A Hybrid Algorithm Based on Evolution Strategies and Instance-Based Learning, Used in Two-dimensional Fitting of Brightness Profiles in Galaxy Images

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1 A Hybrid Algorithm Based on Evolution Strategies and Instance-Based Learning, Used in Two-dimensional Fitting of Brightness Profiles in Galaxy Images Juan Carlos Gomez 1 and Olac Fuentes 2 1 INAOE, Computer Science Department, Luis Enrique Erro No. 1, Tonantzintla, Puebla 72000, Mexico jcgc@inaoep.mx 2 University of Texas at El Paso, Computer Science Department, 500 West University Avenue, El Paso 79968, Texas, USA ofuentes@utep.edu Abstract. The hybridization of optimization techniques can exploit the strengths of different approaches and avoid their weaknesses. In this work we present a hybrid optimization algorithm based on the combination of Evolution Strategies (ES) and Locally Weighted Linear Regression (LWLR). In this hybrid a local algorithm (LWLR) proposes a new solution that is used by a global algorithm (ES) to produce new better solutions. This new hybrid is applied in solving an interesting and difficult problem in astronomy, the two-dimensional fitting of brightness profiles in galaxy images. The use of standardized fitting functions is arguably the most powerful method for measuring the large-scale features (e.g. brightness distribution) and structure of galaxies, specifying parameters that can provide insight into the formation and evolution of galaxies. Here we employ the hybrid algorithm ES+LWLR to find models that describe the bidimensional brightness profiles for a set of optical galactic images. Models are created using two functions: de Vaucoleurs and exponential, which produce models that are expressed as sets of concentric generalized ellipses that represent the brightness profiles of the images. The problem can be seen as an optimization problem because we need to minimize the difference between the flux from the model and the flux from the original optical image, following a normalized Euclidean distance. We solved this optimization problem using our hybrid algorithm ES+LWLR. We have obtained results for a set of 100 galaxies, showing that hybrid algorithm is very well suited to solve this problem. 1 Introduction Galaxies encompass an enormous set of phenomena in the universe, from star formation to cosmology subjects. Thus, study of galaxies is essential to understand many basic questions about the cosmos. Also, there is a huge amount of

2 2 astronomical data in images and spectra in surveys (SDSS, 2MASS, etc.) obtained from modern observatories, and it is important to automatically analyze such information in order to extract important physical knowledge. A very useful way to quantify galaxies and extract knowledge from data is to fit images or spectra with parametric functions [9][11]. The use of standardized fitting functions is arguably the most powerful method for measuring the large-scale features (e.g. brightness profiles) and structure of galaxies (e.g. morphologies), specifying parameters that can provide insight into the formation and evolution of galaxies, since the functions yield a variety of parameters that can be easily compared with the results of theoretical models [4]. Galaxies are composed of distinct elements: stars, gas, dust, planets and dark matter. Old stars are normally present in the central part of a galaxy (also called bulge), while young star, gas and dust are usually in the outer parts (called disk) and dark matter is normally surrounded the galaxy (called halo). Each element contributes in a different way to the light that one galaxy emits; stars producing the light and gas, dust and dark matter dispersing or diffracting it. Galaxy brightness profile describes how this light is distributed over the surface of a galaxy [7]. Thus, studying the brightness profile will lead to understand many subjects about the formation, composition and evolution of galaxies [11]. For example, elliptical galaxies are normally composed only by a bulge and a dark matter halo, which means that ellipticals are old, because they only contains old stars in the central part and they produces a very intense brightness in this part. On the other hand spiral galaxies are usually composed by a bulge, a disk and a dark matter halo, which means spirals are younger than ellipticals because spirals still contain young stars, and gas and dust to produce new stars, where these elements are normally present in the spiral arms. In this case, galaxies present a brilliant central part and a less brilliant disc (with spiral arms) surrounding the bulge. Thus, fitting of galaxy brightness profiles provides a reasonably detailed description of the radial light distribution with a small number of parameters. Nevertheless, ideally fitting functions would be based upon the physics of the formation and evolutionary processes in galaxies. Unfortunately, these processes are neither simple nor well understood, so the most commonly used functions are derived empirically. Here we propose a machine learning [8] hybrid algorithm to automatically find models for galaxy brightness profiles; exploring the search space of possible solutions in order to find the best set of parameters that produces a model able to describe the brightness distribution in a galaxy. Hybridization is referred to a merge or mixture among two or more algorithms, implemented or developed trying to exploit the advantages and avoid the weakness of each particular algorithm. Using such schema, for example in a non-simple search space in an optimization problem, we might initially employ a global algorithm to identify regions of high potential, and then switch, using appropriate switching metrics, to local techniques to rapidly converge on the local minimum. Through hybridization, the optimization strategy can be fitted

