Open-form discrete choice modelsbin.t.u-tokyo.ac.jp/model15/lecture/Yaginuma.pdf• Hard and slow...

Open-form discrete choice models

3.##Genealogy#of#DCMs#(again) 28

Non-GEV (Open-form) Probit model

3.##Difference#of#GEV#and#Non;GEV 30

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 第7回行動モデル夏の学校 18

多項ロジット（MNL）モデル（2）

ε～IIDガンベル分布

busrail

busbus

carcar

XrailUXbusUXcarU

Cjj

i

VViP

expexp

シェア型モデルであるため直感的にわかりやすい closed-formのため計算コストが安い

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')


多項プロビット（MNP）モデル（3） busrail

busbus

carcar

XrailUXbusUXcarU

ε～多変量正規分布

Open-formのため計算コストが高い Identificationの問題から推定可能なパラメータは限られており解釈が困難

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.

3.#Structured#Covariance#MNP#(1)# 31

•  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

3.#Structured#Covariance#MNP#(2)## 32

Tokyo Metropolitan has highly dense railway network ! ⇒ route overlapping problem

Railway line：about 130 Station：about 1800 Passengers：40miliion/day Cong. rate: max over 200%

3.#Structured#Covariance#MNP#(3)## 33

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

εi = εiLength +εi

Route

Error of depend on route length

Error of route specific

Ui =Vi +εi

The Operations Research Society of Japan

NII-Electronic Library Service

Overlap# correlaJon


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

Error of depend on route length Error of route specific

Variance-Covariance structure in Error term

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

Error of route specific ・independent of each route（cov＝0）

Lr :length of route r Lrq :overlap length between route r and q σ2 :variance of unit length

Simplify#use##cov.#raJo

Estimate only cov. ratio !


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

Ex：Oomiya to Kanda station

Estimation results

To achieve a high prediction accuracy by the relaxation of route overlap（Obs ±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

Non-GEV (Open-form) Mixed Logit

Ui =Vi +ηi +ν i

Mixed#Logit#Model## 37

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

η 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) ⇒ MNL ・any G function (GEV Kernel) ⇒ NL, PCL, CNL, GNL…

Ucar = βXcar + +νcarUbus = βXbus +σ transitηtransit +νbusUrail = βX rail +σ transitηtransit +ν rail

Error#Component:#NL#(1) 38

Approximation of Nested Logit (NL)


ミックストロジット（MXL）モデル（5）

222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit

railtransittransitrailrail

bustransittransitbusbus

carcarcar

XUXUXU

1,0Ntransit

Nested自動車

バス

鉄道

NLモデルとは違う!!



222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit



carcarcar

XUXUXU

1,0Ntransit

Nested自動車

バス

鉄道


IID Gamble ⇒ Logit

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

Car Bus Rail

Transit nest

Prail =eVrail+σ transitηtransit

eVcar + eVbus+σ transitηtransit + eVrail+σ transitηtransitηtransit∫ f ηtransit( )dηtransit

Prail =1N

eVrail+σ transitηtranistN

eVcar + eVbus+σ transitηtranistN

+ eVrail+σ transitηtranistN

N∑

Normal ⇒ nest

Choice prob. (open-form)

Choice prob. (Simulated)

Error#Component:#NL#(2) 39

Approximation of Nested Logit (NL)



222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit



carcarcar

XUXUXU

1,0Ntransit

Nested自動車

バス

鉄道


Note that variance-covariance matrix is inconsistent with normal NL

Car Bus Rail

Transit nest

0 0 00 σ transit

2 σ transit2

0 σ transit2 σ transit

2

!

"

####

$

%

&&&&

+

σ 2 0 00 σ 2 00 0 σ 2

!

"

####

$

%

&&&&=

σ 2 0 00 σ transit

2 +σ 2 σ transit2


2 +σ 2

!

"

####

$

%

&&&&

η

σ 2 0 00 σ 2 σ transit

2

0 σ transit2 σ 2

!

"

####

$

%

&&&&

v ε

Normal NL

Approximated NL based on MXL

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

σ road2 σ road

2 0

σ road2 σ road

2 +σ transit2 σ transit

2


2

!

"

####

$

%

&&&&

+

σ 2 0 00 σ 2 00 0 σ 2

!

