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366Vol.36,No.620106ACTAAUTOMATICASINICAJune,2010RBF11(Radialbasisfunction,RBF),.(Sensitivityanalysis,SA),RBF,RBF,;RBF,.,RBF,(Minimalresourceallocationnetworks,MRAN)RBF(Generalizedgrowingandpruningradialbasisfunction,GGAP-RBF).,,RBF,DOI10.3724/SP.J.1004.2010.00865OptimalStructureDesignforRBFNNStructureQIAOJun-Fei1HANHong-Gui1AbstractDuetothefactthattheconventionalradialbasisfunction(RBF)neuralnetworkcannotchangethestructureon-line,anewdynamicstructureRBF(D-RBF)neuralnetworkisdesignedinthispaper.D-RBFisbasedonthesensitivityanalysis(SA)methodtoanalyzetheoutputvaluesofthehiddennodesforthenetworkoutput,thenthehiddennodesintheRBFneuralnetworkcanbeinsertedorpruned.The¯nalstructureofD-RBFisnottoolargeorsmallfortheobjectives,andtheconvergenceofthedynamicprocessisinvestigatedinthispaper.Thegrad-descendmethodfortheparameteradjustingensurestheconvergenceofD-RBFneuralnetwork.ThestructureoftheRBFneuralnetworkisself-organizing,andtheparametersareself-adaptive.Intheend,D-RBFisusedforthenon-linearfunctionsapproximationandthenon-linearsystemsmodelling.TheresultsshowthatthisproposedD-RBFobtainsfavorableself-adaptiveandapproximatingability.Especially,comparisonswiththeminimalresourceallocationnetworks(MRAN)andthegeneralizedgrowingandpruningRBF(GGAP-RBF)revealthattheproposedalgorithmismoree®ectiveingeneralizationand¯nallyneuralnetworkstructure.KeywordsRadialbasisfunction(RBF)neuralnetwork,dynamicdesign,dynamicstructureRBF(D-RBF),chemicaloxygendemand(COD)modelling(Radialbasisfunction,RBF),[1][2][3¡4].,.RBF.,RBF2009-08-272009-10-23ManuscriptreceivedAugust27,2009;acceptedOctober23,2009(863)(2007AA04Z160,2009AA04Z155),(60674066,60873043),(200800050004),(4092010)SupportedbyNationalHighTechnologyResearchandDe-velopmentProgramofChina(863Program)(2007AA04Z160,2009AA04Z155),NationalNaturalScienceFoundationofChina(60674066,60873043),thePh.D.ProgramFoundationofMin-istryofEducationofChina(20080050004),andNaturalScienceFoundationofBeijing(4092010)1.1001241.CollegeofElectronicandControlEngineeing,BeijingUni-versityofTechnology,Beijing100124:1)[5¡6],,,.2)[7¡8],RBF,.3),[9]RBF,.,RBF,,;[10]RBF,,.,,.,.Lu(Minimalresourcealloca-tionnetworks,MRAN)[11],MRAN86636RBF,RBF,RBF[12].HuangRBF(Gener-alizedgrowingandpruningradialbasisfunction,GGAP-RBF)[13],GGAP-RBF,,,GGAP-RBF,,.