An Optimization of Granular Network by Evolutionary Methods

Transcription

1 An Optimization of Granlar Networ by Evoltionary Methods YUN-HEE HAN, KEUN-CHANG KWAK* Dept. of Control, Instrmentation, and Robot Engineering Chosn University 375 Seos-dong, Dong-g, Gwangj, Soth Korea Abstract: - In this paper, we present an optimization method of GN (Granlar Networ) based on evoltionary methods sch as PSO (Particle Swarm optimization) and GA (Genetic Algorithm). The GN is constrcted by lingistic model sing CFCM (Contet-based Fzzy C-Means) clstering algorithm while forming a nified conceptal and compting platform of granlar compting. This networ performs relationship between fzzy sets defined in the inpt and otpt space while bilding information granles, and accomplishes ser-centric system. Here, the nmber of clster obtained in each contet and fzzification factor are optimized by PSO and GA. Finally, we compare and analyze the predication performance between the presented networs and other models for coaglant dosing process in a water prification plant. Key-Words: - granlar networ, information granles, contet-based fzzy c-means, particle swarm optimization, genetic algorithm Introdction Granlar compting forms a nified conceptal and compting platform. Yet, it directly benefits form the already eisting and well-established concepts of information granles formed in set theory, fzzy sets, rogh sets and others []. For forming notional and calclative platform of GC (Granlar Compting) related with lingistic model sing fzzy clstering directly, we develop a design methodology of GN (Granlar Networ) [2-3]. This networ indicates relationship among fzzy congregating forming from inpt and otpt space and epressing information granles. The lingistic contet forming this relationship is admitted by a developer of the system, and information granles are constrcted by sing CFCM (Contet-based Fzzy c-means) clstering [4]. However, this networ is difficlt to find the nmber of clster generated by each contet and fzzification factor related to fzzy clstering. Therefore, we perform the optimization of GN sing GA (Genetic algorithm) or PSO (Particle Swarm optimization) which is one of evoltionary comptation methods respectively and compare these performances. GA encodes each point a parameter space into a binary bit string called a chromosome, and each point is associated with a fitness vale that is sally eqal to the objective fnction evalated at the point. [6-8]. PSO is based on social behavior of bird flocing or fish schooling [9-3]. These evoltionary methods have featres that se parallel processing and objective fnction for solving problem. Finally, we demonstrate the speriority and effectiveness of predication performance for coaglant dosing process in a water prification plant. 2 Granlar Networ (GN) In this section, we describe the concept of GN based on lingistic model introdced by Pedrycz [2]. The GN belongs to a category of fzzy modeling sing directly basic idea of fzzy clstering [3]. This clstering techniqe bilds information granles in the form of fzzy sets and develops clsters by preserving the homogeneity of the clstered patterns associated with the inpt and otpt space [4-5]. The nmerical formla of this membership matri U of clstering is compted as follows i = f c j= c c i j 2 ( m ) where m [, ] is a fzzification factor. Here the vale of f prodced the membership degree between 0 and. The f = T( d ) represents a level of involvement of the th data in the assmed contet (fzzy set) of the otpt space. Fzzy set in otpt space is defined by T : D [0,]. This is a niverse of discorse of otpt variable. For this reason, we modify the reqirements of the membership matri as follows () ISSN: ISBN:

