A sampling of alternatives approach for route
choice models
Emma Frejinger
Centre for Transport Studies, KTH, Stockholm
Joint work with Michel Bierlaire and Moshe Ben-Akiva
Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models – p. 1/34
Short about CTS at KTH
• A collaboration between
• KTH (Transport and Location Analysis, Transport
and Logistics)
• VTI Transport Economics
• SIKA, WSP, JIBS
• 10 year financing
• VINNOVA, Rail adm., Road adm., Rikstrafiken
• The partners
• Model development, appraisal methodology, applied
appraisal and analysis
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Outline
• Introduction to route choice modeling and choice set
generation
• Sampling of alternatives and derivation of sampling
correction
• Stochastic path generation
• Expanded path size attribute
• Numerical results
• Conclusions and future work
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Introduction
Given
• a uni-modal transportation network
• an origin-destination pair
• link and path attributes
• socio-economic characteristics of a traveler
identify the route a (given) traveler would select
• Important problem in e.g. intelligent transport systems,
GPS navigation and transportation planning
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Introduction
• Most simple route choice model: travelers use the
shortest path with respect to any generalized cost
function
• Behaviorally unrealistic
• Random utility models
• Here we consider estimation of random utility models
based on disaggregate revealed preferences data
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Introduction
Network
Trips
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Introduction
Network
Trips
Path
Observations
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Introduction
Trips
Network
Choice sets
Path
Observations
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Introduction
Trips
Network
Choice sets
Path
Observations
Description of
correlation
Route choice model
Seminar, Namur, February 2009, A sampling of alternatives approach for route choice models – p. 6/34
Introduction
generation
formation
model
Route choice Choice set
Path
Set of all paths U from o to d
Deterministic
Stochastic
M⊆U
Mn ⊆ U
Deterministic
P (i|Cn ) P (i) =
Probabilistic
X
P (i|Cn )P (Cn )
Cn ∈Gn
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Introduction
• Underlying assumption in existing approaches: the
actual choice set is generated
• Empirical results suggest that this is not always true
• In order to avoid bias in the econometric model we
propose
• True choice set = universal set U
• Too large
• Sampling of alternatives
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Sampling of Alternatives
• Multinomial Logit model can be consistently estimated
on a subset of alternatives (McFadden, 1978)
q(Cn |i)P (i)
eµVin +ln q(Cn |i)
P (i|Cn ) = X
= X
q(Cn |j)P (j)
eµVjn +ln q(Cn |j)
j∈Cn
j∈Cn
Cn : set of sampled alternatives
q(Cn |j): probability of sampling Cn given that j is the
chosen alternative
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Importance Sampling of Alternatives
• Attractive paths have higher probability of being
sampled than unattractive paths
• Path utilities must be corrected in order to obtain
unbiased estimation results
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MNL Route Choice Models
• Path Size Logit (Ben-Akiva and Ramming, 1998 and
Ben-Akiva and Bierlaire, 1999) and C-Logit (Cascetta
et al. 1996)
• Additional attribute in the deterministic utilities
capturing correlation among alternatives
• These attributes should reflect the true correlation
structure
• Here we use the Path Size Logit with an Expanded
Path Size attribute
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Sampling Correction
• Based on Ben-Akiva (1993)
• Sampling protocol
1. A set Cen is generated by drawing Rn paths with
replacement from the universal set of paths U
2. Add chosen path to Cen
• Outcome of sampling: (e
k1n , e
k2n , . . . , e
kJn ) and
P
e
j∈U kjn = Rn
Y
R
!
e
n
e
e
e
P (k1n , k2n , . . . , kJn ) = Q
q(j)kjn
e
kjn !
j∈U
j∈U
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Sampling Correction
• Alternative j appears kjn = e
kjn + δjc in Cen
• Let Cn = {j ∈ U | kj > 0}
Y
ki
R!
ki −1
kj
e
Y
q(i)
q(j) = KCn
q(Cn |i) = q(Cn |i) =
q(i)
(ki − 1)!
kj !
j∈Cn
j6=i
j∈Cn
j6=i
KCn =
Q R!
j∈Cn
kj !
Q
kj
q(j)
j∈Cn
k
i
µVin +ln( q(i)
)
e
P (i|Cn ) = X
e
“ k ”
j
µVjn +ln q(j)
j∈Cn
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Sampling Correction
• General methodology valid for any definition of U if
• it exists an algorithm generating any path in U with
non zero probability
• path sampling probabilities can be computed
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Stochastic Path Generation
• Biased (towards shortest path) random walk algorithm
• A measure of distance xℓ ∈ [0, 1] to the shortest path
for link ℓ = (v, w)
SP (v, sd )
xℓ =
C(ℓ) + SP (w, sd )
• Parametrized function mapping [0, 1] into itself (inspired
from the Kumaraswamy distribution)
ω(ℓ|b1 , b2 ) = 1 − 1 −
b1 b2
xℓ
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Stochastic Path Generation
1.0
ω(ℓ|b1 , b2 )
b1 = 1, 2, 5, 10, 30
b2 = 1
1
2
5
10 30
0
0
xℓ
1.0
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Stochastic Path Generation
Initialize
Loop
v = so , Γ = ∅
While v 6= sd perform the following
For each link ℓ = (v, w) ∈ Ev , where Ev is the set of
outgoing links from v, we compute the weights ω(ℓ|b1 , b2 )
Weights
For each link ℓ = (v, w) ∈ Ev , we compute
Probability
ω(ℓ|b1 , b2 )
q(ℓ|Ev , b1 , b2 ) = P
m∈Ev ω(m|b1 , b2 )
Randomly select a link (v, w∗ ) in Ev based on the
above probability distribution
Draw
Update path
Next node
Γ = Γ ∪ (v, w∗ )
v = w∗ .
