Nonlinear Modeling Method Based on RBF Neural Network Trained by AFSA with Adaptive Adjustment

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1 Advances n Engneerng Research (AER), volue 48 3rd Workshop on Advanced Research and Technology n Industry Applcatons (WARTIA 27) Nonlnear Modelng Method Based on RBF Neural Network Traned by AFSA wth Adaptve Adustent u-sheng Gan,a, Zh-bn Chen 2,b,Mng-gong Wu 2,c 2 Jng College, an, Shaanx, 723, Chna Ar Traffc Control and Navgaton College, Ar Force Engneerng Unversty, an, Shaanx, 75, Chna a ganxusheng23@63.co, b zhbn_chen@sna.co, c g_w32@63.co Keywords: Radal bass functon; Neural network; Artfcal fsh swar algorth; Nonlnear functon Abstract. To prove the nonlnear odelng capablty of RBF neural network, an Artfcal Fsh Swar Algorth (AFSA) tranng algorth wth an adaptve echans s proposed. In the tranng algorth, the search step sze and vsble doan of AFSA algorth can be adusted dynacally accordng to the convergence characterstcs of artfcal fsh swar, and then the proved AFSA algorth s used to optze the paraeters of RBF neural network. The exaple shows that, the proposed odel s a better approxaton perforance for the nonlnear functon. Introducton Radal Bass Functon (RBF) neural network s an abstracton sulaton of bran neural network. and shows a satsfactory odelng effect wth good convergence. Presently, the algorths of RBF neural network anly have the orthogonal least square, hybrd recursve, RAN, etc.. In recent years, the ntellgent algorths are also present [][2]. In the paper, Artfcal Fsh Swar Algorth (AFSA) s ntroduced to tran RBF network for the proveent of perforance. The experent shows the feasblty and effcency of the algorths. Adaptve artfcal fsh swar algorth AFSA frst proposed by L aole et al n 2, s a novel swar ntellgence optzaton algorth that sulates the foragng behavor of fsh swar. Its basc prncple s to learn fro the foragng, clusterng and pleup behavor of fsh, start fro the botto behavor of the sngle artfcal fsh, through local optzaton of each ndvdual n fsh swar, reaches the purpose that the global optu value eerges n the swar. Once the algorth appears, the algorth has pay attenton, and puts forward soe proved AFSA algorth, s wdely used n pattern recognton, optzng paraeters n the feld [3][4]. The foragng behavor lad the foundaton of the convergence of the algorth; the clusterng behavor can prove the stablty and global perforance of the algorth convergence; the pleup behavor can enhances speed and global perforance of the algorth convergence. The evaluaton f ts behavor also provdes the guarantee for the speed and stablty of the algorth convergence. In general, the value range of each paraeter n the algorth was very tolerant wthout the requreent of the ntal value of the algorth. In vew of the artfcal fsh odel descrbed above and ts behavor, each artfcal fsh explore the envronents at present (ncludng: the change of the partners and the obectve functon), so as to choose a behavor, and ultately, the artfcal fshes gather around several local extreu. Norally, for solvng the proble of axu proble, the artfcal fsh wth larger food concentraton value s around the extree value range that the target value s larger, whch helps to Copyrght 27, the Authors. Publshed by Atlants Press. Ths s an open access artcle under the CC BY-NC lcense ( 34