[14](Particleswarmoptimization,PSO)RBF,PSORBF,.PSO,.Huang[15](Recursiveorthogonalleastsquarealgorithm,ROLSA)PSORBF(Minimumvolumecoveringhyperspheres-recursiveorthogonalleastsquare{particleswarmoptimization,MVHC-ROLS-PSO),[14],.Chen[16](Orthogonalforwardselection,OFS)RBF,(Leave-one-out,LOO)RBF,,(OFS-LOO)|(Repeatedweightedboostingsearch,RWBS),.,RBF,RBF.,RBF(DynamicRBFNeuralNetwork,D-RBF),RBF,.RBF,.,.1RBF1.1RBFRBF,RBF()yyy=KXk=1®kÁk(xxx)(1),K,xxx(x1;¢¢¢;xm),®kk,Ákk,:Ák(xxx)=e¡kxxx¡¹kk¾2k(2),¹¹¹k,¾k.,.1.2RBF(Sensitivityanalysis,SA)[17],,,.[18¡20],.RBF,,,.RBF,,,1..,:Sh=varh[E(yyyjZh=®hÁh(xxx))var(yyy)(3),ZZZ=[Z1;Z2;¢¢¢;Zk],yyy,yyyZZZyyy=f(Z1;Z2;¢¢¢;Zk),varh[E(yyyjZh=®hÁh(xxx))]Zh®hÁhyyy,var(yyy)yyy,Sh®hÁhyyy.®hÁh(®hÁh[ah;bh]):Zh=ah+bh2+bh¡ah¼arcsin(sin(!hs))(4),!h,,[17]yyy=f(s)=+1X¡1(Acos(!js)+Bsin(!js))(5),¡¼s¼,Aj=1=2¼R1¡1f(s)£cos(!js)ds,Bj=1=2¼R1¡1f(s)sin(!js)ds.,(5)6:RBF867var(yyy)=21Xj=1(A2j+B2j)(6)varZh[E(yyyjZh=®hÁh)]=2+1Xj=1(A2j!h+B2j!h)(7)Sh=+1Pj=1(A2j!h+B2j!h)+1Pj=1(A2j+B2j)(8)1RBFFig.1ThestructureofRBFneuralnetwork,,.Th=(A2k!h+B2k!h)+1Pj=1(A2j+B2j)(9)ThSTh=ThKPi=1Ti(10),(N)E=12NNXj=1e2j(11)RBF:1..2..3.(9)(10)®hÁh(xxx),STh.4.STh²1,;K,t,²1j,K+1j®K+1(t)=¸£®j(t)¹¹¹K+1(t)=¹¹¹j(t)¾K+1(t)=¾j(t)(12)®0j(t)=(1¡¸)£®j(t)¹¹¹0j(t)=¹¹¹j(t);¾0j(t)=¾j(t)(13),¸(0;0:3)(),(16)»(18).5.STh²2,;t,²2i,iii,i,ii®0ii=®ii(t)=®i(t)Ái(x)Áii(x)(14)¹¹¹0ii(t)=¹¹¹ii(t);¾0ii(t)=¾ii(t)(15)(16)»(18).6.(11)([21]):®i(t+1)=®i(t)¡´1STi@E@®i(t)(16)86836¹¹¹i(t+1)=¹¹¹i(t)¡´2STi@E@¹¹¹i(t)(17)¾i(t+1)=¾i(t)¡´3STi@E@¾i(t)(18),´1,´2,´3.7..,,.,,.,[20],;,tt,[9¡16]t;,;.1.3RBF,,,;,D-RBF,.tK,eK(t),Ed.1)t(j),,RBFe0K+1(t)=K+1Xk=1®kÁk(xxx(t))¡yd(t)=KXk=1®kÁk(xxx(t))+®K+1ÁK+1(xxx(t))¡yd(t)(19)(11)(12)®0jÁj(xxx(t))+®K+1ÁK+1(xxx(t))=®jÁj(xxx(t))(20)e0K+1(t)e0K+1(t)=KXk=1;K6=j®kÁk(xxx(t))+®jÁj(xxx(t))¡yd(t)=eK(t)(21),,.2)t(i,ii),RBF:e0K¡1(t)=KXk=1®kÁk(xxx(t))¡yd(t)¡®iÁi(xxx(t))(22)(14)(15),(22)e0K¡1=KXk=1;k6=ii®kÁk(xxx(t))¡yd(t)¡®iÁi(xxx(t))+®0iiÁiii(xxx(t))=KXk=1;k6=ii®kÁk(xxx(t))¡yd(t)¡®iÁi(xxx(t))+(®ii+®iÁi(xxx(t))Áii(xxx(t)))Áii(xxx(t))=KXk=1;k6=ii®kÁk(xxx(t))¡yd(t)¡®iÁi(xxx(t))+®iiÁii(xxx(t))+®iÁi(xxx(t))=KXk=1;k6=ii®kÁk(xxx(t))¡yd(t)+®iiÁii(xxx(t))=eK(t)(23),,.,RBF.3)(16)»(18);[21],RBF,,.,RBF(D-RBF),D-RBF.2D-RBF,,,RBF.D-RBF6:RBF869(Chemicaloxygendemand,COD),;(MRAN)[11]GGAP-RBF[13],.2.1SIFRFy=x22+sin3x2+2x21sin(4x1)+x1sin4x2(24)y=1:234((x1¡0:5)2+(x2¡0:5)2)£(0:75¡(x1¡0:5)2¡(x2¡0:5)2)(25),¡1x11,¡1x21,SIFRF[22].800,400,400.3,2-3-1,,¹¹¹k=[¡2;0;2;¡2;0;2],¾k=1.,SIF2,SIF3,4.RF5,RF6,7.MRANGGAP-RBF(D-RBF,[11][13])1.25,RBF(D-RBF),,36,D-RBF,.47D-RBF,0.015.1D-RBFMRANGGAP-RBF,,MRAND-RBF;MRANGGAP-RBFD-RBF,,.,,MRANGGAP-RBFD-RBF;D-RBF,.2.2,.D-RBF(COD),COD,.2SIFFig.2TheneuronsleftinthelearningprocessofSIF3D-RBFSIFFig.3TheapproximatingresultsofSIFb