基于支持向量机的磁化曲线拟合新方法

61OO31TM402B1001-8425200607-0008-051[12][3][4]2090VapnikSVM[5]SVMSVM22.112.2VCVapnikVCVCVCh2hhVChVCyxdx1x2α1y1α2y2αsysK(x2x)K(x1x)K(xsx)1Fig.1DiagramofSVM!#$%%&#TRANSFORMER’()*!+,)-.(*#$%%&7VCVCVCVCVCnVCn+1VCVC2.3Kxx'Mercer!Kxx'gxgx'dxdx'≥0g∈L211Kxx'=((xx')+c)dc≥0d∈N22RBFKxx'=exp(-||x-x'||22σ2)33BKxx'=B2N+1||x-x'||44Kxx'=sin(N+12)(xx')sin12(x-x')55sigmoidKxx'=tanh(k(xx')+v)k0v062.4!T={(x1,y1),(xlyl)}∈(X×Y)lxi∈X=Rdyi∈Y=Ri=1lminw∈Rn,ζ(*)∈R2l,b∈Rτ(w,ζ(*)=12||w||2+C1lli=1(ζi+ζi*)7s.t.((wφ(xi))+b)-yi≤ε+ζii=12l8yi-((wφ(xi))+b)≤ε+ζii=12l9ζi(*)≥0i=01l10***LagrangeL=(w,b,ζ,α)=12||w||2+Clli=1(ζi+ζi*)-li=1(ηiζi+ηi*ζi*)-li=1αi(ε+ζi+yi-(wφ(xi))-b)-li=1αi(ε+ζi*-yi+(wφ(xi))+b)11Lagrangeαi*ηi*≥0bwhζ(*)0▽bL=li=1(αi-αi*)=012▽wL=w-li=1(αi-αi*)φ(xi)=013▽ξ(*)L=Cl-αi(*)-ηi(*)=014minα(*)∈R2l12li,j=1(αi*-αi)(αj*-αj)Kxixj+εli=1(αi*+αi)-li=1yi(αi*-αi)15s.t.li=1(αi*-αi)=0160≤αiαi*≤Cli=12l17Kxx'=(φ(x)φ(x'))f(x)=Σxi∈SV(αi*-αi)Kxix+b18SVbαjb=yj-li=1(αi*-αi)Kxixj+ε19αk*b=yk-li=1(αi*-αi)Kxixk-ε2033.1εMatlabαT=[αlTαl*T]αlT=[α1αl]αlT=[α1*αl*]15943s.t.[vl-vl]α=0220≤αi(*)≤Ci=12l23v=[111]1×lεl=[εεε]1×lTY=[y1y2yl]T3.2[6][7]SSVM[8]RSVM[9]LS-SVMLS-SVMSuykensVandewalbLS-SVM7～10CSVMminw∈Rn,ζi∈R2l,b∈Rτ(w,ζi)=12||w||2+rli=1!ζi224s.t.yi((wφ(xi))+b)=1-ζii=12l25ζi≥0i=01l26Lagrangee=[111]T1×lIl×lα=[α1αl]Q=[qij]l×lqij=yiyjK(xixj)rf(x)=Σxi∈SVαK(xixj)+b28LSSVM44.112εC32143184.2MTP1-7.5[4]2142a2b2c2d11①Cε②εC③Cε①②εC③2e2SVMSVM52[1][2]SVMSVMSVM1O72Table2FittingmagneticcurveformulasofSVMfx=486.71*exp(-1/2*(u-1)ˆ2-700*exp(-1/2*(u-23/30)ˆ2-526.8*exp(-1/2*(u-19/36)ˆ2)+700*exp(-1/2*(u-49/180)ˆ2)+700*exp(-1/2*(u-1/9)ˆ2)+700*exp(-1/2*Uˆ2)+700*exp(-1/2*(u+11/180)ˆ2)-473.6*exp(-1/2*(u+1/12)ˆ2)-700*exp(-1/2*(u+7/60)ˆ2)-700*exp(-1/2*(u+31/180)ˆ2)-700*exp(-1/2*(u+49/180)ˆ2)+127.5*exp(-1/2*(u+23/45)ˆ2)+700*exp(-1/2*(u+23/30)ˆ2)-304.7*exp(-1/2*(u+1)ˆ2)gx=-486.71*exp(-1/2*(u+1)ˆ2+700*exp(-1/2*