2 Advances n Engneerng Research (AER), volue 48 acqure the global extreu range; and generally ore artfcal fshes gather around the extree value range that the target value s larger, whch helps to udge and get the global extreu. In foragng behavor of artfcal fsh, the ndvdual always try to ove toward a better drecton, whch lad the foundaton of the convergence of the algorth. The artfcal fsh randoly nspect the state of a pont n the range of ts vson. If t s better than the current state, ove forward one step along the drecton of the state to reach the state ; f the state s not better than the state, t contnues to randoly nspect the state wthn the range of ts vson, f the nuber of nspectons reach a certan nuber of tes try - nuber, and has not yet found a better state, perfor the rando walk. In ths process, the step sze Step and the vsble doan Vsual have a great nfluence on the speed and precson of convergence of fsh swar algorth. Set properly t to fall nto a local nu or not up to the precson. In foragng behavor, f the ndvdual of artfcal fsh dd not fnd a better state, a new state s randoly chosen wthout akng full use of the nforaton obtaned, resultng n the ncreasng coputaton and slow convergence. The change of vsual doan have great nfluence on the algorth convergence. As Vsual s saller, the foragng behavor and rando walk of the artfcal fsh s ore pronent, the local search ablty of the algorth s strong; as Vsual s larger, the pleup behavor and clusterng behavor of the artfcal fsh s ore pronent, the global search ablty and fast convergence of the algorth s strong. In addton, the nfleunce of Step can not be gnored, Step s large, the convergence s fast, but occasonally there are the oscllatons; Step s sall, the convergence s slow, but the accuracy s very hgh. Fro the analyss, ore dffcult the functon s optzed, stronger the the global search ablty s requred. Once the approxately optal poston s located, t s requred to prove the local search ablty and strengthen the fne search. Therefore, the ethod that dynacally adust Vsual and Step can be used to prove the perforance of AFSA algorth. At the early stage of the algorth, the larger Vsual and Step can be used to enhance the global search capablty and convergence speed of the algorth, n order that the artfcal fsh can perfor the coarse search wthn a large range, wth the progresson of the search, Vsual and Step are gradually reduced. At the late stage of the algorth, the global search at the early stage s gradually translated nto the local search, and located and searched near the optal solutons, t can prove the local search ablty and accuracy of optzaton algorth. Vsual and Step can be adusted dynacally by Vsual = Vsual a + Vsualn Step = Step a + Stepn () s a = exp[ 3 ( tt ) ] where Vsual n =., Step n =.2, Step = Vsual 8, t s the current nuber of teratons, T s the axu nuber of teratons. Usually the ntal value Vsual s Z 4 ( Z s the axu value of the search range). The Vsual and Step functon s coposed of 3 stages, at the early stage of the algorth, keep the axu value, and then gradually change fro large to sall, fnally keep the nu value. Ths ethod s a good balance between the global search capablty and local search capablty, speeds up the convergence, proves the precson of the algorth. The change rate s of the value of the functon a fro large to sall s an nteger greater than l, usually the values range s [,3]. Fg. shows the varaton curve of the functon a when T = and s s taken as 3, 6, 2 respectvely. Bref ntroducton of RBF neural network RBF neural network can approxate any nonlnear functon and process the regularty that s dffcult of the analyss n the syste, and has good generalzaton ablty wth fast convergence speed. 342

