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32,6200712HighwayEngineeringVol.32,No.6Dec.,2007[]20070301[](1979),,,,,,(,100084)[]LogitLogitIIA,LogitLogit,,Logit,[]Logit;Loigt;;;Logit[]U491.2+54[]A[]10021205(2007)06009206ReviewoferrorstructureextensionofdiscretechoicemodelsintravelmodechoiceWUShijiang,SHIQixin,LUHuapu(InstituteofTransportationEngineering,Beijing100084,China)[Abstract]Multinomiallogitmodelsarebasedontheassumptionthaterrortermsareindependentlyi2denticallydistributedoveralternatives,whichcausestheindependenceofirrelevantalternativesinmulti2nomiallogitmodels.Thispaperrelaxestherigidassumptionofmultinomiallogitmodels,andsystemati2callyreviewstheextensionsofnon2independentidenticalerrorstructure,independentnon2identicalerrorstructureandnon2independentnon2identicalerrorstructureaimingtodevelopmoreflexibleerrorstructuretoreflecttheheteroscedasticityandcorrelationamongtherandomutilityofalternatives.[Keywords]multinomiallogitmodel;nestedlogitmodel;generalizedextremevaluemodel;het2eroscedasticextremevaluemodel;mixedlogitmodel(DiscreteChoiceModelDisaggregateChoiceModel),,,,()[1],,[2]Logit(MultinormalLogit,MNL)MNLGumbel(independentidenticaldistribution,IID),MNLMNLIIA(IndependenceofIrrelevantAlterna2tives),MNL[1]MNLIIAMNL,MNLIIA,MNL(closedform),MNL,,Logit(NestedLogit,NL)[3](General2izedExtremeValue,GEV)[4](Het2eroscedasticExtremeValue,HEV)[5]Logit6,:(MixedLogit,ML;MixedMultinomialLogit,MMNL)[6],,,1,,i(i=1.2,,J)n(n=1.2,,N)Uin,ni,in,:UinUjnPji(1)inUin,Vin,in,,Uin:Uin=Vin+in(2),,niPn(i):Pn(i)=P(UinUjn,Pji)=P(Vin+inVjn+jn,Pji)=P(jnVin-Vjn+in,Pji)(3)n=(1n,2n,,jn)f(x1,x2,,xj),Pn(i)in:Pn(iin)=Vin-V1n+in-Vin-V(i-1)n+in-Vin-V(i+1)n+in-Vin-VJn+in-f(x1,,in,,xJ)dx1,,dxi-1dxi+1dxJ(4)(4)in(-,+),Pn(i):Pn(i)=+-Vin-V1n+in-Vin-V(i-1)n+in-Vin-V(i+1)n+in-Vin-VJn+in-f(x1,,in,,xJ)dx1dxi-1dxi+1dxJdin(5)f(x1,x2,,xJ),(5),niPn(i)f(x1,x2,,xJ),,,Gumbel(1)LogitProbitGumbel,Gumbel(LocationParame2ter)0(ScaleParameter)1,(5)Pn(i)(closedform),Logit(MNL),:Pn(i)=exp(Vin)Jj=1exp(Vjn)(6),(5)Pn(i)n=(1n,2n,,jn)(J),Pn(i):Pn(i)=+-Vin-V1n+in-Vin-V(i-1)n+in-Vin-V(i+1)n+in-Vin-Vjn+in-1(2)J||exp-12(x1,xJ)-1(x1,,xJ)dx1dxi-1dxi+1dxJdin(7)(7)Probit(MNP),MNP,2MNLGumelIIA,Gumbel,3932Gumel,Gumbel1,Gumbel(GeneralizedExtremeValuefamilyofmodels,GEV)Logit(NestedLogit,NL)[3],NLNL(Nest),,NLMNL,MNLIIANL,ni:Pn(i)=PimPm=expVinmjNmexpVjnmexpmlnjNmexpVmmkMexpklnjNkexpVink(8)(8):Pimmi,Pmm,Nmm,mm,,0m1m=0,m,m=1,mNL1,(8)NLMNL,mm1,NLMNLNL,,NLNL,NL,Voshva(1997)Logit(CrossNestedLogit,CNL)[7],WenKoppel2man(2001)Logit(GeneralizedNestedLogit,