3 3 to suit the specific characteristics of a problem, thereby enhancing the overall robustness and efficiency of the optimization process. Here we use a different way to perform hybridization: a local algorithm produces one proposed solution that is evaluated and used to produce new solutions by a global algorithm; in this manner an exchange of possible solutions between algorithms occurs. Hybridization is done in each iteration of the global algorithm Evolution Strategies (ES) [10], adding a new solution approximated by the instance-based algorithm Locally Weighted Linear Regression (LWLR) [1]to the set of solutions in ES. We have obtained fitting for a set of 100 galaxy images, from spiral and elliptical galaxies, using hybrid algorithm, showing that ES+LWLR is a well suited method to solve this problem. The rest of the paper is structured as follows: in Section 2 a brief description of theory for brightness profile and description of the problem is presented, the hybrid algorithm is shown and explained in Section 3, Section 4 includes the general description of the optimization process, results are presented in Section 5 and Section 6 includes conclusions and future work. 2 Brightness Profile Surface brightness in a galaxy is literally defined as how much light emits the galaxy [7], and luminosity is defined as the total energy received by unit of area by unit of time. Then, the surface brightness of an astronomical source is the ratio of the source s luminosity F and the solid angle (Ω) subtended by the source. B = F Ω (1) The surface brightness distribution in elliptical galaxies depends essentially only on the distance from the centre and the orientation of the major and minor axis. If we consider that elliptical galaxies are only composed by a bulge, and if r is the radius along with the major axis, the surface brightness I(r) is well described by de Vaucoleurs law r 1/4 [6]: [ I b = I e exp 3.3 [( r 1/4 r e ) ]] 1 where r is the distance from the galactic center, r e is the mean ratio of the galaxy brightness (the radius where half of the total brightness lies), and I e is the surface brightness for r = r e. Although de Vaucoleurs law is a purely empirical relation, it still gives a remarkably good representation of observed light distribution in bulges. However, in the outer regions of elliptical galaxies changes in light distribution may often occur. Different galaxies differ widely in this respect indicating that the structure of ellipticals is not as simple as it might appear. (2)

4 4 Fig. 1. Example of an observed galaxy image and its modelled brightness profile Spiral galaxies, and according with observation some ellipticals, are also composed by a disc. The surface brightness profile for a disc galaxy has an exponential distribution: ) I d = I 0 exp ( rr0 (3) where I 0 is the central surface brightness and r 0 is the radial scale length. Finally, surface brightness distribution in elliptical and spiral galaxies can be described in a more general way as the sum of equations 2 and 3, which is an approximation of the profile using concentric ellipses [7]. I = I b + I d (4) In fact, it is not expected that equations 2 and 3 fit all the profile measured in the radial range of the galaxy, because sometimes sky subtraction errors in external regions of galaxy can distort the profile. Also the fitting process does not allow the presence of structures, it fits across arms and bars as if they were noise in the data, since models are based only in concentric ellipses. An example of a galaxy image and its corresponding generated brightness profile using the previous equations is shown in Figure Fitting In order to fit a galaxy image, it is necessary to define a model of its brightness profile that matches the brightness distribution of the original image.

