Structural estimation for a route choice model with …bin.t.u-tokyo.ac.jp/model16/lecture/Oyama.pdfPath candidate set Large Small Pyo et al. (2001); Bierlaire et al. (2013) Oyama,

Structural estimation for a route choice model with uncertain measurement

Yuki Oyama* & Eiji Hato

*PhD student Behavior in Networks studies unit Department of Urban Engineering

School of Engineering, The University of Tokyo

September 25, 2016

Oyama, Y. (The University of Tokyo) Sep. 25, 2016 15th Behavior Modeling summer course

Jury 29, 2016 2

Outline Outline Outline

1.  Introduction

2.  Link-based route measurement model

3.  Structural estimation method

4.  Numerical examples

5.  Conclusions


Jury 29, 2016 3


1.  Introduction




5.  Conclusions


Jury 29, 2016 4

Motivation

Route choice analysis Introduction

GPS data

Route measurement model

Route choice data

Route choice model

Estimated parameters

Parameter estimation results largely depend on accuracy of route measurement


Jury 29, 2016 5

Motivation

Route choice analysis Introduction

GPS data


Route choice data

Route choice model


Parameter estimation results largely depend on accuracy of route measurement


Jury 29, 2016 6

1000 200 400 800(m)

N

Pedestrian route choice analysis Measurement uncertainty; Dense and high-resolution network

Motivation Introduction

•  Dense network

•  Spatial dependence of Measurement errors ü  Along riverü  Wide streetü  Narrow streetü  With arcade


Jury 29, 2016 7

Literature review

Route measurement models (1) Introduction

Sequential approach infers the true location at each data in chorological order

: GPS data


Jury 29, 2016 8

Literature review


Sequential approach can output meaningless paths


Jury 29, 2016 9

Literature review


Path-based probabilistic approach evaluates path likelihood regarding all GPS data included in a trip

50% 40% 10%

Pyo et al. (2001); Bierlaire et al. (2013)

Path 1 Path 2 Path 3

Domain of Data Relevance

Likelihood


Jury 29, 2016 10

Literature review


Path-based probabilistic approach suffers with trade-off between computational efficiency and measurement accuracy

Path candidate set

Large Small

Pyo et al. (2001); Bierlaire et al. (2013)


Jury 29, 2016 11

Literature review


−200 −100 0 100 200

0.00

0.01

0.02

0.03

0.04

0.05 x̂ j

t − x j = d

σ =10

σ = 50

σ =100

Distance from the true location

Mea

sure

men

t pro

bab

ility

p(x̂ j | x j ;σ ) =x̂ j − x jσ 2 exp −

x̂ j − x j2

2σ 2

"

#

$$

%

&

''

Assuming error variance constant on network distort measurement probabilities

PDF of GPS measurement error:

σ : Large σ : Small


Jury 29, 2016 12

Literature review Introduction

Route measurement models (3) Bayesian approach incorporates behavioral models into measurement models

50% 40% 10% Measurement

25% 60% 15% Prior (Route choice model)

16% 78% 6% Posterior

Chen et al. (2013); Danalet et al. (2014)


Jury 29, 2016 13


Route measurement models (3) Bayesian approach has a problem regarding parameter inconsistency

GPS data


Route choice data

Route choice model


Given parameters

Route choice model

Prob.

•  Arbitrary•  Previous study•  Other data


Jury 29, 2016 14


Route measurement models

•  Challenges: –  Disconnected path

•  Not suitable for route choice models

–  Setting of the measurement parameter•  Possible to miss the true path•  Ignorance of spatial difference distorts path likelihood

–  Parameter inconsistency of route choice model•  Estimated parameter includes biases regarding initial parameter


Jury 29, 2016 15

Framework

Framework Introduction

1.  Link-based route measurement model –  Matching each decomposed trip data to a link–  Estimating a measurement parameter for each link–  Incorporating a link-based route choice model as prior

2.  Structural estimation method –  Parameters at convergence satisfy the parameter consistency of

the route choice model


Jury 29, 2016 16


1.  Introduction




5.  Conclusions


Jury 29, 2016 17

Problem & Notation

Problem & Notation

•  GPS data –  Pair of coordinates　　　　　　with error variance–  Timestamp–  A given trip