2 c N U ( f ) = i [ 0,] i = f and 0< i < N i (2) i= = The lingistic contets to obtain f are generated throgh a series of trianglar membership fnctions along the domain of an otpt variable and a /2 overlap between sccessive fzzy sets. These contets are atomatically prodced by probabilistic distribtion of the otpt space. The center of clster generated from each contet is epressed as follows i = N = m i N = m i Fig. shows the architectre of GN. The premise parameter consists of the clster centers obtained throgh CFCM clstering. The conseqent parameter is composed of lingistic contets prodced in otpt space. The networ otpt Y is compted by fzzy nmber as follows = W t zt (3) Y (4) The ncertainty throgh a bondary vale of pper bond and lower bond can be epressed as follows lower bond : pper bond : i c t y y p = zt wt + t= p + = zt wt+ + t= z w w 0 0 w 3 Optimization by evoltionary methods 3. Optimization by PSO PSO method is one of wide category of swarm intelligence methods for solving the optimization problems. PSO algorithm proposed by Kennedy is performed by social behavior of bird flocing or fish schooling [9]. The character of PSO easily can handle fitness fnction for solving comple problems. Frthermore, it can control relationship between global and local search. Here, each particle adjsts information of location with eperience of them and neighborhood. It can form the answer of optimm in shortly time. As the velocity of particle movement of PSO is only demanded, it is easy to be embodiment and brevity of a theory [9-3]. Basic element of PSO is simply as follows Particle: individal belonged swarm. Swarm: a set of particles. Pbest: particle had located information of optimm. Gbest: particle had located information of optimm in Pbest. Velocity: an operator of PSO, velocity of movement in particles. The velocity is compted as follows v ( t+ ) = w( t) v ( t) + c r ( pbest ( t) ( t)) + c r ( gbest ( t) ( )) (5) 2 2 t where (t) is position of dimension of particle j at time t. w is inertia weight factor. v ( t ) is velocity of particle j at time t. c and c 2 are cognitive and social acceleration factors respectively. r and r2 are random nmbers niformly distribted in the range(0,), pbest ( t ) is best position fond by particle j. gbest ( t ) is best position fond by the whole swarm. ti tc p pi pc z t w t w p z p Contet-based Contets centers Y Fig. Architectre of granlar networ The optimization stage sing PSO algorithm is as follows [Step ] Set the initial parameters of PSO: the size of swarm (30), the nmber of ma iteration (30), a dimension (46), recognition (2), sociality (2), the range of velocity of movement [ v ma, v ma ], the range of clster ( 2< c i 9 ), the range of fzzification Coefficient (.5< m 3). [Step 2] Compte the otpt vales of GN. [Step 3] Compte the fitness fnction from each particle. Here, we se RMSE (Root Mean Sqare Error) ISSN: ISBN:

3 between the networ otpt and actal otpt on training data and test data. F= Q trnrmse + Q chrmse (6) [Step 4] Adjst scaling by F= F min(f) to maintain the positive vales. [Step 5] Compte the localization information of particle as follows ( t) = v ( t) + ( t ) (7) [Step 6] If it satisfied with condition of a conclsion, stop search process, otherwise go to the step3. [Step 3] Compte the fitness fnction from each particle. Here, we se RMSE between the networ otpt and actal otpt on training data and test data. [Step 4] Select two individals from the poplation with probabilities proportional to their fitness vales. [Step 5] Adjst scaling by F= F min(f) to maintain the positive vales. [Step 6] Perform crossover with a probability eqal to the crossover rate. [Step 7] Perform mtation with a probability eqal to the mtation rate. [Step 8] If generation reaches maimm iteration cont, stop search process, otherwise go to the Step 3. The comptational flow chart by GA in the design of GN is shown in Fig. 3. The comptational flow chart by PSO in the design of GN is shown in Fig. 2. Fig. 2 Flow chart by PSO in the design of GN 3.2 Optimization by GA GA has been sed for optimizing the parameters of control system that are comple and difficlt to solve by conventional optimization methods. The GA belongs to a class of poplation-based stochastic search algorithm that is inspired from principles of natral evoltion [6-8]. The optimization stage sing GA algorithm is as follows [Step ] Set the initial parameters of GA: the maimm nmber of generations (30), poplation size (30), probability of crossover (0.97), probability of mtation (0.0), the range of clster ( 2< c i 9 ), the range of fzzification coefficient (.5< m 3). [Step 2] Compte the otpt vales of GN Fig. 3 Flow chart by GA in the design of GN 4 Eperimental Reslts The goal of this comprehensive eperimentation is to demonstrate how the GA and PSO contribte effectively to the optimization design of the GN, respectively. In this eperiment, we apply the proposed GA-based lingistic model and PSO-based GN to coaglant dosing process in a water prification plant, respectively. The field test data of this process to be modeled is obtaeined at the Amsa water prification plant, Seol, Korea, having a water prification capacity of,320,000 ton/day. We se the sccessive 346 samples among jar-test data for one year. The inpt consists of for variables, inclding the trbidity of raw water, temperatre, ph, and alalinity. The otpt variable is PAC (Poli-Alminm Chloride) widely sed as a coaglant. In order to evalate the resltant model, we ISSN: ISBN:

