Optmal Desgn of onlnear Fuzzy Model by Means of Independent Fuzzy Scatter Partton Keon-Jun Park, Hyung-Kl Kang and Yong-Kab Km *, Department of Informaton and Communcaton Engneerng, Wonkwang Unversty, 344-2, Shnyong-dong, Iksan-s, Chonbuk, 570-749 South Korea brd75@wonkwang.ac.kr, ykm@wonkwang.ac.kr Abstract. We ntroduce a optmal desgn of fuzzy model by means of ndependent fuzzy scatter partton to contruct the nonlnear model. The fuzzy rules of fuzzy model are generated by parttonng the nput space n the fuzzy scatter form. The premse parameters of the rules are determned by membersh matr by means of FCM clusterng algorthm that has ndependent fuzzfcaton factors. The consequence part of the rules s represented n the form of polynomal functons. And the optmal process s conducted by PSO to desgn the optmal model. The proposed model s evaluated usng the data wdely used n nonlnear process. Keywords: Fuzzy Scatter Partton, Fuzzy Model, Fuzzy C-Means Clusterng Algorthm, Independent Fuzzfcaton Factor, Partcle Swarm Optmzaton. 1 Introducton Fuzzy sets have been wdely nvestgated and the fuzzy model s a popular computng framework based on the concepts of fuzzy sets, fuzzy f-then rules, and fuzzy reasonng [1]. It has found successful applcatons n a wde varety of felds. Lngustc modelng [2] and fuzzy relaton equaton-based approach [3] were proposed as prmordal dentfcaton methods for fuzzy models. The general class of Sugeno-Takag models [4] gave rse to more sophstcated rule-based systems. In fuzzy modelng, the structure and parameter dentfcaton are usually concerned [5],[6]. The desgners fnd t dffcult to develop adequate fuzzy rules and membersh functons to reflect the essence of the data. The generaton of fuzzy rules has the problem that the number of fuzzy rules eponentally ncreases. In ths paper, we ntroduce a fuzzy model based on ndependent fuzzy scatter partton of nput space. Independent fuzzy partton realzed wth fuzzy c-means (FCM) clusterng [7] help determne the fuzzy rules of fuzzy model. The premse part of the rules s realzed wth the ad of the scatter partton of nput space generated by FCM clusterng algorthms. The number of the partton of nput space equals the number of clusters and the ndvdual parttoned spaces descrbe the rules. The consequence part of the rules s represented by polynomal functons. We also * Correspondng author : ykm@wonkwang.ac.kr AICT 2013, ASTL Vol. 26, pp. 183-188, 2013 SERSC 2013 183
Proceedngs, The 1st Internatonal Conference on Advanced Informaton and Computer Technology optmze the parameters of the fuzzy model usng partcle swarm optmzaton (PSO) algorthm [8]. The proposed model s evaluated wth numercal epermentaton to deal wth the nonlnear process. 2 Independent Fuzzy Scatter Partton-based Fuzzy Model 2.1 Premse Identfcaton The premse part of the FIS s developed by means of the fuzzy c-means clusterng algorthm [7]. Ths algorthm dvdes the nput space by the clusters and each parttoned local space represents the fuzzy rules. Therefore, the number of clusters s equal to the number of rules. Ths algorthm s amed at the formaton of c clusters (relatons) n R n. Consder the set X, whch conssts of data ponts treated as vectors located n some n-dmensonal Eucldean space, that s X={ 1, 2,, }, p R n. In clusterng we assgn patterns p X nto c clusters, whch are represented by ts prototypes v R n. The assgnment to ndvdual clusters s epressed n terms of the partton matr U = [u ] where c u 1 0 u p 1 1, 1 p, 1 c (1). (2) The obectve functon Q gudng the clusterng s epressed as a sum of the dstances of ndvdual data from the prototypes v 1, v 2,, and v c, Q c 1 p 1 u m p v 2. (3) Here denotes the Eucldean dstance; m stands for a fuzzfcaton coeffcent, m >1.0. The resultng partton matr s denoted by U = [u ]. The mnmzaton of Q s realzed through successve teratons by adustng both the prototypes and entres of the partton matr, that s mn Q(U, v 1, v 2,, v c ). The correspondng formulas used n an teratve fashon read as follows. m m v u p u (4) p 1 p 1 u 1 c 1 p p v v 2 m 1. (5) 184