GNL)[8]Voshva(1997)CNL,,,1,CNLni:Pn(i)=mPimPm=mimexp(Vin1jNmjmexpVjn1jNmjmexpVjn1kjNkjkexpVjn1(9)(9):CNL,imm,:mim=10im1(10)WenKoppelman(2001)GNLCNL,CNLmm,GNLni:Pn(i)=mPimPm=mimexpVin1mjNmjmexpVjn1mjNmjmexpVjn1mmkjNkjkexpVjn1mm(11)WenKoppelmanGNLmm,iim,GNL,GNL,GNL,NLCNLGNL,Bresnahan(1997)PD(PrinciplesofDifferentia2tion)[9]Chu(1989)PCL(PairedCombi2natorialLogit)[10]Small(1987)OGEV(OrderedGeneralizedExtremeValue)[11]496,:3MNLNL,Gumbel,MNL,,Horowitz(1981),,[12],,Bhat(1995)(HeteroscedasticExtremeValue,HEV)[13]HEViGumbel(ScaleParameter)i,HEV,ni:Pn(i)=+-jC,jiVin-Vjn+iwj(w)dw(12)(12):w=in/i,C,iiGumbel,(Ó)(Ó)Gumbel(1),:(t)=exp-exp-t(t)=exp-texp-exp-t(13)Bhat(1995)HEV,,MNLIIA,,iHEVi,HEVMNLHensher(1998)[14]Hensher(1999)[15]Zeng(2000)[16]HEVHEV,Recher(1995)OddballAlternative[17]SwaitStacy(1996)ParameterizedHeterscedasitcMNL(PHMNL)[18],,,HEV,4Logit,,Logit(MixedLogit,ML;MixedMultinomialLogit,MMNL)Logit(LogitKernel)(FlexibleDisturbance,FlexibleErrorComponent,FlexibleErrorStructure)Logit(RandomParameterLogit)[19]Logit,Logit,McFaddenTrain(2000)[6]Ben-Akiva(2001)[20]McFaddenTrain(2000)MNL,,Logit,,ni:Pn(i)=exp(Xin)Jj=1exp(Xjn)fb,d(14)(14):b,f(Ó),f(Ó),f(Ó),ReveltTrain(1998)[21]McFaddenTrain(2000)Hens2herGreene(2003)[22]Logit,Majken(2003)Twin2PeakedLogit[23],Ben2Akiva(2001),,n:n=Fnn+vn=FnTn+vn(15)Fn(JM),nM(M1),TTn,TChol2esky,n,vnIIDGumbel(J1)n,ni5932:Pn(in)=expXin+FinTnJj=1expXjn+FjnTn(16)Pninni,FjnFnj(j=1,2,,J)n,ni:Pn(i)=nPnin3fndn=nexpXin+FinTnJj=1exp(Xjn+FjnTn)f(n)dn(17)Ben2Akiva(2001)FnT,(17)Fn(14),(17)(14),(17)TT(14)Logit,(15),:Cov(n)=Var(FnTn+vin)=FnTTFn+I2/6(18)(18):I,Cov(n)ii,;ijij,2080,Logit,BoydMellman(1980)[24]CardellDunbar(1980)[25],,Logit,,,Logit,Train(1986)[26]Ben2Akiva(1993)[27],Logit,Logit,Bhat(2001)[28]Train(2002)[29],Hensher(2003)[22]Hess(2005)[30]Hess(2007)[31]5,,MNL,,[][1].()[M].:.2006.[2].[M].:,2006.[3]WilliamsH.C.W.L..OntheFormationofTravelDe2mandModelsandEconomicEvaluationMeasuresofUserBenefit[J].EnvironmentandPlanning,1977,Vol.9A:285344.[4]McFaddenD..ModelingtheChoiceofResidentialLoca2tion[J].TransportationResearchPartB,1978,No.672:7277.[5]Bhat,ChandraR.AHeteroscedasticExtremeValueMod2elofIntercityTravelModeChoice[J].TransportationResearchPartB,1995,Vol.29:471483.[6]DanielMcFadden,KennethTrain.MixedMNLModelsforDiscreteResponse[J].JournalofAppliedEconometrics,2000,Vol.15,No.5:447470.[7]VoshvaP..ApplicationofCrossNestedLogitModeltoModechoiceinTelAvivIsraelMetropolitanArea[C].TransportationResearchRecord,1997,No.1607:615.[8]WenC.H.,KoppelmanF.S..TheGeneralizedNestedLogitModel[J]