"

####

$

%

&&&&

=

σ road2 +σ 2 σ road

2 0

σ road2 σ road

2 +σ transit2 +σ 2 σ transit

2


2 +σ 2

!

"

####

$

%

&&&&

Error#Component:#CNL# 40

Approximation of Cross Nested Logit (CNL)



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自動車

バス

鉄道

CNLモデルとは違う!!2008/9/20 第7回行動モデル夏の学校 34


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自動車

バス

鉄道

CNLモデルとは違う!!

η

v

ε



222

222

2

2

2

2

22

22

00

00

000000

00

000

transittransit

transittransit

transittransit

transittransit



carcarcar

XUXUXU

1,0Ntransit

Nested自動車

バス

鉄道


Car Bus Rail

Transit nest

Road nest

Describe the nest (covariance) using structured η.

η

Uzorn1 = βX zorn1 +σ η1 +σ η2 +ν zorn1Uzorn2 = βX zorn2 +σ η1 +σ η2 +σ η3 +ν zorn2Uzorn3 = βX zorn3 +σ η2 +σ η3 +σ η4 +ν zorn3Uzorn4 = βX zorn4 +σ η3 +σ η4 +ν zorn4

Error#Component:#SCL 41

Approximation of Spatial Correlation Logit



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

その他１：空間的な相関

O



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

その他１：空間的な相関

O

2σ 02 σ 0

2 0

σ 02 3σ 0

2 σ 02 0

0 σ 02 3σ 0

2 σ 02

0 0 σ 02 2σ 0

2

!

"

######

$

%

&&&&&&

+

σ 2 0 0 00 σ 2 0 00 0 σ 2 00 0 0 σ 2

!

"

#####

$

%

&&&&&

=

σ 02 +σ 2 σ 0

2 0 0

σ 02 2σ 0

2 +σ 2 σ 02 0

0 σ 02 2σ 0

2 +σ 2 σ 02

0 0 σ 02 σ 0

2 +σ 2

!

"

######

$

%

&&&&&&

η v ε

Describe the spatial correlation using structured η.

zorn1

zorn2

zorn3

zorn4

Error#Component:#HL# 42

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



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

異分散

Identificationの問題で，一つのσは0に固定する必要あり



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

異分散

Identificationの問題で，一つのσは0に固定する必要あり

σ car2 +σ 2 0 0

0 σ bus2 +σ 2 0

0 0 σ rail2 +σ 2

!

"

####

$

%

&&&&

ε

・Rail: High travel time reliability 　⇒Error variance is small

※consider only heteroscedasitc（IID assumption is not relaxed）

・Car: Low travel time reliability 　⇒Error variance is large

Assume heteroscedastic in error σ bus2 =1FIX !



2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

2120

11

10

,

,GroupGroup

GroupGroup

嗜好の異質性：ランダム係数モデル（1）βは母集団で同一⇔嗜好は母集団で同質と仮定

嗜好には異質性（個人差）が存在

観測異質性非観測異質性

女性の定数項：α0 男性の定数項：α0+ α1

女性のﾊﾟﾗﾒｰﾀ：β2 男性のﾊﾟﾗﾒｰﾀ：β1Group1のﾊﾟﾗﾒｰﾀﾍﾞｸﾄﾙ：Group2のﾊﾟﾗﾒｰﾀﾍﾞｸﾄﾙ：

ncarncarnncar TU ,,,

ncarncarncar TU ,,,

アプリオリ・マーケット

セグメンテーション

Random#Coefficient#(1) 43

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



2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

2120

11

10

,

,GroupGroup

GroupGroup







ncarncarncar TU ,,,


セグメンテーション2008/9/20 第7回行動モデル夏の学校 38


2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

2120

11

10

,

,GroupGroup

GroupGroup







ncarncarncar TU ,,,



Segmentation (observable heterogeneity)

・Constant by gender：・parameter by gender：



2,,

21

20

2,

1,,

11

10

1,

,,2,10,

,,110,

*1***

Groupncarncar

GroupGroupGroupncar

Groupncarncar

GroupGroupGroupncar

ncarncarnncarnncar

ncarncarnncar

TU

TU

TmaleTmaleU

TmaleU

2120

11

10

,

,GroupGroup

GroupGroup







ncarncarncar TU ,,,



Female’s constant; α0

male’s constant:α0+α1

Female’s parameter: β2

male’s parameter:β1

※ 1 - malen = femalen

Random#Coefficient#(2) 44

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).