(u+23/30)ˆ2+526.8*exp(-1/2*(u+19/36)ˆ2)-700*exp(-1/2*(u+49/180)ˆ2)-700*exp(-1/2*(u+1/9)ˆ2)-700*exp(-1/2*uˆ2)-700*exp(-1/2*(u-11/180)ˆ2)+473.6*exp(-1/2*(u-1/12)ˆ2)+700*exp(-1/2*(u-7/60)ˆ2)+700*exp(-1/2*(u-31/180)ˆ2)+700*exp(-1/2*(u-49/180)ˆ2)-127.5*exp(-1/2*(u-23/45)ˆ2)-700*exp(-1/2*(u-23/30)ˆ2)+304.7*exp(-1/2*(u-1)ˆ2)2Fig.2Fittingmagneticcurveunderdifferentparameters×!--φ/mWbi/A21.510.50-0.5-1-1.5-2(e)RBFC=700ε=0.1-15-10-5051015φ/mWb21.510.50-0.5-1-1.5-2(c)RBFC=1000ε=0.1-15-10-5051015i/Ai/A21.510.50-0.5-1-1.5-2(d)C=1000ε=0.1-15-10-5051015φ/mWb21.510.50-0.5-1-1.5-2(b)RBFC=1000ε=0.3-15-10-5051015i/Aφ/mWb21.510.50-0.5-1-1.5-2-15-10-5051015i/A(a)RBFC=1000ε=0.6φ/mWbSVMSVM[1].[J].199634931-39.[2].[J].200239510-13.[3].[J].2003261063-66.[4].[J].20032738-11.[5]VapnikVN.Thenatureofstatisticalleaningtheory[M].NewYorkSpringer-Verlag,1999.[6]PlattJ.Fasttrainingofsupportvectormachinesusingsequentialminimaloptimization[C].AdvancesinKemelMethods-SupportVectorLearning,Cambridge,1999185-208.[7]LeeYJ,MangasarianOL.SSVMasmoothsupportvectormachines[R].WisconsinUnivcrsityofWiscon-εCRBF0.8100012RBF0.6100013RBF0.3100014RBF0.350013RBF0.11000140.11000141Table1Comparisonofdata11!,!#!$$:、,º:;;;:TM402:B:1001-8425(2006)07-0012-051、,、。,、。、、,。,、、、、,,()。,。!#$%%&#!#$%&’()’()*!+,)-.(*#$%%&sin,1999.[8]LeeYJ,MangasarianOL.RSTV:reducedsupportvectormachines[R].Wisconsin:UniversityofWisconQsin,2RRR.[9]SuySensJTUVVandeWaXXeJ.LeastsYuaressupportvectormachines[J].ZeuraX[rocessingLetters,1999,9(\):29\-\RR.*+,&-..-/0(+.123425670/+.-89:5;+7=+32/%:25.?+8.256781-/+!#$%&!’(&)*+,-&!./0&)12!3&4!5/%&4)6!789:3&4!;0&4(SouthWestJiaotongUniversity,]hengdu61RR\1,]hina)@A=.578.:!#$%&’’&()*’+,-%,.)/(*’&001.2*345*6,5177,.’2*0’,.)40+&*&’.458%,.)*.*)194’&,&57.*5*’*6:;+*)*’+,6&54779&*6:B+C,253=:!#$%&’();*+,,’-.)/-’(#/01$);2#3$)-1//+.);4(+56-1’$:2RR5-R5-2\:(1976-),,,,;(1957-),,,,、;(1969-),,,,、;(1977-),,,,。