open-form discrete choice models - 都市生活学・ネッ...

Open-formdiscretechoicemodels

GenealogyofDCMs(again) 26

DifferenceofGEVandNon-GEV 27

Multinomial Logit (MNL)

Luce(1959), McFadden(1974) Not consider correlation of choice alternatives (IIA)

Easy and fast estimation High operability (easy evaluation for new additional choice alternative benefit of IIA)

2008/9/20 7 18

MNL2

IIDbusrail

busbus

carcar

XrailUXbusUXcarU

Cjj

i

VViP

expexp

closed-form

Cov(U)

2

22

2

2

2

600

0000

railVbusVcarVcarVcarP

expexpexpexp

Multinomial Probit (MNP)

Thurstone(1927) Consider correlation of choice alternates based on Variance-Covariance matrix

Hard and slow estimation (need calculation of multi-dimensional interrelation depend on N of alternatives')

2008/9/20 7 16

MNP3

busrail

busbus

carcar

XrailUXbusUXcarU

Open-formIdentification

1211

1

21

exp2

1

1

1

J

V V

Jii

i

Jii

JddiP

GEV model (Closed-form) Non-GEV model (Open-form)

Non-GEV model has high power of expression, however parameter estimation cost is high.

StructuredCovarianceMNP(1) 28

Proposed new probit type railway route choice model considering overlapping problem. (1993, 1998)

This model applied to practical demand forecasting in real Tokyo network, and it used for decision making of railway policy toward 2015. (2000)

Multinomial Probit with Structured Covariance for Route Choice Behavior, Transportation Research Part B, Vol.31, No.3, pp195-207, 1997.

Prof.Morichi Prof.Yai Prof.Iwakura


Tokyo Metropolitan has highly dense railway network ! route overlapping problem

Railway lineabout 130Stationabout 1800Passengers40miliion/dayCong. rate: max over 200%


In the overlap network that has correlation between routes, Logit model is susceptible to error by IIA property.

Probit is better ? Difficult to setting covariance matrix for each OD pair

structured covariance by divide into two error Difficult to parameter estimation. (multi-dimensional Integral)

reduce computational time using simulation methods

i = iLength +i

Route

Error of depend on route length

Error of route specific

Ui =Vi +i

The Operations Research Society of Japan

NII-Electronic Library Service

Overlapcorrelation


Error of depend on route lengthVariance of route utility increases in proportion to the route length.

Error of depend on route lengthError of route specific

Variance-Covariance structure in Error term

Covariance between routes increases in proportion to the length of route overlap.

Error of route specificindependent of each routecov0

Lr :length of route rLrq :overlap length between route r and q2 :variance of unit length

Simplifyusecov.ratio

Estimate only cov. ratio !


Apply to the SCMNL for The 18th master plan for urban railway network in TMA (2000)

ExOomiya to Kanda station

Estimation results

To achieve a high prediction accuracy by the relaxation of route overlapObs 10% in all route

: Shonan-Shinjuku line

: Takasaki line

: Keihin-Tohoku line

: Utsunomiya line

: Yamanote line

(1) Omiya (2) Akabane (3) Ueno (4) Tokyo

Saitama Pref.

Inner Yamanote line

(1)

(2) (3)

(4)

Prediction results

Ui =Vi +i + i

MixedLogit Model 33

Mixed Loigt (Train 2000)High flexible structure using two error term.Utility function

Error Component: approximate to any GEV model Random Coefficient: Consider the heterogeneity

dist.: basically assume Normal dist. In the case of normal distribution takes a non-realistic value, it can assume a variety of probability distribution (triangular distribution, cutting normal distribution, lognormal distribution, Rayleigh distribution, etc.).

v dist.: assume any G function IID Gamble (Logit Kernel) MNLany G function (GEV Kernel) NL, PCL, CNL, GNL

Ucar = Xcar + +carUbus = Xbus + transittransit +busUrail = X rail + transittransit + rail

ErrorComponent:NL(1) 34

Approximation of Nested Logit (NL)

2008/9/20 7 33

MXL5

222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit

railtransittransitrailrail

bustransittransitbusbus

carcarcar

XUXUXU

1,0Ntransit

Nested

NL!!

2008/9/20 7 33

MXL5

222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit



carcarcar

XUXUXU

1,0Ntransit

Nested

NL!!