3 Advances n Engneerng Research (AER), volue 48 At present, t has been successfully appled n the doan such as nonlnear functon approxaton, te seres analyss, data classfcaton, pattern recognton, nforaton processng, age processng, syste odelng, control and fault dagnoss and so on. The convergence of RBF neural network learnng s fast. When one or ore adustable paraeters of the network (the weghts or thresholds) have an effect on any output, such a network can be called the global approxaton network. For each nput, each weght value of the network need to be adusted, so as to leads to the slow learnng speed of global approxaton network. BP neural network s a typcal exaple. If there are only a few connecton weghts affected the output for a certan local regon n the nput space, then the network can be called as the local approxaton network. falar local approxaton network ncludes: RBF network, cerebellar odel artculaton controller (CMAC) network, B splne network etc...8 a.6.4 s=3 s=6 s= t/t Fg. Varaton curve of the functon a Tranng flow of RBF neural network based on adaptve AFSA algorth The realzaton steps of WNN traned by adaptve AFSA follows as [5]:. Intalze the artfcal fsh swar sze M, the ntal poston of each artfcal fsh, the vson doanvsual, the step sze Step, the crowdng degree δ, axu repeated attepts factor try - nuber, the axu nuber Maxter of teratons; 2. set the ntal teratons tes nu =, randoly generate M ndvduals of the artfcal fsh n the feasble doan of control varables, forng the ntal fsh swar, naely produce M group paraeters ncludng: the center, the wdth and connecton weght of the network, and these paraeters can be taken as the rando nuber n the nterval [,]. 3. Calculate the ftness of each artfcal fsh, copare wth the state on bulletn board, f better, then assgn t to a bulletn board; 4. Calculate the vson dean Vsual and step sze Step accordng to the forula (); 5. Update ther poston by each artfcal fsh through the foragng, clusterng, pleup and rando behavor; 6. Judge whether t has reached the axu nuber of teratons, f satsfed, algorth s ternated the and the results are outputed (.e., the value on bulletn board); otherwse nu +, turn to step 3; 7. The optal soluton obtaned by proved AFSA algorth s decoded as the optal paraeters of RBF neural network, whch can get the tranng odel of RBF neural network. Nuercal exaple For a nonlnear pecewse functon.4x+ 4 x 2 f( x) = x< (2).5( ) 2 3.2e x cos(.6x +.6 x) x Radoly take 48 saple ponts n [-, ]. The experental paraeter settngs: the sze of the fsh swar s 4, the axu nuber of teratons Maxter=, Try-nuber=, Vsual =2.95, 343

4 Advances n Engneerng Research (AER), volue 48 Step=.5, δ =.68, accordng to the experence forula, the nuber of the nodes n hdden layer can be taken as 5. After the odel s traned, Fg.2 and Fg.3 gves the approxate effect and error, the tranng convergence curve s shown n Fg.4, "---" denotes the approxaton curve, " " denotes the reference curve. Table shows the coparson of approxaton perforance between two odels. It can be seen fro the experent result, the overall perforance of RBF neural network usng adaptve AFSA algorth has a good proveent fro the accuracy and convergence, ths shows that the global optzaton technology based on adaptve AFSA algorth, can effectvely reduce the syste error caused by the nonlnear characterstcs of the syste, prove the perforance of RBF neural network approxaton. Table Coparson of approxaton result between 2 algorth Model perforance RBFNN Adaptve AFSA-RBFNN Approxaton MSE Tranng te (s) f(x) x Fg.2 Approxate effect.5 f(x) x Fg.3 Approxate error M S E t Fg.4 Convergence curve of RBFNN based on adaptve AFSA Conclusons A RBF neural network odel based on adaptve AFSA algorth s proposed. In proved AFS algorth, the step sze of search and the vsble doan of the artfcal fsh can be adusted n an adaptve for n ters of three stages. Nuercal exaple ndcates that the proposed RBF neural network odel based on adaptve AFSA algorth has a good approxaton perforance for the nonlnear functon. References [] S. Chert, S. A. Bllngs, P. M. Grant. Recursve hybrd algorth for nonlnear syste dentfcaton usng radal bass functon networks. Internatonal Journal of Control, 55(5), (992), 5-7 [2 ] S. Chert, P. M. Crant, C. F. N. Cown. Orthogonal least square algorth for radal bass functon networks. IEEE Transacton on Neural Networks, 2(2), (99),

5 Advances n Engneerng Research (AER), volue 48 [3] C. R. Wang, C. L. Zhou, J. W. Ma. An proved artfcal fsh swar algorth and ts applcaton n feedforward neural networks. Proceedngs of the Fourth Internatonal Conference on Machne Learnng and Cybernetcs, (25), [4]. L. L, Z. J. Shao, J.. Qan. An optzng ethod based on autonoous anats: fsh-swar algorth. Systes Engneerng Theory & Practce, 22(), (22), [5]. L. L, J.. Qan. Studes on artfcal fsh swar optzaton algorth based on decoposton and coordnaton technques. Journal of Crcuts and Syste, 8(), (23),