•  Network –  Node　 : the horizontal position–  Link　　　　　　　: the vector of spatial attributes–  Network connection 　　 : 1/0

v ∈Va = (vu,vd )∈ A

δ(a ' | a)

G = (V,A)xv = {xlat, xlon}

ya

Link-based route measurement model

m̂ = (x̂, τ̂ )x̂ = (x̂lat, x̂lon )

τ̂

m̂ = (m̂1,..., m̂n,..., m̂N )

σ

Matching row GPS data 　　to the transportation network m̂ G


Jury 29, 2016 18

Link-based route measurement Flow

p(m̂t | at ) : Measurement probability of　　 given ; measurement equation m̂t atp(at | at−1) : Prior probability of　 given ; system equation 　 at−1at

Matching all data observed within a period to the same link

GPS data of a tripm̂1 m̂2 m̂n m̂N・・・・・・

m̂n = (x̂n, ˆn)m̂ = (m̂1,..., m̂n,..., m̂N )

Data decompopsition・・・・・・m̂J

1m̂11 m̂12 m̂JT

t =1

m̂t = (m̂1t,..., m̂jt ,..., m̂J

t )m̂ = (m̂1,..., m̂t,..., m̂T )

Estimation:

Path inference:

Link probabilities

a1 aTa2 ・・・ = [a1,...,at,...,aT ]

p(at | m̂t,at 1) p(m̂t | at; at )p(at | at 1; )

a1 a2 ataT



Jury 29, 2016

Link probability

19

Link probability

The probability of given measurements and state m̂t at−1

•  Candidate set: –  Calculate link probabilities for all

A(at−1) = {at |δ(at | at−1) =1}

m̂1t

m̂2t

m̂3t

at 1

1

2

3: GPS data

at ∈ A(at−1)

m̂t = {m̂1t, m̂2

t , m̂3t}

p(at | m̂t,at−1)

at



Jury 29, 2016

Measurement equation

20

Measurement equation

The probability of measurements given m̂t at

p(m̂t | at ) = p(x̂t | at )

•  Assumption:–  Timestamp has no measurement error; –  Measurement probability of data is independent from each other–  Traveler moves at the constant speed on the same link

τ̂

p(x̂1t,..., x̂J

t | at;σ at) = p(x̂ j

t | at;σ at)

j=1

J

∏

= p(x̂ jt | x j

t ,at;σ at)

x j∈at∫

j=1

J

∏ p(x j | at )dx j

PDF of GPS measurement error: Rayleigh distribution (van Diggelen, 2007)

p(x̂ jt | x j

t ,at;σ at) =

x̂ jt − x jσ at2 exp −

x̂ jt − x j

2

2σ at2

"

#

$$

%

&

''

p(m̂t | at;σ at)



Jury 29, 2016 21

Estimation of measurement parameter Measurement equation

σ at

Measurement likelihood maximization

σ at= argmax

σp(m̂t | at;σ )

Where,

p(m̂t | at;σ at) =

x̂ jt − x jσ at2 exp −

x̂ jt − x j

2

2σ at2

"

#

$$

%

&

''⋅ p(x j | at )

)

*+

,+

-

.+

/+x j∈at∫

j=1

J

∏ dx j

Link-based map matching can regard error variance as a link peculiar variable



Jury 29, 2016

System equation

22

System equation

The prior probability of given a state at

p(at | at−1;θ )at−1

Link-based route choice model

at 1

A(at 1)

1

2

3

u(at | at−1) = v(at | at−1)+ε(at ) =θyat |at−1 +ε(at )

Utility function:

v(⋅) : Deterministic component of utility

ε(⋅) : Probabilistic component of utility (i.i.d. gumbel distribution)

yat |at−1 : Vector of explanatory variables

θ : Vector of parameters



Jury 29, 2016

System equation

23

System equation

The prior probability of given a state at

p(at | at−1;θ )at−1

Link-based route choice model

at 1

A(at 1)