5 5 Thus, let q = [r e, I e, I 0, r 0, i 1, i 2 ] be a vector of brightness parameters, a model for a brightness profile is constructed as follows: m = I b (q) + I d (q) (5) where I b (q) and I d (q) are the equations 2 and 3 applied with the parameters in q. This model produces an artificial image m of size n m that represent certain brightness distribution. Then, the final goal in this fitting task is to approximate, efficiently and automatically, the best combination for the following parameters: r e, mean ratio of the galaxy brightness; I e, surface brightness in r = r e ; I 0, central surface brightness; r 0, radial scale length and two angels i 1 and i 2 which are the rotation angles in x and z axis, that produce a model that matches the brightness distribution of the observed image. 3 Optimization The process starts with an observed image o, which first is resized to a pixel size. This process is done to simplify the fitting task and it is necessary to standardize all the data (the observed and the simulated). Thus, let o be the observed image variable of size , let m be the simulated image with the same dimensionality as o. The goal of the optimization process is to obtain a model m that maximizes the following function: f(m) = (m i,j o i,j ) max 2 (6) i=1 j=1 where max is the maximum difference that can exist between images. This is the fitness function used by our hybrid algorithm. The fitness function represents the similarity between both images, and its values range is [0-1], with 1 as a perfect match and 0 as totally different images. At the end, the simulated image that maximizes such equation is the one that was produced by the set of brightness parameters we were looking for. 3.1 Hybrid Algorithm Our hybrid algorithm, that we called ES+LWLR, is based in the main idea of using the individuals in each generation of ES as the training set for the learning algorithm LWLR [1]. Such algorithm will be used to predict a new individual, which is expected to be closer to the global solution. LWLR uses implicit knowledge in the ES population about the target function to predict a new individual that is potentially better than the ones in ES. LWLR creates local linear models of the target function around a query point h q (original image), and local models achieve an approximation finer of the target function than those based on global models, reaching a more accurate prediction.

6 6 Literally LWLR is exploiting the current population configuration for predicting directly a new solution. In this work we implemented a modified version of (µ + λ) ES [2][3] that includes some changes to the canonical version. We create µ = 7 parent individuals and λ = 14 children individuals, we use discrete recombination and traditional mutation, but we also include a new way to create offspring with the average operator and a dynamical mutation [5] for strategy parameter vectors based on a simple, but effective and easy to understand, multiplication by constant factors. We employed LWLR in the following way: given a query point h q, to predict its output parameters y q, we find the k closest examples in the training, and assign to each of them a weight given by the inverse of its distance to the query point: w i = 1 h q h i i = 1,..., k where h i is the i-st closest example. This measure is called relevance. Let W, the weight matrix, be a diagonal matrix with entries w 1,..., w k. Let H be a matrix whose rows are the vectors h 1,..., h k, the input parameters of the examples in the training set that are closest to h q, with the addition of a 1 in the last column. Let Y be a matrix whose rows are the vectors y 1,..., y k, the output parameters of these examples. Then the weighted training data are given by Z = W H and the weighted target function is V = W Y. Then we use the estimator for the target function y q = h T q Z V, where Z is the pseudoinverse of Z. Merging both ideas of ES and LWLR we obtain the ES+LWLR algorithm, where we have: first, the initial population of ES x i i = 1,..., µ, where each x i = [r e,i, I e,i, I 0,i, r 0,i, i 1,i, i 2,i ], and their corresponding strategy parameters vectors σ i i = 1,..., µ are formed by randomly generated values. Then, each x i is passed to a module that following equation 5 creates the simulated galaxy image m i. Afterwards, each m i is evaluated using the fitness function 6. The next step consists of an iterative process: if some model has obtained a good match with the original image we stop the process, otherwise we create λ new individuals x j j = 1,..., λ using recombination, mutation and average operators. Then we create their corresponding models m j and evaluate them using the fitness function. The next step consist in the hybridization, here we pass the (µ + λ) population, the m k k = 1,..., µ + λ models and the original image o to the LWLR module, where LWLR takes the 7 closest models to the original images an predict the output vector of parameters y q. This vector is evaluated using the fitness function and returned to ES, replacing the least fit individual of the (µ+λ) population. Afterward we select the best µ individuals from the total population and return to compare the new models with the observed image. The pseudocode of the ES+LWLR algorithm for this problem is the following: 1. Create µ = 7 parent vectors x i i = 1,..., µ, where each vector contains 6 parameters x i = [r e,i, I e,i, I 0,i, r 0,i, i 1,i, i 2,i ], and their corresponding strategy parameter vectors σ i i = 1,..., µ with the same dimensionality as x i. Each

