5404第四章___径向基函数神经网络

4RadialBasisFunctionNeuralNetworkRBFBPRBF4.1RBF4.24.34.4RBF4.5RBFRBF4.1RBF(RBF)4.14.1RBF4.1mhn−−RBFnhmnTnRxxx∈=)(21LxmhR×∈W0b,,mb[]Tmyy,...,1=y)(∗Φii∑SigmoidBPSigmoidRBFGaussiannRBF4.1ici∗()⋅Φi4.14.34.2(1)Gaussian()22itietδ−=Φ4.1(2)Reflectedsigmoid()2211δtiet+=Φ4.2(3)Multiquadric()()0,122+=Φαδαiitt4.34.2RBF4.14.3iδSpreadiδ4.1RBFk∑=−Φ=hiiikwy1)(icx4.4RBFi()iiw,,δicRBF4.34.36x)(icx−ΦiRBFRBF4.3RBF4.2RBRBFon-centeroff-surround()()()SpikeSpike70HubelWiesel(on-centeroff-surround)(off-centeronsurround)4.44.4()()icxx−=ΦGi4.5xic()⋅G4.14.34.3RBF4.3.1N)(N21XXXL=s)(21NyyyL=tF()iXFyi=4.6RBF4.4RBFRBFNRBFRBFNTN)(21L=WiwiXNiL21=j)(jicX−Φ=jijh4.7()⋅ΦjjjXc=RBFRBF[]ijh=HNNR×∈HRBF∑∑==−Φ==NjjNjjijwwhf11)()(jiicXX4.8)(iXfyi=TNTyyy)(21L==tyTN)(21L=WHWy=4.8NNR×∈HNRyHW1−=4.9Micchelli[Micc1986]N21XXX,,,LHRBF0(1)NH(2)4.3.2(){}NiRRySni,...,2,1,=×∈=iXFF()()()∑=−=NiiSFyFE1221iX4.10[Tikh1977]()221DFFER=4.11D()()()FEFEFERSλ+=4.12λ7[Pogg1990]()()∑==NiiGwXF1,iXX4.13()iXX,GDD~GreeniwGreen()iXX,GDDGreenXiX()()iiXXXX−=GG,4.14DGreenGaussian4.54.5(1)(2)ff(3)4.124.4RBFRBF4.4RBFhiciδRBFRBF1)OLSROLS2(K-meansKohenenSOFM[Koho1984])(RAN)OLSRBF(RBF)RBFRBFRBFOLSRBFRANRBFN21XXX,,,LNyyyL21j)(⋅Φj4.4.1RBFMoodyDarken[MoDa1989]meansk−RBFhkk)(),...,()(kkkh21ccc)(),...,()(21kwkwkwhmeansk−RBFiciδ(1)h1=khh(2)Njhik,...,2,1,,...,2,1,)(==−ijcX(3)jX()hikii,...,2,1,)(min=−=ijjcXXjXi)(kwi∈jX(4)hiNkkwii,...,2,1,1)1()(==+∑∈xixciNi)(kwi(5))()1(kkiicc≠+(2)(6)(6)iidκδ=idi)(minkdijiijcc−=≠κ[]Th=wLMSiXNiL21=j4.7[]ijh=Hˆ4.15hNR×∈HˆRBF[]Th=wwHyˆˆ=4.16yyˆ−=ε[]TNyyy,...,,21=yHˆwwHyyyˆˆ−=−=ε4.17wLMSyHw+=ˆ4.18+HˆHˆTHHHHTˆ)ˆˆ(ˆ1−+=4.194.4.2RBF[Hayk2001]BPRBF∑==NjjjeE1221β4.20jβje()()∑=Φ−=−=hiiijjjwyFye1jjXX4.21()XFiciriw()()icXXX−Φ=∇iiicrwFi22)(4.22()232)(icXXX−Φ=∇iiirrwFi4.23()XXiwfiΦ=∇)(4.24iciriw()()∑=−Φ=∆Njjijjiierw12ijicXXcβη4.25()∑=−Φ=∆Njijjiiierwr123ijjcXXβη4.26()∑=Φ=∆Njijjiew1jXβη4.27()jXiΦijXη4.4.3OLSRBFChenOLS[ChCo1991]4.3.1MicchelliNNR×∈HyHNHNyHNNM≤MNR×∈Hˆεε−0wHyˆ4.280wwHyˆ−wHˆ4.28HˆRBFHˆRBFyNN21xxxL∑==Niia1ixy4.29Tix2iTixyxia=Ni,...,2,1=4.30∑==NiiTa12ixyy4.31∑==NiTia121yyxi4.32M∑∑====MiTiMiiAagg121yyxi4.3310≤≤AgMAgNM=1=Ag4.29ixH(OLS)HHGram-SchmidtGram-Schmidt(1)HHNN12111PPP,,,LNHNE(2)yN12111PPP,,,Lyk1Pk1k1PyPyT(k1Py)k1Pk1P1E(3)4.4.11−NGram-Schmidt1E1N22212PPP−,,,L(4)yj2P(5)M4.5RBFRBFRBFRBFRBF(RAN)RBF(MRBF)RBFRBFRBFPlatt(RAN)[Platt1991]RBFYingweiMRBF[YiSu1997aYiSu1997b]RANWilson[WiBl1997]MLPMLP[PoIo1992]sigmoid[AtJi1997]RBFBPRBFRBFRBFRBFRBFRBFRBFRBF4.5.11−−hmRBFmhii()xiφGaussian()22