Optmal Desgn of onlnear Fuzzy Model by Means of Independent Fuzzy Scatter Partton The resultng partton matr obtaned from (5) becomes the frng strengths of fuzzy rules. The dentfcaton of the concluson parts of the rules deals wth a selecton of ther structure that s followed by the determnaton of the respectve parameters of the local functons occurrng there. The concluson s epressed as follows. R : If and 1 d 1 d and s F Then y f (,, ). (6) Type 1 (Smplfed Inference): f a 0 Type 2 (Lnear Inference): f a 0 d k 1 a k k Where R s the -th rule, k represents the nput varables, F s a membersh grades (matr) obtaned by usng FCM clusterng algorthm, a s are coeffcent of polynomal functon. The calculatons of the numerc output of the model, based on the actvaton (matchng) levels of the rules there, are carred out n the well-known format y * n 1 w p y n 1 w p n 1 wˆ y. p (7) 3 Optmzaton of the Proposed Model Partcle swarm optmzaton (PSO) [8] was proposed by Kennedy, Eberhart to smulate socal behavor by representng the movement of a brd flock or fsh school. PSO s a computatonal algorthm that optmzes a gven problem by teratvely tryng to mprove canddate solutons (partcles). PSO algorthm optmzes a gven problem by havng a swarm of partcles and movng these partcles around n the search space. Each partcle's movement s affected by ts local best postons and s also guded toward the global best postons n the search-space over the partcle's poston and velocty. And these postons are updated as better postons. The swarm move toward the best solutons. In order to optmze the parameters of the proposed model, we determned the fuzzfcaton factors of the clusters composed of the premse part of the fuzzy rules. 4 Epermental Studes We dscuss numercal eample n order to evaluate the advantages and the effectveness of the proposed approach. The tme seres data (296 nput-output pars) comng from the gas furnace nonlnear process has been ntensvely studed n the prevous lterature [9]. The delayed terms of methane gas flow rate u(t) and carbon dode densty y(t) are used as nput varables organzed n a vector format as [u(t-3), 186
Proceedngs, The 1st Internatonal Conference on Advanced Informaton and Computer Technology u(t-2), u(t-1), y(t-3), y(t-2), y(t-1)]. y(t) s the output varable. The frst part of the data set (consstng of 148 pars) was used for tranng purposes. The remanng part of the seres serves as a testng data set. We consder the MSE as a performance nde. We construct the model for a two-dmensonal system by confgurng 2-nput 1- output system usng u(t-3) and y(t-1) as nputs. And we epermented wth the model usng the parameters outlned n Table 1. Table 1. Intal parameter of PSO Parameters Value Generaton 150 Swarm sze 50 V ma 20% Inerta weght [04. 