IID Gamble Logit

Describe the nest (covariance) using structured . Ex: model choice

CarBus

RailTransit nest

Prail =eVrail+ transittransit

eVcar + eVbus+ transittransit + eVrail+ transittransittransit f transit( )dtransit

Prail =1N

eVrail+ transittranistN

eVcar + eVbus+ transittranistN

+ eVrail+ transittranistN

N

Normal nest

Choice prob.(open-form)

Choice prob.(Simulated)

ErrorComponent:NL(2) 35

Approximation of Nested Logit (NL)

2008/9/20 7 33

MXL5

222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit



carcarcar

XUXUXU

1,0Ntransit

Nested

NL!!

Note that variance-covariance matrix is inconsistent with normal NL

CarBus

RailTransit nest

0 0 00 transit

2 transit2

0 transit2 transit

2

!

"

####

$

%

&&&&

+

2 0 00 2 00 0 2

!

"

####

$

%

&&&&=

2 0 00 transit

2 + 2 transit2

0 transit2 transit

2 + 2

!

"

####

$

%

&&&&

2 0 00 2 transit

2

0 transit2 2

!

"

####

$

%

&&&&

v

Normal NL

Approximated NL based on MXL

Diagonal elements (variance of Bus and Rail) is bigger than transit

road2 road

2 0

road2 road

2 + transit2 transit

2

0 transit2 transit

2

!

"

####

$

%

&&&&

+

2 0 00 2 00 0 2

!

"

####

$

%

&&&&

=

road2 + 2 road

2 0

road2 road

2 + transit2 + 2 transit

2

0 transit2 transit

2 + 2

!

"

####

$

%

&&&&

ErrorComponent:CNL 36

Approximation of Cross Nested Logit (CNL)

2008/9/20 7 34

MXL6

222

22222

222

2

2

2

22

2222

22

0

0

000000

0

0

transittransit

transitroadtransitroad

roadroad

transittransit


roadroad


busroadroadtransittransitbusbus

carroadroadcarcar

XUXUXU

1,0, Nroadtransit

Cross-Nested

CNL!!2008/9/20 7 34

MXL6

222

22222

222

2

2

2

22

2222

22

0

0

000000

0

0

transittransit


roadroad

transittransit


roadroad


busroadroadtransittransitbusbus

carroadroadcarcar

XUXUXU

1,0, Nroadtransit

Cross-Nested

CNL!!

v

2008/9/20 7 33

MXL5

222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit



carcarcar

XUXUXU

1,0Ntransit

Nested

NL!!

CarBus

RailTransit nest

Road nest

Describe the nest (covariance) using structured .

Uzorn1 = X zorn1 + 1 + 2 + zorn1Uzorn2 = X zorn2 + 1 + 2 + 3 + zorn2Uzorn3 = X zorn3 + 2 + 3 + 4 + zorn3Uzorn4 = X zorn4 + 3 + 4 + zorn4

ErrorComponent:SCL 37

Approximation of Spatial Correlation Logit

2008/9/20 7 35

MXL7

220

20

20

220

20

20

220

20

20

220

2

2

2

2

20

20

20

20

20

20

20

20

20

20

0020

0200

000000000000

0020

0200

43044

3302033

2201022

11011

zonezonezone

zonezonezone

zonezonezone

zonezonezone

XUXUXUXU

1,0,, 321 N

O

2008/9/20 7 35

MXL7

220

20

20

220

20

20

220

20

20

220

2

2

2

2

20

20

20

20

20

20

20

20

20

20

0020

0200

000000000000

0020

0200

43044

3302033

2201022

11011

zonezonezone

zonezonezone

zonezonezone

zonezonezone

XUXUXUXU

1,0,, 321 N

O

2 02 0

2 0

02 3 0

2 02 0

0 02 3 0

2 02

0 0 02 2 0

2

!

"

######

$

%

&&&&&&

+

2 0 0 00 2 0 00 0 2 00 0 0 2

!

"

#####

$

%

&&&&&

=

02 + 2 0

2 0 0

02 2 0

2 + 2 02 0

0 02 2 0

2 + 2 02

0 0 02 0

2 + 2

!

"

######

$

%

&&&&&&

v

Describe the spatial correlation using structured .

zorn

1 zorn

2 zorn

3 zorn

4

ErrorComponent:HL 38

Approximation of heteroscedastic LogitAssume the different error variance in each alternatives'Identification problem occur in the case of not fixed one of the parameters to zero at least.