7 7 parameter is chosen through a random process and satisfying the constraints of the problem. 2. For each vector x i produce a simulated galaxy image m i 3. For each m i compute the fitness function 256 f(m i ) = max j=1 l=1 (m i,j,l o j,l ) 2 4. If some model m i fits good the observed galaxy brightness profile terminate, otherwise continue next step 5. Create new λ = 14 child individuals in the following way: Create 10% of λ population using discrete recombination, from two parents x a and x b : x k,j = x a,j or x b,j σ k,j = σ a,j or σ b,j Create 10% of λ population using average operator, from two parents x a and x b, and a random number d between 0-1: x k = dx a + (1 d)x b σ k = dσ a + (1 d)σ b Create 80% of λ population using mutation: x k = x a + N(0, σ a ) 6. For each vector x k produce a simulated galaxy image m k k = 1,..., λ 7. For each m k compute the fitness function f(m k ) = max j=1 l=1 (m k,j,l o j,l) 2 8. Merge µ and λ populations to obtain a (µ + λ) population 9. Sort the merged population by fitness function, in descending order 10. Pass entire population x k, its corresponding models m k k = 1,..., µ+λ and original observed image o to LWLR module where: (a) Calculate relevance for all the models w i = 1 o m i i = 1,..., µ + λ (b) Select the k = 7 closest examples to form matrix W

8 8 (c) Transform each simulated image m j j = 1,..., k in a row for matrix H (d) Transform o in a row (e) Transform each individual x j j = 1,..., k in a row for matrix Y (f) Do Z = W H (g) Do V = W Y (h) Do y = o T Z V (i) Return y 11. Do x µ = y 12. Select the best µ individuals from the sorted population 13. Mutate strategy parameter vectors: { σ 0.3σi if x i = i is a child 3.3σ i if x i is a parent where these values have been selected experimentally. 14. If some model m i fits good the observed galaxy brightness profile or the maximum number of generations is reached terminate, otherwise return to 5 4 Results In order to evaluate the performance of the hybrid algorithm ES+LWLR we have obtained results for a set of 100 galaxies. Also, we made a comparison with ES algorithm alone to have a best scenario about the improvements reached with the hybrid algorithm. The fitting of one galaxy image takes on average 12 minutes on a PC with a PIV 3 Ghz processor and 512 MB of RAM, using MatLab. Computational time can be improved if we employ a compiler language such C, C++ or FORTRAN rather than an interpreter one. The fitness function describes the fitness of each individual by measuring the normalized Euclidean distance with respect to the observed image o, the function is ranged from 0 to 1, with 1 as a perfect match and 0 as totally different images. We say a model m matches perfectly the observed image when f(m ) = 0, but

9 9 Galaxy Original Image Best Model Difference Algorithm F.E. f(m) NGC2768 ES ES+LWLR NGC2903 ES ES+LWLR NGC3031 ES ES+LWLR NGC3344 ES ES+LWLR NGC4564 ES ES+LWLR Table 1. Comparison between ES and ES+LWLR for a set of 5 galaxy images (F.E. means Function Evaluations).

