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357Vol.35,No.720097ACTAAUTOMATICASINICAJuly,2009ººº-1;211,,,º-,º-(Robustwaveletº-supportvectormachine,RWº-SVM).,,º-(Wº-SVM).,RWº-SVM.,,,TP391ProductSalesForecastingModelBasedonRobustWaveletººº-SupportVectorMachineWUQi1;2YANHong-Sen1WANGBin1AbstractAimingatthenormalGaussiandistributionalnoise,greaterbreadthnoiseandodditypointnoiseofproductsalesseriesandcombingadesignedrobustlossfunctionwithwaveletkernelfunction,weproposeanewwaveletº-supportvec-tormachine,namedasrobustwaveletº-supportvectormachine(RWº-SVM).TheRWº-SVM,whichhasastrongerrobustnessandsimplerdualoptimizationproblemthanstandardwavelet-supportvectormachine(Wº-SVM),caninhibitsometypesofnoiseanddisturbingodditypointnoiseofproductsalesseriese®ectively.Finally,theRWº-SVMisappliedtotheforecastsofcarsales,andtheresultsshowthattheforecastingmodelbasedontheproposedRWº-SVMise®ectiveandfeasible.KeywordsSupportvectormachine(SVM),waveletkernelfunction,robustlossfunction,forecasting,,,,.,(Noise).,,,.,(Whitenoise),.(Gaussiannoise).,,2008-06-052008-11-14ReceivedJune5,2008;inrevisedformNovember14,2008(863)(2007AA04Z112),(50875046,60574062)SupportedbyNationalHighTechnologyResearchandDevelop-mentProgramofChina(863Program)(2007AA04Z112)andNa-tionalNaturalScienceFoundationofChina(50875046,60574062)1.()2100962.2100961.KeyLaboratoryofMeasurementandControlofComplexSys-temsofEngineering(SchoolofAutomation,SoutheastUniversity),MinistryofEducation,Nanjing2100962.SchoolofMechanicalEngineering,SoutheastUniversity,Nanjing210096DOI:10.3724/SP.J.1004.2009.01027102835(Autoregressiveintegratedmovingaverage,ARIMA)(Autoregressiveconditionalheteroskedasticitymodel,ARCH)[1¡2].,[3].,,\;,Holt-Winter[4].,,,,,,.,().,,.,,,.[5]Vapnik,,,,,[6¡11].YanXu[11],.,,,,,,,..,,,,º-(Robustwaveletº-supportvectormachine,RWº-SVM).,.1ººº-,.RossiVillab[8],,.ShimHwang[12],,.[13],..,,,.,,.,,,[5].,[14],,.-,,Laplace,.,.,Laplace,.,-,,Laplace,,,,.,(),,,.,,,,,º-(RWº-SVM).1.1,,,.,Laplace()[14].,,,,L(»)=8:0;j»j·12(j»j¡)2;j»j·¹¹(j»j¡)¡12¹2;j»j¹(1),+¹=¹,¸0,¹¸0.:1)j»j·,,,;2)j»j·¹,,;3)j»j¹,Laplace,.1.2,,,,,.,,.,Hilbert-Schmidt,Mercer,.1(Mercer)[15¡17].L2(Rn)KKK(xxx;xxx0),'(xxx)6=0R'2(xxx)dxxx1,ZZKKK(xxx;xxx0)'(xxx)'(xxx0)dxxxdxxx0¸0;xxx;xxx02Rn(2)(1)KKK(xxx;xxx0).2()[15¡17].L2(Rn)KKK(xxx;xxx0),xxx;xxx02Rn,KKK(xxx;xxx0)=KKK(xxx¡xxx0),Mercer,7:º-1029.12.,3.3[15¡17].KKK(xxx;xxx0)=KKK(xxx¡xxx0),KKK(xxx).F[xxx](!!!)=(2¼)¡n2ZRnexp(¡j(!!!¢xxx))KKK(xxx)dxxx¸0;!!!2Rn;xxx2Rn(3),,.4[15¡17].