2008/9/20 7 37

MXL9

22

22

22

2

2

2

2

2

2

000000

000000

000000

rail

bus

car

rail

bus

car

railrailrailrailrail

busbusbusbusbus

carcarcarcarcar

XUXUXU

1,0,, Nrailbuscar

Identification0

2008/9/20 7 37

MXL9

22

22

22

2

2

2

2

2

2

000000

000000

000000

rail

bus

car

rail

bus

car

railrailrailrailrail

busbusbusbusbus

carcarcarcarcar

XUXUXU

1,0,, Nrailbuscar

Identification0

car2 + 2 0 0

0 bus2 + 2 0

0 0 rail2 + 2

!

"

####

$

%

&&&&

Rail: High travel time reliabilityError variance is small

consider only heteroscedasitcIID assumption is not relaxed

Car: Low travel time reliabilityError variance is large

Assume heteroscedastic in error bus2 =1FIX !

2008/9/20 7 38

MXL10

2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

21

20

11

10

,

,GroupGroup

GroupGroup

1

0 0+ 1

2 1Group1Group2

ncarncarnncar TU ,,,

ncarncarncar TU ,,,

RandomCoefficient(1) 39

Taste heterogeneity of decision makerParameters defined homogeneously in population. However, decision maker n has different taste ( = heterogeneity)

2008/9/20 7 38

MXL10

2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

21

20

11

10

,

,GroupGroup

GroupGroup

1

0 0+ 1

2 1Group1Group2


ncarncarncar TU ,,,

2008/9/20 7 38

MXL10

2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

21

20

11

10

,

,GroupGroup

GroupGroup

1

0 0+ 1

2 1Group1Group2


ncarncarncar TU ,,,

Segmentation (observable heterogeneity) Constant by gender

parameter by gender

2008/9/20 7 38

MXL10

2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

21

20

11

10

,

,GroupGroup

GroupGroup

1

0 0+ 1

2 1Group1Group2


ncarncarncar TU ,,,

Females constant; 0

males constant:0+1

Females parameter: 2

males parameter:1

1 - malen = femalen

RandomCoefficient(2) 40

Parameter distribution (unobservable heterogeneity)Assume the heterogeneity of parameter In the case of parameter following Normal dist., we estimate the dist.s hyper-parameter (mean and variance).

2008/9/20 7 39

MXL11

nrailnrailnnrailnrail

nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

2

2,Nn

parameterunknown:,

nn income10

2008/9/20 7 39

MXL11


nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

2

2,Nn

parameterunknown:,

nn income10

2008/9/20 7 39

MXL11


nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

2

2,Nn

parameterunknown:,

nn income10

2008/9/20 7 39

MXL11


nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

2

2,Nn

parameterunknown:,

nn income10 Hyper-parameter can describe using observable variables

2008/9/20 7 39

MXL11


nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

2

2,Nn

parameterunknown:,

nn income10

depend on observable income variable2008/9/20 7 39

MXL11


nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

2

2,Nn

parameterunknown:,

nn income10

Summaryofopen-formmodels 41

Strengths

v Describe correlation between alternatives by EC MNP: all alternatives (relax and reduce by structuring) MXL: depend on approximated model

v Describe heterogeneity by RC Segmentation, parameter distribution

Limitations

v High calculation cost in parameter estimation Open-form model has high dimensional integration. Recently, proposed high speed estimation methodsEx: Bayesian estimation (MCMC) see Trains book

MACML: analytical integration by Bhat et al.(2011)

Reference 42

Yai,T.,Iwakura,S.,&Morichi,S.:Multinomialprobit withstructuredcovarianceforroutechoicebehavior.TransportationResearchPartB:Methodological,31(3),pp.195-207,1997.

Morichi,S.,Iwakura,S.,Morishige,S.,Itoh,M.,&Hayasaki,S.:Tokyometropolitanrailnetworklong-rangeplanforthe21stcentury.TransportationResearchBoard,(01-0475),2001.

Train,K.E.:Discretechoicemethodswithsimulation.Cambridgeuniversitypress,2009.

Revelt,D.,&Train,K.:.Mixed logitwithrepeatedchoices:households'choicesofapplianceefficiencylevel.Reviewofeconomicsandstatistics,80(4),pp.647-657,1998.

Ben-Akiva,M.,Bolduc,D.,&Walker,J.:Specification,identificationandestimationofthelogitkernel(orcontinuousmixedlogit)model.DepartmentofCivilEngineeringManuscript,MIT,2001.

Walker,J.L.,Ben-Akiva,M.,&Bolduc,D.:Identificationofparametersinnormalerrorcomponentlogit-mixture(NECLM)models.JournalofAppliedEconometrics,22(6),pp.1095-1125,2007.

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