irieicxx−−=φmR∈xRBFmR∈iciir()RBF()()∑==hiiiwf1,xθx(h)φ4.34iwi(h)θRBFh()Rf∈(h)θx,(h)θx[]Th=w)(hNethRBF{}iiirw,,ic=θiRBF{}hii,...,2,1,==θ(h)θ()xgmRD⊂RBF()()(h)θx,fD()xg()()()ε−=xθxθ(h)(h)gfP,4.35()()()()()∫∈−=−Dxdgfgfxxθxxθx(h)(h)2,,ε()xg4.35()()()ε−=∑=NiifyP12,(h)i(h)θxθ4.36()iy,ixNi,...,2,1=D∈ixRyi∈iyN[]TN21x...xxX,,,=[]TNyyy,...,,21=Yi[]TiNiiooo,...,,21=io22iijrcxijeo−−=jxih∑=−=hiiw1ihoYe4.37he4.36()hTh(h)eeθ=P()0=(h)θPRBF()(h)θP()(h)θPRBF!h*(h)θ()(h)θPRBF(h)θ*(h)θδ−*(h)(h)θθδ*(h)(h)θθ=RBFi0≠ioDD0=ioi0=iwi0≠ioD0=io4.5.2RBF1)RBFhhwhchr∑−=−−=11,1hiijijjhowye[]∑−=−−−−−==11,12,11,1,...,,hiiTNhhhweeei1hoYe1−hh1hhoeehw−=−()hThhT1h1hT1h(h)oooeeeθ22hhwwP+−=−−−4.38()(h)θPhchrhwhhTh1hhThhocoecoc∂∂+∂∂−=∂∂−222hhwwP4.39hTh1hThooeohhhhhrwrwrP∂∂+∂∂−=∂∂−2224.40()hh1hooeThhwwP−−=∂∂−24.414.394.40[])(,),(),(2212hNh2h1hThcxcxcxco−⋅⋅⋅−−=∂∂hNhhhooor[]222213,,,2hNh2h1Thcxcxcxo−⋅⋅⋅−−=∂∂hNhhhhooorr[]()()∑∑==−−−−=−−−=−−⋅⋅⋅−−−=∂∂NihiihhhNihihihihhhhhNhhhhoerwoowerwwooorwP1,21,12212)(4)(4)(,),(),(4hihih1hhNh2h1hcxcxoecxcxcxc4.42[]()()∑∑==−−−−=−−−=−−⋅⋅⋅−−−=∂∂NihiihhhNihihihhihhhhNhhhhhoerwoweorwwooorwrP12,31,12322221344,,,4hihih1hhNh2h1cxcxoecxcxcx4.434.424.430hioix(0,ihe)hcixhr(0,ihe)hcixhrhchrhwhchr0=∂∂hcP0=∂∂hrP0=∂∂hwP4.424.434.410)(41,2=−−∑=Nihiihhhoerwhicx4.440412,2=−−∑=Nihiihhhoerwhicx4.45()021,1=−∑=−Nihihihihoowe4.464.444.454.41∑∑===NihiihNihiihoeoe1,1,ihxc4.47012,=−∑=Nihiihoehicx4.48hThhT1hoooe−=hw4.49N∑==NiNC11ix∑==Njhjjhhiihioeoek1,,4.504.49∑==Niik1ihxc4.51hcNikixhhioihe,ixhchhc4.48hrhohh1hhoeehw−=−1−hhohwhrhohehrhohe0=hTheoihiihodo,',=2iicx−=idhrh[]TNhhhhoooo','2,'1,',...,,=heid0=−icx0=−icxhr4.491he−hohwhRBFRBFRBF21−hRBF)1(−hNeths≥hs=RBF()xg4.1()xg)1(−hNet)(hNet)1(−hNet)(hNet()xg)(hNet1−h)1(−hNet)(hNet2−h)1(−hNet)1(−hNet()xg)1(−hNet)(hNet()()()∑∑=−===hjjjhiiivwg111xxxϕφ()()0111=−∑∑=−=hjjjhiiivwxxϕφ4.51)1(−hNetiw)1(−hNet()xiφ()xjϕiw()xjϕ1−h()xiφ()xjϕ()xiφ(1)()xjϕ()xiφ(20)()xjϕ(20)()()()()011111=−=−∑∑∑−==−=hiijihjjjhiiivwvwxxxφϕφ()xiφiivw=1,...,2,1−=hi)(hNet1−h)1(−hNet(2)()xjϕ()xjϕ()xiφ()xjϕ()xiφ2−h)1(−hNet()xgmR∈x()xg()xg4.1{}1,...,2,1,−===−hi*iii*1)(hθθθθ)1(−hNet)(hNet4.2{}1,...,2,1,−===−hi*iii*1)(hθθθθ)1(−hNet)(hNet)1(−hNet)(hNet1−h*1)(hθ−hi()θx,firiw()232),(icxxθx−=∇iiirrwfiφ()xθxiwfiφ=∇),(iriw()∑=−=∆Njiiijiirwer123,2ijjcxxφη()∑==∆Njijiiew1,jxφηη1−h)(,jxhhjhweφ−≈h()∑=−−≈∆Njhhhhrwr122322hjjcxxφη()∑=−≈∆Njhhhww12jxφη0∆hrhw∆hw1+t)()1(trtrhh+)()1(twtwhh+h4.20=hohw4.3{}1,...,2,1,**)1(−===−hiiiihθθθθ)1(−hNet)(hNet)1(−hNet)(hNetkhhjδ−*jkccδ−*jkrrδ−+*jhjδ−*kkc