Ã(xxxi),xxxi;xxx0i2Rd,xxx0xxx0id,aaal,mmmmmm0d,aaa2Rl,mmm;mmm02Rd,MercerKKK(xxx;xxx0)=lYi=1õxxxi¡mmmai¶Ãµxxx0i¡mmm0ai¶(4)KKK(xxx;xxx0)=lYi=1õxxxi¡xxx0iai¶(5).(5)mmm,xxx0i,.,.1[17].MorletÃ(xxx)=cos(!0xxxi)exp(¡kxxxi2k=2);!02RKKK(xxx;xxx0)=lYi=1cosµ!0£xxxi¡xxx0iai¶expµ¡kxxxi¡xxx0ik22a2i¶;xxx0xxx0id(6).,MexicanÃ(xxx)=(1¡kxxxik2)expµ¡kxxxi2k2¶(7)41,MexicanKKK(xxx;xxx0)=lYi=1µ1¡kxxxi¡xxx0ik2a2i¶expµ¡kxxxi¡xxx0ik22a2i¶;xxx0xxx0id(8)1.3ººº-,b,.,,b,.,b.b,.b,[18],:º-¹f(xxx)=¹¢¹xxx,º-f(xxx)=¢xxx+b.T=fxxxi;yigli=1,xxxi2Rd,yi2R,xxxid.¹xxx=(xxxT;´)THilbert,¹xxx1=(xxxT1;´)T¹xxx2=(xxxT2;´)T¹xxx1¢¹xxx2=xxx1¢xxx2+´2,¹=(=´)T,¹´,¹f(xxx)=¹¢¹xxx=¢xxx+(b=´)´=¢xxx+b=f(xxx).,bº-¹f(xxx),º-f(xxx).,T=f(xxxi;yi)gli=1,xxxi2Rd,yi2R,xxxid,b,º-,(1),º-(RWº-SVM),min»»»(¤);b;12(k)+C¢Ãº¢+1lXi2I112(»2i+»¤2i)+1lXi2I2¹(»i+»¤i)!s:t:8:yi¡¢xxxi¡b·+»i¢xxxi+b¡yi·+»¤i»i;»¤i¸0;º2(0;1];¸0;i=1;¢¢¢;l(9),º2(0;1];,;»i,»¤i(i=1;¢¢¢;l),;I10j»ij,j»¤ij·¹,I2¹j»ij,j»¤ij.RWº-SVM,LagrangeL(®®®(¤);¯;»»»¤;;´´´¤)=12(k)+C¢Ãº¢+1lXi2I112(»2i+»¤2i)+1lXi2I2¹(»i+»¤i)!¡¯¢¡lXi=1(´i»i+´¤i»¤i)¡lXi=1®i(+»i+yi¡¢xxxi¡b)¡lXi=1®¤i(+»¤i¡yi+¢xxxi+b)(10),®¤i,´¤i,¯¸0.»»»¤,rwL=0,rbL=0,rL=0,r»(¤)iL=0.rwL=0rbL=01030358:=lXi=1(®¤i¡®i)xxxib=lXi=1(®¤i¡®i)(11)Karush-Kuhn-Tucker(KKT),(9)min®;®¤12lXi=1lXj=1(®¤i¡®i)(®¤j¡®j)(KKK(xxx;xxx0)+1)¡lXi=1yi(®¤i¡®i)+l2ClXi=1(®2i¡®¤2i)s:t:8:eeeT(®®®+®®®¤)·C¢º;0·®i;®¤i·minµCl;C¢¹l¶(12),:0·®i;®¤i·C=l;:0·®i;®¤i·C¹=l.,minµCl;C¢¹l¶®i,®¤i.,min®;®¤12[(®®®¤)T;®®®T]264Q+ElC¡Q¡QQ+ElC375®®®®®®¤#+[¡yyyT;yyyT]®®®®®®¤#s:t:8:eeeT(®®®+®®®¤)·C¢º;0·®i;®¤i·minµCl;C¢¹l¶(13),Qij=K(xxxi;xxxj)+1,i=1;¢¢¢;l,eee=[1;¢¢¢;1]T,eeel;El;®®®;®®®¤Lagrange,l.(13)min12¹®®®TH¹®®®+¹yyyT¹®®®s:t:8:eeeT(®®®+®®®¤)·C¢º;0·®i;®¤i·minµCl;C¢¹l¶(14),¹®®®=®®®¤®®®#,H=264Q+ElC¡Q¡QQ+ElC375,¹yyy=¡yyyyyy#.(5)(11)RWº-SVMf(xxx)=¢xxx+b,RWº-SVMf(xxx)=¢xxx+b=lXi=1(®¤i¡®i)lYj=1õxj¡xijai+1¶(15),xjxxxj,xijij.º-(Wº-SVM),RWº-SVM,.RWº-SVM(15)b,b.,,.2.®®®(¤)=(®1;¢¢¢;®l;®¤1;¢¢¢;®¤l)T(10),1)®i=®¤i,(xij;yi)-;2)®i2µ0;minµCl;Cl¢¹¶¶,®¤i=0®i=0,®¤i2µ0;minµCl;Cl¢¹¶¶,(xij;yi)-;3)®i=minµCl;Cl¢¹¶,®¤i=0®i=0,®¤i=minµCl;Cl¢¹¶,(xij;yi)-..®®®(¤)=(®1;¢¢¢;®l;®¤1;¢¢¢;®¤l)T(10),KKT,,®i(+»i¡yi+¢xxxi+b)=0(16)®¤i(+»¤i+yi¡¢xxxi¡b)=0(17)µminµCl;Cl¢¹¶¡®i¶»i=0(18)µminµCl;Cl¢¹¶¡®¤i¶»¤i=0(19)1)»3).1)®i=®¤i=0,(18)(19),»i=»¤i=0,(1),(xij;yi)-.2)®i2µ0;minµCl;Cl¢¹¶¶,®¤i=0®i=0,®¤i2µ0;minµCl;Cl¢¹¶¶,(18)(19),»i=»¤i=0,(16)(17),yi=¢xxxi+b+;(xij;yi)-.3)®i=minµCl;Cl¢¹¶,®¤i=0,(18)»i¸0;»¤i=0(20)(16)yi=+»i+¢xxxi+b¸+¢xxxi+b(21)®i=0,®¤i=minµCl;Cl¢¹¶,(19)»¤i¸0;»i=0(