0.9] Acceleraton constants 2.0 We epermented by optmzng the parameters of the fuzzfcaton factors of the clusters n two methods usng PSO. One tunng method s a method that all clusters have the same fuzzfcaton factor and another tunng method s a method that each cluster has the ndependent fuzzfcaton factor. Table 2 summarzes the performance nde for tranng and testng data by settng the number of clusters and nference type. Here, PI and E_PI stand for the performance nde for the tranng data set and the testng data set, respectvely. Ths table shows that the proposed approach has better results. From Table 2, we selected the best model wth fve rules (clusters) wth lnear nference that ehbts PI = 0.019 and E_PI = 0.285. Table 2. Performance of the proposed model (a) Smplfed Inference Same fuzzfcaton factor Independent fuzzfcaton factors o. of Clusters PI E_PI PI E_PI 2 2.225 2.431 2.286 2.220 3 0.946 1.376 0.904 1.397 4 0.564 1.280 0.598 1.017 5 0.592 1.161 0.555 0.784 (b) Lnear Inference Same fuzzfcaton factor Independent fuzzfcaton factors o. of Clusters PI E_PI PI E_PI 2 0.022 0.335 0.022 0.329 3 0.022 0.337 0.022 0.304 4 0.020 0.335 0.020 0.295 5 0.018 0.315 0.019 0.285 186
Optmal Desgn of onlnear Fuzzy Model by Means of Independent Fuzzy Scatter Partton Fgure 1 shows ndependent fuzzy-parttoned nput spaces usng FCM clusterng algorthm for the selected model. Fgure 2(a) depct that the selected model has three clusters because the other two clusters n the selected model ddn t have the patterns. (a) Local spaces (b) Membersh matr Fg. 1. Independent parttoned nput spaces usng FCM clusterng algorthm. Fgure 2 presents the optmzaton procedure for fve partcles and performance nde for the selected model usng PSO. The model outputs of tranng data and testng data for the selected model are presented n fgure 3. (a) Partcles Fg. 2. Optmzaton process. (b) Performance nde (a) Tranng data Fg. 3. Orgnal and model outputs. (b) Testng data 188
Proceedngs, The 1st Internatonal Conference on Advanced Informaton and Computer Technology 5 Conclusons In ths paper, we ntroduced a fuzzy model based on ndependent fuzzy scatter partton of nput space. The nput spaces of the proposed model were dvded as the scatter form usng FCM clusterng algorthm to generate the rules of the system for nonlnear process. And the proposed fuzzy model was optmzed by PSO to fnd the best values of the fuzzfcaton factors to reflect the characterstcs of the data. By ths method, we could allevate the problem of the curse of dmensonalty and desgn the fuzzy model that s compact and smple. From the results n the prevous secton, we were able to desgn preferred model wth a very small number of rules that has the appromaton abltes and the generalzaton capabltes. Acknowledgements. Ths research was supported by Basc Scence Research Program through the atonal Research Foundaton of Korea (RF) funded by the Mnstry of Educaton (2013R1A1A2011835). References 1. Jang, J.S.R, Mzutan, E., Sun, C.T.: euro-fuzzy and Soft Computng, A Computatonal Approach to Learnng and Machne Intellgence. Prentce Hall, J, (1997) 2. Tong, R.M.: Synthess of fuzzy models for ndustral processes. Int. J Gen Syst. 4, 143--162 (1978) 3. Pedrycz, W.: umercal and applcaton aspects of fuzzy relatonal equatons. Fuzzy Sets Syst. 11, 1--18 (1983) 4. Takag, T., Sugeno, M.: Fuzzy dentfcaton of systems and ts applcatons to modelng and control. IEEE Trans Syst, Cybern. SMC-15(1), 116--132 (1985) 5. Mana, Y., and Benreeb, M.: ew Condton of Stablsaton for Contnuous Takag-Sugeno Fuzzy System based on Fuzzy Lyapunov Functon, Internatonal Journal of Control and Automaton, 4, 3, 51--63, (2011) 6. Park, K. J., Lee, D. Y. and Lee, J. P., Desgn of FCM-based fuzzy neural networks and ts optmzaton for pattern recognton. Communcatons n Computer an Informaton Scence, 261 CCIS, 438--444, (2011) 7. Bezdek, J. C.: Pattern Recognton wth Fuzzy Obectve Functon Algorthms, Plenum Press, ew York, (1981) 8. Kennedy, J., Eberhart, R., Partcle Swarm Optmzaton. Proceedngs of IEEE Internatonal Conference on eural etworks, 4, 1942-1948, (1995) 9. Bo, G.E.P., Jenkns, G.M.: Tme Seres Analyss: Forecastng and Control, 2nd ed., Holden-Day, San Francsco, CA (1976) 188