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187©Vol.18No.720067JournalofSystemSimulationJuly,2006•1785•410075RBFElmanSVRElmanRBFSVR(783.5%)ElmanTP391.9A1004-731X(2006)07-1785-04SupportVectorMachinesForTimeSeriesRegressionandPredictionDONGHui,FUHe-lin,LENGWu-ming(CivilArchitectualEngineeringCollege,CentralSouthUniversity,Changsha410075,China)Abstract:Amethodforpredictingtimeseriesbasedonsupportvectormachineswasproposed.Thetimeseries,includingsimulateddataandlandslidedeformationdatasets,werepreformedforregressionandpredictionbysupportvectormachine,RBFnetworks,andelmanrecurrentneuralnetworks.Acomparisonofthesethreemethodswasmadebasedontheirpredictingability.Theresultsshowthat:whennoiselevelislowerinsimulatedexperiment,supportvectormachineisperfectrelatively,andtheElmanandRBFnetworkareofmoreinstability,ontheotherhand,withthehighernoiselevels,thegreaterrelativeerroroftwonetworksmodelsismade.Forlandslidedatasetsprediction,theneuralnetworksarelimitedtopredictshorttermnonlineartimeseriesintermsoftheiraccuracy,whereassupportvectormachinehasahigherprecisionintheshorttermandlongterm.Keyword:supportvectormachine;regression;elmanrecurrentnetwork;landslidedeformation1[1](SupportVectorMachinesSVM)[6]RBFElman1(SVR)VapnikVC2005-05-082005-07-19(200331880201)(1976-),GISk},{kkyxnkRx∈nRyk∈ϕHbxxfk+=))(,()(ϕωϕHRn→nR∈ωbωϕ2125.0)(5.0][][ωω+=+=∑=kikempregeCfRfRkkkyxfe−=)()(⋅Cε}|)(|,0max{|)(|εε−−=−xfyxfyεε∑=−−=kiempxfykfR11|)(|][∑=++=kiiiCJ1*2)(||||21minξξω**(,())..(,()),0iiiiiiyxbstxbyωϕεξωϕεξξξ−−≤+⎧⎪+−≤+⎨⎪≥⎩(1)187Vol.18No.720067July,2006•1786•Cε(1)**,1*111max()()(,)2()()kiijjijijkkiiiiiiJKxxyyαααααεαε====−−−−++−∑∑∑⎪⎩⎪⎨⎧≤≤=−∑=Ctsiikiii*1*,00)(..αααα(2)∑=−=kiiiix1*)()(ϕααω∑=+−=kiiiibxxKxf1*),()()(αα(3))()(),(jijixxxxKϕϕ⋅=ϕϕωMercer),(jixxK(1)djijixxxxK)1(),(+⋅=,2,1=d(2)Sigmoid])(tanh[),(cxxbxxKjiji+⋅=(3)22(,)exp(||||/2)ijijKxxxxp=−−KKT(karush-kuhn-tucker)⎩⎨⎧∈=−+∈=+−),0(0)(),0(0)(*CxfyCxfyiiiiiiαεαεb2(RBF)ElmanRBFElman2.1},{kkyx,,2,1=k,x[-10,10]yδ+=xxc/)sin(sin41δσ1SVRεCε[4-5])(*)5.01.0(~xrangep−d)5.01.0(~−dpElman1(1.0=σ)SVRElmanRBF9C25ε0.1(p)0.031256TFsigmoidradbasTFlinearlinear2SVRRBFElman2SVRElmanSVRRBFElman2MSEmaxminSVR0.0473854.790610.000368RBF0.0554638.674260.006852Elman0.05514710.79930.002854b),(1xxK),(2xxK),(mxxK1x2xmx##1ω2ωmω#y1187Vol.18No.720067July,2006•1787•2.25[1]255935606672.2.1Takensmττt∆t∆1(1=∆t)()),,(21mttttxxxx−−−=}{ttxy=RRfm→:mmm5),,,(21nxxxKKK⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=−+−−+1113221nmnmnmmxxxxxxxxxX#%##⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=++nmmxxxY#22.2.2∑−=+⋅−=mnitiiitbxxKy1*)()(KKααnmt,,1+=},,,{211nmnmnmnxxxxK+−+−+−=1+n∑−=+−++⋅−=mnimniiinbxxKy11*1)()(KKαα},,,{211nmnmnmnxxxxK+−+−+−=}ˆ,,,,{1322++−+−+−=nnmnmnmnxxxxxK1ˆ+nx1+nl∑−=+−++⋅−=mnilmniiilnbxxKy1*)()(KKαα2.2.372(a)(0,1.0=σ);(b)SVR;(c)RBF;(d)Elman(x)y=sinc187Vol.18No.720067July,2006•1788•)(RBFpCεMatlabRBF16×Elman35()33(y)3SVRRBFElman6030.029.830.5729.920.2730.070.236131.030.591.3230.950.1631.361.166232.032.702.1831.581.3132.180.566333.034.735.2434.243.7634.464.426442.040.353.9336.6712.736.6312.86547.044.575.1736.9821.339.3716.26661.050.9616.546.5123.842.9929.53svr,rbf,elman2.2.466%783.5%SVR(7)RBPElman4SVR4SVRRBF6RBFElman(2(d)3)Elman(356)()63(1)SVRElmanSVRRBF(4)SVR(7)(2))(SVRSVR(3)RBPElman(4)()[2][1].[M].:,2000.[2]AJSmola,BScholkopf.Atutorialonsupportvectorregression[D].RoyalHollowayCollege,UniversityofLondon,UK,1998.[3]UThissen,RvanBrakel,APdeWeijer,etal.Usingsupportvectormachinesfortimeseriesprediction[J].ChemometricsandIntelligentLaboratorySystems(S0899-7667),2003,69:35-49.[4]CherkasskyV,MaY.Comparisonofmodelselectionforregression[J].NeuralComputation(S0169-7439)2003,15(7):1691-1714.[5]SmolaA,MtirataN,ScholkopfB,MullerK.Asymptoticallyoptimalchoiceofε-lossforsupportvectormachines[C]//proceedingsofICANN,1998.[6]VVapnik.TheNatureofSatisticalLearningTheory[M].SpringerVerlag,1995.[7]VVapnik.SatisticalLearningTheory[M].Wiley,1998.