10 10 Algorithm Function Evaluations f(m) Standard Success (average) (average) deviation (%) ES ES+LWLR Table 2. Comparison between ES and ES+LWLR for a set of 100 galaxy images since we are matching models with observational data, we do not expect to reach the real maximum. Rather, we are interested in obtaining good approximations to the observed flux distribution. After a set of experiments we determined that a value of 0.96 for the fitness function is good enough and the model can be acceptable, a bigger value for this threshold would lead to a better fit, albeit at an increase in computation time. We illustrate the fitting of brightness profiles with a sample of 5 examples, presented in Table 1 that shows 5 galaxy images and their corresponding models approximated by ES+LWLR or ES. The first column indicates the name of the galaxy, the second, third and fourth columns show the original, the model and the difference images respectively, the fifth one indicates the algorithm (ES or ES+LWLR), the sixth presents the total number of function calls needed by the algorithm to reach convergence, and the last one shows the value for the cost function for the maximum found for each algorithm. We can observe that, as stated before, structure features (such as bridges, tails or spiral arms) of the galaxies are not fitted by the models, but the general distribution of brightness and angles are approximated very closely. Better fit is obtained in the central part, because basically in all the galaxies the centre is formed by a bulge, which can be fitted very well using the de Vaucoleurs law; outer parts of galaxies are less fitted because the models are known not to be as accurate to describe details about structures. In Table 2 we present the summarized results for a sample of 100 galaxy images, comparing behaviors of ES and ES+LWLR algorithms. The employed set in this case was formed using 85 spiral galaxies and 15 elliptical galaxies. We can see in the table that both algorithms have similar behaviors in terms of accuracy and average value for fitness function, because both of them present very similar values: 85% and for ES and 86% and for ES+LWLR. Nevertheless, we also observe that ES+LWLR has a better performance since it presents a less number of function evaluations than ES. 5 Conclusions In this work we have solved the problem of two-dimensional fitting of brightness profiles for spiral and elliptical galaxies using a hybrid algorithm, based on Evolution Strategies and Locally Weighted Linear Regression, an instance based method, this new algorithm is called ES+LWLR. The hybrid algorithm achieved very good results, because was able to find an acceptable solution for almost all

11 11 the cases in the galaxy images set. The ES+LWLR algorithm shows that knowledge generated by ES, in form of proposed solutions or individuals within a population can be employed to breed a new solution or individual that could be potentially better than those in the present population. This improved solution inserted in the set of current individuals helps to improve the global fitness of the population. Literally, LWLR is exploiting current population configuration for predicting directly a new solution. Next step with this algorithm is a version where more than one individual is produced by LWLR, taking various training sets extracted from the current population in a random way. 6 Acknowledgments This work was partially supported by CONACYT, Mexico, under grants C /A-1 and References [1] Atkenson, C.G., Moore, A.W., Schaal, S.:Locally Weighted Learning. Artificial Intelligence Review 11 (1997) [2] Back, T., Hammel, U., Schwefel, H.P.: Evolutionary Computation: Comments on the History and Current State. IEEE Transactions on Evolutionary Computation 1 (1997) 3 17 [3] Back, T., Fogel, D., Michalewicz, Z.: Handbook of Evolutionary Computation. Oxford University Press. London (1997) [4] Baggett, W.E., Baggett, S.M., Anderson, K.S.J.: Bulge-Disk Decomposition of 659 Spiral and Lenticular Galaxy Brightness Profiles. Astronomical Journal 116 (1998) [5] Beyer, H.G.: Toward a Theory of Evolution Strategies: Self-Adaptation. Evolutionary Computation 3 (1996) [6] de Vaucouleurs, G.: Recherches sur les Nebuleuses Extraglactiques. Ann. d Astrophysics 11 (1948) 247 [7] Karttunen, H., Kroger, P., Oja, H., Poutanen, M., Donner, K.J.: Fundamental Astronomy. Springer-Verlag. Berlin, Heidelberg (2000) [8] Mitchell, M.: Machine Learning. McGraw-Hill. New York (1997) [9] Peng, C.Y., Ho, L.C., Impey, C.D., Rix, H.W.: Detailed Structural Decomposition of Galaxy Images. Astrophysical Journal 124 (2002) [10] Rechenberg, I.: Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. Frommann-Holzboog, Stuttgart (1973) [11] Wadadekar, Y., Robbason, B., Kembhavi, A.: Two-dimensional Galaxy Image Decomposition. Astronomical Journal 117 (1999)