1

2

3

Choice probability option:

p(at | at−1) =ev(at |at−1 )

ev(at |at−1 )at∈A(at−1 )∑

Markov model

Recursive logit model

p(at | at−1) =ev(at |at−1 )+V

d (at )

ev(at |at−1 )+Vd (at )

at∈A(at−1 )∑

Fosgerau et al. (2013)

And others: e.g.,

Mai et al. (2015); Mai (2016); Oyama et al. (2016)



Jury 29, 2016

Link inference

24

Link inference

at−1•  Link (posterior) probability:

–  The probability of given measurements and a state

•  Link inference: –  Link likelihood maximization subject to switching condition

p(at | m̂t,at−1)∝ p(m̂t | at;σ at

)p(at | at−1;θ )

m̂tat

at = argmaxat∈A(at−1 )

p(at | m̂t,at−1)

s.t., maxa∈A(at )

p(m̂t+1 | a;σ a )> γ



Jury 29, 2016 25


1.  Introduction

2.  Link-based map matching algorithm



5.  Conclusions


Jury 29, 2016 26

Methodology Structural estimation method

A fixed point problem Need to solve a fixed point problem of route measurement and estimation

Route measurement

Route choice probability

Measurement probability

Path

Route choice model

LL(θ;ψ) = ln p(at | at−1;θ )δati

t=2

T

∏i∏#

$%

&

'(

θ = argmaxθ

LL(θ;ψ)

Where,

δati =

1, if ai,t ∈ψi

0, otherwise

"#$

p(at | at−1; !θ )

p(m̂t | at;σ at)

ψ

|θ − !θ |<ζs.t.,


Jury 29, 2016 27

Methodology

Structural estimation A method for parameter estimation of models with fixed point problem

Structural estimation method

•  NFXP (Nested Fixed Point)•  NPL (Nested Pseudo Likelihood)•  MPEC (Mathematical Programming with Equilibrium Constraint)•  …

Rust (1987) Aguirregabiria and Mira (2002)

Su and Judd (2012)

Structural estimation for route choice model with uncertain data

•  Solving a fixed problem regarding parameter of route choice model

•  Inner problem: Route measurement model •  Outer problem: Parameter estimation of route choice model


Jury 29, 2016 28

Methodology

No

Yes

pi (at | m̂it,at 1) =

p(m̂it | at; at )p(at | at 1; (h) )


at

ai,t = argmaxat

pi (at | m̂it,at 1; (h) )

(h+1) = argmaxLL( ; (h) )

(h)

(h+1) (h) <

(h)

h := h+1

GPS datah = 1,

Measurement model Behavior model

m̂

m̂i = (m̂i1,..., m̂i

t,..., m̂iT )

i(h) = [ai,1,...,ai,t,...,ai,T ]

maxLL( ) = log pi (at | at 1; ) ai,t(h )

t=2

T

i

I

ai,t(h) =

1, if ai,t i(h)

0, otherwise

(h) =

= (h), = (h)

Structural estimation A estimation method for solving a fixed point problem of route choice parameter

Structural estimation method

No

Yes




at

ai,t = argmaxat

pi (at | m̂it,at 1; (h) )

(h+1) = argmaxLL( ; (h) )

(h)

(h+1) (h) <

(h)

h := h+1

GPS datah = 1,

Measurement model Behavior model

m̂

m̂i = (m̂i1,..., m̂i

t,..., m̂iT )

i(h) = [ai,1,...,ai,t,...,ai,T ]

maxLL( ) = log pi (at | at 1; ) ai,t(h )

t=2

T

i

I

ai,t(h) =

1, if ai,t i(h)

0, otherwise

(h) =

= (h), = (h)


Jury 29, 2016 29


1.  Introduction




5.  Conclusions


Jury 29, 2016 30

Twins experiments | Simulation Twins experiment

DCaCCa

x

y

O

D

x = 40 x = 80o

dy = 90

y = 60

y = 30

x = 120

*(continuous cost: / discrete cost: / variance: )

(2/1/5) (1/1/5) (2/1/5)

(2/1/10) (2/1/10) (2/1/10)

(1/0/20) (1/0/20) (1/0/20)

(2/0/5)

(2/1/5) (2/1/5) (2/1/5) (3/1/5)

(3/1/5) (2/0/5) (2/1/5) (3/1/5)