4 divide the data sets into training and checing data sets. Here we choose 73 training sets for model constrction, while the remaining data sets are sed for model validation. To se GA and PSO to find the optimized parameters, we first confine the search domain sch as the nmber of clster from 2 to 9 each contet and fzzification factor from.5 to 3, respectively. We sed 8-bit binary coding for each variable. In the case of PSO, each swarm contains 30 particles. Also, we linearly sed inertia weight factor redced from 0.9 to 0.4. Each generation in GA implementation contains 30 individals. Frthermore, we sed a simple one-point crossover scheme with the crossover rate eqal to 0.97 and niform mtation with the mtation rate eqal to 0.0. We also apply elitism to eep the best two individals across generations. Fig. 4 shows the lingistic contet obtained from otpt variable when the nmber of contet is 8. The fleible contets are prodced by histogram, probability density fnction, and conditional density fnction in order. Fig. 5 Variation of fitness vales by generation of PSO and GA, respectively (a) Fig. 4 Generation of lingistic contets Fig. 5 show the best vales of the objective fnction of both GA and PSO. Since we sed elitism to eep the best two individals at each generation, the best crve is monotonically increasing with respect to generation nmbers. Fig. 6 (a) visalizes the clsters estimated in each contet after performing PSO. Here we finally obtained the best parameters (the nmber of clster: c = [ ], fzzification factor: m =.872) when the nmber of lingistic contet is 9. Fig. 6 (b) visalizes the clsters estimated in each contet after performing GA. Here we finally obtained the best parameters (the nmber of clster: c = [ ], fzzification factor: m =.80) when the nmber of lingistic contet is 8. (b) Fig. 6 Clsters generated by each contet (a)pso (b) GA Fig. 7 shows the performance comparison between the desired and model otpt for training data and checing data, respectively. As shown in Fig. 7, it is obvios that the proposed methods have a good prediction performance. Table lists the comparison reslts of RMSE of training and checing data, respectively. As listed in Table, the reslts obtained by the proposed method yielded better performance than the previos approaches. ISSN: ISBN:

5 two approaches (PSO and GA) is demonstrated throgh remarable generalization and approimation capability. In confronting real-world compting problems, it is advantageos to se several compting techniqes as performed in this paper synergistically rather than eclsively, reslting in constrction of complementary hybrid intelligent system. Acnowledgment (a) This research was financially spported by the MEST (Ministry of Edcation, Science Technology) and KIAT (Korea Institte for the Advancement of Technology) throgh the hman resorce training project for regional innovation. (b) Fig. 7 Approimation and generation capability (a) PSO (b) GA Table Performance comparison of RMSE RMSE RMSE (training) (checing ) Linear regression Neral networs The proposed method RBFN[4] Conclsion GA PSO We have proposed the optimized granlar networ based on the fndamental concept of information granles being regarded as semantically meaningfl conceptal entities that are crcial to the overall framewor of ser-centric modeling. Based on the eperiments for simlation eamples, the effectiveness of the proposed References: [] W. Pedrycz, A. Sowron, and V. Kreinovich, Handboo of Granlar Compting, John Wiley & Sons, 2008 [2] W. Pedrycz and A. V. Vasilaos, Lingistic models and lingistic modeling, IEEE Trans. on Systems, Man, and Cybernetics-Part C, Vol.29, No.6, 999, pp [3] W. Pedrycz and K. C. Kwa, Lingistic models as framewor of ser-centric system modeling, IEEE Trans. on Systems, Man, and Cybernetics-Part A, Vol.36, No.4, 2006, pp [4] W. Pedrycz, Conditional fzzy c-means, Pattern Recognition Letters, Vol.7, 996, pp [5] W. Pedrycz and K. C. Kwa, The development of incremental models, IEEE Trans. on Fzzy Systems, Vol.5, No.3, 2007, pp [6] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 989 [7] S. K. Oh, W. Pedrycz, and H. S. Par, Genetically optimized fzzy polynomial neral networs, IEEE Trans. on Fzzy Systems, Vol.4, No., 2006, pp [8] W. Pedrycz and M. Reformat, Genetically optimized logic models, Fzzy Sets and Systems, Vol.50, 2005, pp [9] J. Kennedy and R. Eberhart, Particle swarm optimization, IEEE Int. Conf. Neral Networs, Vol. Ⅳ, 995, pp [0] M. A. Abido, Optimal design of power system stabilizers sing particle swarm optimization, IEEE Trans, Energy Conversion, Vol.7, No.3, 2002, pp ISSN: ISBN:

6 [] K. F. Parsopolos, On the comptation of all global minimizes throgh particle swarm optimization, IEEE Trans. Evoltionary Comptation, Vol.8, No.3, 2004, pp [2] J. Kennedy, The particle swarm: Social adaptation of nowledge, IEEE Int. Conf. Evoltionary Comptation, 997, pp [3] S. Panda, N. P. Padhy, Comparison of particle swarm optimization and genetic algorithm for TCSC-based controller design, International Jornal of Compter Science and Engineering, Vol., No., 2007, pp [4] W. Pedrycz, Conditional fzzy clstering in the design of radial basis fnction neral networs, IEEE Tans. on Neral Networs, Vol.9, No.4, 999, pp ISSN: ISBN: