts n the dgtal spectrum, P(λ ) s a pont generated by the model, f(λ ) s a pont n the orgnal data, n g s the number of Gaussan functons, and α s a constant. The frst term of the ftness functon corresponds to the root mean squared error and the second term s a penalty, whch favors ndvduals wth fewer Gaussan functons. 5. EXPERIMENTAL RESULTS In ths secton we detal the result of applyng GaES to a smulated spectrum and a real dgtal optcal spectrum obtaned from Davs database [7]. We generate a smulated spectrum P(λ) wth 5 spectral lnes G (λ) s wth parameters [λ o, A, W ]= [396.8x x x x x x x x x x1-7 ] and usng the Planck s functon wth a temperature of 6,5K for contnuum approxmaton. GaES-planck found the temperature and smulated spectral lnes wth an rms error of 4.38x1-8 as shown n fgure 7. The GaES ran for 5 generatons, but good ftness values were attaned after about 15 generatons, as shown n fgure 8.

4 Fgure 7. Smulated Spectrum and GaES Planck ft model. Fgure 9. Dgtal spectrum of a star of type A8V and GaES-Quadratc ft model. The evoluton strategy ran for 5 generatons. We used n all the experments a populaton of 5 ndvduals, mplyng 5, evaluatons of the objectve functon. Also, we used two dfferent recombnaton methods: dscrete recombnaton on 1% of the parent populaton and ntermedate recombnaton on another 1% of the parent populaton. Mutaton s the most mportant operator n ES and we used t on 68% of the parent populaton. We used elmnaton-mutaton on 1% and ntellgentmutaton on % of the parent populaton. Fgure 8. Ftnees of GaES-Planck for a smulated spectrum. Real data possess nherent nose, and ths s a problem for tradtonal technques, but ES have shown to be nsenstve to nosy data. We appled our method to the analyss of a real dgtal optcal spectrum of a star of type A8V n the 38-5 nm wavelength range. In the frst experment we used the GaES-Quadratc model, where t acheved to an rms error of.74, fndng 8 Gaussan functons as shown n fgure 9. In the second experment, we used the GaES-Gaussan model, where t acheved an rms error of.7 and 8 Gaussan functons as shown n fgure 1. In the last experment we used the GaES-Planck model achevng an rms error of.199 and fndng 8 Gaussan functons as shown n fgure 11. Fgure 1. Dgtal spectrum of a star of type A8V and GaES Gaussan ft model. The behavor of each GaES s shown n fgures 1, 13 and 14. As can be seen n the fgures, good ftness values can be attaned quckly after a few hundred teratons; after that, progress becomes slower. Ths behavor s typcal of stochastc optmzaton algorthms.

5 Fgure 11. Dgtal spectrum of a star of type A8V and GaES-Planck ft model. Fgure 14. Ftness functon of GaES-Planck. For comparson we made an experment usng the Nelder-Mead Smplex drect search from the MatLab Optmzaton Toolbox gvng the ntal parameters as shown n fgure 15. After 46,9 teratons and 5, evaluatons of objectve functon, we acheved an rms error of.379 as shown n fgure 16. We use the same ftness functon from ES to evaluate the objectve functon and ts behavor s shown n fgure 17. The run tme was smlar for both algorthms. One dsadvantage of ths method s the dependence on ntal search parameters, but a combnaton of ths algorthm wth the ES to fnd the ntal search parameters may be an alternate soluton to reduce the search tme and to acheve a better accuracy. Fgure 1. Ftness functon of GaES-Quadratc. Fgure 15. Dgtal spectrum of a star of type A8V and Smplex-Planck ntal ft model. Fgure 13. Ftness functon of GaES-Gaussan.

6 7. REFERENCES [1] Levenberg, K., A Method for the Soluton of Certan Problems n Last Squares, Quart. Appl. Math. Vol., pp , [] Marquardt, D., An Algorthm for Least Squares Estmaton of Nonlnear Parameters, SIAM J. Appl. Math. Vol. 11, pp , [3] Nelder, J.A. and R. Mead, A Smplex Method for Functon Mnmzaton, Computer J., Vol.7, pp , Fgure 16. Dgtal spectrum of a star of type A8V and Smplex-Planck ft model. [4] S. W. McIntosh, D. A. Dver, P. G. Judge, P. Charbonneau, J. Ireland, and J. C. Brown. Spectral decomposton by genetc forward modellng, Astronomy & Astrophyscs Supplement Seres, 13, 1998, [5] Tomas Bäck, Evolutonary Algorthms n Theory and Practce, Oxford Unversty Press, [6] I. Rechenberg, Evolutonsstratege: Optmerung technscher Systeme nach Prnzpen der bologschen Evoluton, (Stuttgart: Frommann-Holzboog Verlag), [7] D. R. Slva, A New Lbrary of Stellar Optcal Spectra, Astrophyscal Journal Supplement Seres Vol. 81, p , 199 Fgure 17. Ftness functon of Smplex Method. [8] H. Karttunen, P. Kröger, H. Oja, M. Poutanen, K. J. Donner. Fundamental Astronomy: Second Enlarged Edton. Sprnger Verlag, CONCLUSIONS In ths paper we have presented an approach to ft a model of Gaussan functons to dgtal spectra n order to fnd spectral lnes and background usng ES. Our expermental results show that applyng ths method to a dgtal stellar spectrum provdes the followng advantages: It does not requre user nput regardng search parameter ntalzaton nor the number of Gaussan functons. It s stable n the presence of nosy data. It s less senstve to local mnma than gradent-based methods. Future work wll extend the expermental results to more spectra and compare the results wth theoretcal models. Also, we wll attempt to combne ES wth standard optmzaton methods to reduce the search tme.