(3/1/5) (3/1/5) (3/1/5) (2/1/5)

(2/0/5) (2/0/5)

a

x

y

x

y

x

y

(a) (b)

: True path : Measurement

(c)

v(a | k) =θ1TTa +θ2CCa +θ3DCa +θ4UTa|k!θ = [−0.1,−2,−1.5,−4]True parameter:

Settings

t = 30sPeriod interval:

τ̂ j − τ̂ j−1 =10sData generation:

Numerical examples


Jury 29, 2016 31

Table: Measurement accuracy and the difference of the parameter from the true value

Twins experiments | Measurement results Which model improves the route measurement accuracy ?

Twins experiment Numerical examples


Jury 29, 2016 32

Twins experiments | Estimation results Does structural estimation method improve the parameter estimation results ?

Twins experiment Numerical examples


Jury 29, 2016 33

1000 200 400 800(m)

N

Real data Matsuyama Probe Person data in 2007, 30 pedestrians, 729 locations

Real data Numerical examples


Jury 29, 2016 34

Real data | Model specification

•  Route choice model: (static) Markov model

•  Target: Pedestrian trip in city center

•  Utility function:

v(a | k) =θ1TTa +θ2CUa +θ3DUa +θ4UTa|k

CU :

DU :

Sidewalk width (m)

Arcade dummy variable


TT : Travel time (min.)

UT : U-turn dummy variable


Jury 29, 2016 35

Real data | Parameter estimation results Real data Numerical examples

•  Travel time ( ) seems to be significant from the result of one-way model, however,•  Structural estimation results show that links with arcade ( ) are the most likely to be

passed by pedestrians; travel time ( ) is not significant•  Other t-values and rho-square ( ) indicate that the structural estimation improves

parameter estimation results

Travel time (min.)Sidewalk width (m)

With arcadeU-turn

θ1θ3

θ1ρ2


Jury 29, 2016 36

Real data | Estimation results of error variance Estimated measurement error variance is dependent on each link


~ 20 (or no measurement)

20 ~ 30

30 ~ 50

50 ~ 100

100 ~

Estimation result of σ


Jury 29, 2016 37


1.  Introduction




5.  Conclusions


Jury 29, 2016 38

Conclusions and Future work Conclusions

•  Conclusions –  A link-based measurement model with route choice model–  Estimation of measurement parameter for each link–  Structural estimation method for solving a fixed point problem

regarding route choice parameters

•  Future work –  Comparison of computational efficiency with previous

measurement models–  Alternatives and utility of pedestrian link choice–  Characteristics of the fixed point problem

Conclusions


Jury 29, 2016

Thank you for attention! Questions?

39

[email protected]


Jury 29, 2016 40

References Reference Reference

1.  Bierlaire, M., Chen, J., Newman, J., 2013. A probabilistic map matching method for smartphone GPS data. Transportation Research Part C: Emerging Technologies 26, 78-98.

2.  Bierlaire, M., Frejinger, E., 2008. Route choice modeling with network-free data. Transportation Research Part C: Emerging Technologies 16(2), 187-198.

3.  Chen, J., Bierlaire, M., 2015. Probabilistic multimodal map matching with rich smartphone data. Journal of Intelligent Transportation System 19(2), 134-148.

4.  Danalet, A., Farooq, B., Bierlaire, M., 2014. A bayesian approach to detect pedestrian destination-sequences from WiFi signatures. Transportation Research Part C: Emerging Technologies 44, 146-170.

5.  Fuse, T., Nakanishi, W., 2012. A study on multiple human tracking by integrating pedestrian behavior model (in Japanese). Journal of JSCE Series D3: Infrastructure Planning and Management 68(2), 92-104.

6.  Fosgerau, M., Frejinger, E. and Karlstrm A., 2013. A link-based network route choice model with unrestricted choice set. Transportation Research Part B, 56, pp.70-80.

7.  Hunter, T., Abbeel, P., Bayen, A., 2014. The path inference filter: model-based low-latency map matching of probe vehicle data. IEEE Transactions on Intelligent Transportation Systems 15(2), 507-529.