nrailnrailnnrailnrail

nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn

嗜好の異質性：ランダム係数モデル（2）

2, Nn

β

parameterunknown:,

ＩＩＤガンベル分布

nn income10 としても良い観測異質性と非観測異質性の両方を考慮




nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn


2, Nn

β

parameterunknown:,






nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn


2, Nn

β

parameterunknown:,






nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn


2, Nn

β

parameterunknown:,


nn income10 としても良い観測異質性と非観測異質性の両方を考慮Hyper-parameter can describe using observable variables




nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn


2, Nn

β

parameterunknown:,



βdepend on observable income variable 2008/9/20 第7回行動モデル夏の学校 39



nbusnbusnnbusnbus

ncarncarnncarncar

TTU

TTU

TTU

,,,,

,,,,

,,,,


1,0Nn


2, Nn

β

parameterunknown:,



Summary#of#open;form#models# 45

Strengths

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

! Describe heterogeneity by RC •  Segmentation, parameter distribution…

Limitations

! High calculation cost in parameter estimation •  Open-form model has high dimensional integration. •  Recently, proposed high speed estimation methods Ex: Bayesian estimation (MCMC) ⇒ see Train’s book MACML: analytical integration by Bhat et al.(2011)

References• Ben-Akiva, M. (1973) Structure of passenger travel demand models. Ph.D. Thesis,

Massachusetts Institute of Technology. Dept. of Civil and Environmental Engineering (http://hdl.handle.net/1721.1/14790 ).

• Bhat, C.R. (1995) A heteroscedastic extreme value model of intercity travel mode choice. Transportation Research Part B 29, 471-483.

• Bierlaire, M. (2002) The Network GEV model. Proceedings of the 2nd Swiss Transportation Research Conference, Ascona, Switzerland.

• Cardell, N.S., Dunbar, F.C. (1980) Measuring the societal impacts of automobile downsizing. Transportation Research Part A: General 14, 423-434.

• Castillo, E., Menendez, J.M., Jimenez, P., Rivas, A. (2008) Closed form expressions for choice probabilities in the Weibull case. Transportation Research Part B 42, 373-380.

• Daly, A. (2001) Recursive nested EV model. ITS Working Paper 559, Institute for Transport Studies, University of Leeds.

• Daly, A., Bierlaire, M. (2006) A general and operational representation of Generalised Extreme Value models. Transportation Research Part B: Methodological 40, 285-305.

• Fosgerau, M., McFadden, D., Bierlaire, M. (2013) Choice probability generating functions. Journal of Choice Modelling 8, 1-18.

• Hato, E. (2002) Behaviors in network, Infrastructure Planning Review, 19-1, 13-27 (in Japanese).

• Koppelman, F.S., Wen, C.-H. (2000) The paired combinatorial logit model: properties, estimation and application. Transportation Research Part B: Methodological 34, 75-89.

28

References• Li, B. (2011) The multinomial logit model revisited: A semi-parametric approach in discrete

choice analysis. Transportation Research Part B 45, 461-473.• Luce, R. (1959) Individual Choice Beahviour. John Wiley, New York.• Mattsson, L.-G., Weibull, J.W., Lindberg, P.O. (2014) Extreme values, invariance and choice

probabilities. Transportation Research Part B: Methodological 59, 81-95.• McFadden, D., (1978) Modelling the choice of residential location, in: Karlqvist, A., Lundqvist, L.,

Snickars, F., Weibull, J. (Eds.), Spatial Interaction Theory and Residential Location. North-Holland, Amsterdam.

• McFadden, D. (1989) A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration. Econometrica 57, 995-1026.

• Nakayama, S. (2013) q-generalized logit route choice and network equilibrium model. Proceedings of the 20th International Symposium on Transportation and Traffic Theory (Poster Session).

• Thurstone, L.L. (1927) A law of comparative judgment. Psychological Review 34, 273-286.• Train, K. (2009) Discrete Choice Methods with Simulation, 2nd Edition ed. Cambridge

University Press.• Vovsha, P. (1997) Cross-nested logit model: an application to mode choice in the Tel-Aviv

metropolitan area. Transportation Research Board, Presented at the 76th Annual Meeting, Washington DC.

• Wen, C.-H., Koppelman, F.S. (2001) The generalized nested logit model. Transportation Research Part B 35, 627-641.

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