8.  Pyo, J.S., Shin, D.H., Sung, T.K., 2001. Development of a map matching method using the multiple hypothesis technique. IEEE Proceedings on Intelligent Transportation Systems, 23-27.

9.  Quddus, M.A., Noland, R.B., Ochieng, W.Y., 2005. Validation of map matching algorithms using high precision positioning with GPS. The Journal of Navigation 58(2), 257-271.

10.  Quddus, M.A., Ochieng, W.Y., Noland, R.B., 2007. Current map-matching algorithms for transport applications: State-of-the art and future research directions. Transportation Research Part C: Emerging Technologies 15(5), 312-328.

11.  Quddus, M.A., Ochieng, W.Y., Zhao, L., Noland, R.B., 2003. A general mao matching algorithm for transport telematics applications. GPS Solutions 7(3), 157-167.

12.  Quddus, M.A., Washington, S., 2015. Shortest path and vehicle trajectory aided mapmatching for low frequency GPS data. Transportation Research Part C: Emerging Technologies 55, 328-339.

13.  van Diggelen, F., 2007. GNSS accuracy: lies, damn lies, and statistics. GPS World 18(1), 26-32.14.  Velaga, N.R., Quddus, M.A., Bistow, A.L., 2009. Developing an enhanced weight-based topological map-matching

algorithm for intelligent transport systems. Transportation Research Part C: Emerging Technologies 17, 672-683. 15.  White, C.E., Bernstein, D., Kornhauser, A.L., 2000. Some map matching algorithms for personal navigation assistants.

Transportation Research Part C: Emerging Technologies 8, 91-108.


Jury 29, 2016

Appendix

41 Oyama, Y. (The University of Tokyo) Sep. 25, 2016 15th Behavior Modeling summer course

Jury 29, 2016 42

Previous route measurement models Literature review

Geometric -  Point-to-point -  Point-to-curve -  Curve-to-curve

Introduction

White et al. (2000)

Topological -  Adjacency -  Connectivity -  Vehicle heading Greenfield (2002)

-  Error region -  Fuzzy logic

Probabilistic Ouchieng et al. (2004) Quddus et al. (2006)

-  MHT

-  Measurement equation

Pyo et al. (2006) Bielraire et al. (2013)

70%

30%

Path-based


Jury 29, 2016 43

Methodology

m̂13

m̂23

m̂33

m̂12

m̂22

m̂32

m̂21m̂1

1

m̂31

: Estimated path: More likely path

31

2

4

5

68

11

9

12

10

7

Link switching | errors Difficulties regarding link connectivity because of myopic optimization



Jury 29, 2016 44

Link switching | algorithm Link switching

[at,1,...,at,r,...,at,|A(at 1 )| ]

STEP1: Calculating probabilities

STEP3: Calculating measurement equation at (t+1)

where,

STEP2: Sorting candidates by probabilities

r = 1

r := r+1

r := 1

No

No

YesYes

at = at,r

at+1 = argmaxa

p(a | m̂t+1,at,r )

LLmrJ < r =| A(at 1) |

p(at,1) p(at,r ) p(at,|A(at 1 )| )


p(m̂it | at; at )p(at | at 1; )p(m̂i

t | at; at )p(at | at 1; )at A(at 1 )

LLmr = log p(m̂t+1 | at+1; at+1 )( )



Jury 29, 2016 45

s Twins experiments | iterations

0

2.0

4.0

6.0

1 2 3 4 5 6 7 8

0

20

40

60

80

100

1 2 3 4 5 6 7 8 0

5.0

10.0

15.0

20.0

25.0

1 2 3 4 5 6 7 8

Link

accu

racy

(%)

Sum.

Ave.

Estim

atio

n er

ror:

Estim

atio

n er

ror:

a

: input

: input

#iteration

(wrong parameters)

(no information)

(true parameters): input

= [0, 0, 0, 0]

= [ 0.1, 2, 1.5, 4]

= [ 1.5, 0.1, 2, 10]



Documents

Structural estimation for a route choice model with …bin.t.u-tokyo.ac.jp/model16/lecture/Oyama.pdfPath candidate set Large Small Pyo et al. (2001); Bierlaire et al. (2013) Oyama,