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Stochastic Path Generation
• Probability q(j) of generating path j is
Y
q(j) =
q(ℓ|Ev , b1 , b2 )
ℓ∈Γj
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Expanded Path Size
• Path size attribute should reflect the “true” correlation
structure
• Original formulation
PSCin
X La 1
=
,
Li Man
a∈Γ
Man =
X
δaj
X
δaj Φjn
j∈Cn
i
• Expanded formulation
X La 1
,
EPSin =
EPS
Li Man
a∈Γ
i
EPS
Man
=
j∈Cn
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Expanded Path Size
• Expansion factor

1
Φjn =
1

q(j)Rn
if δjc = 1 or q(j)Rn ≥ 1
otherwise
• Chosen alternative always included
• Duplicates are ignored
• Asymptotically valid
if Rn → +∞ then q(j)Rn ≥ 1 ∀ j ∈ U and
P
EPS
Man ≈ j∈U δaj
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Numerical Results
• Evaluation of the impact on estimation results of
• the sampling correction,
• the definition of the PS attribute,
• the biased random walk algorithm parameters and
• the definition of the postulated model used for
generating the data
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Numerical Results – Data
D
SB
SB
SB
SB
SB
SB
O
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Numerical Results – Data
• Two sets of observations with two postulated Path Size
Logit models
Uin = βPS ln PSUi + βL Lengthi + βSB NbSBi + εin
• βPS = 1, βSB = −0.1, εin i.i.d. EV(0,1)
• βL = −0.3 and βL = −1
• Path size
PSUi
X La 1
X
=
Li
δaj
a∈Γ
i
j∈U
• 3000 observations per dataset
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Numerical Results
0.08
+++
Probability
0.07
βL = −0.3
βL = −1
b
+
0.06
0.05
b
0.04
0.03
b
b
++++
b
b
0.02
b
b
+++++
b
b
b
b
b
++++++
b
b
b
b
b
b
+++++++
b
b
b
b
b
b
b
0.01
++++
b
b
b
b
0
0
5
10
15
20
25
Path number
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Numerical Results – Models
MPNoCorr
S(C) Vin =
MPCorr
S(C) Vin =
MPNoCorr
S(U) Vi =
MPCorr
S(U) Vin =
Corr
MEP
S Vin =
µ βPS ln PSCin + βL Lengthi + βSB NbSBi
kin
C
)
µ βPS ln PSin + βL Lengthi + βSB NbSBi + ln(
q(i)
µ βPS ln PSUi + βL Lengthi + βSB NbSBi
kin
U
)
µ βPS ln PSi + βL Lengthi + βSB NbSBi + ln(
q(i)
kin
)
µ (βPS ln EPSin + βL Lengthi + βSB NbSBi ) + ln(
q(i)
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Numerical Results – Sampling
• Number of draws: 10, 20, 40, 80, 120, 170, 250
• Parameters of the distribution: b1 = 1, 2, 3, 5, 10, 15, 20
with b2 always fixed to one
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Average choice set size
Numerical Results
100
b
+
80
bc
b
+
b
bc
+bc
60
b
+bc
40
b
+bc
20
+bcb
+bc
b
b1 = 1
+
b1 = 2
bc
b1 = 3
0
10
50
90
130
170
Number of draws
210
250
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Numerical Results – Estimation
• Over 300 models have been estimated
Two datasets, five model specifications, different
settings of the sampling algorithm
• Detailed results for one setting: βL = −1 dataset using
40 draws and b1 = 1
• Average size of sampled choice sets: 30.92
• BIOGEME (Bierlaire, 2003, and Bierlaire, 2007)
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Numerical Results
MPNoCorr
S(U)
MPCorr
S(U)
MPNoCorr
S(C)
MPCorr
S(C)
Corr
MEPS
βbPS
-0.108
0.969
0.285
0.397
1.09
Rob. std
0.045
0.0541
0.0785
0.074
0.052
Rob. t-test 1
βbSB
-24.62
-0.57
-9.11
-8.15
1.73
-0.547
-0.0849
-0.52
-0.00941
-0.109
Rob. std
0.0322
0.0262
0.0331
0.0261
0.0281
Rob. t-test -0.1
-13.88
0.58
-12.69
3.47
-0.32
1.04
0.983
1.05
0.945
1.05
0.0314
0.028
0.0316
0.0264
0.0314
1.27
-0.61
1.58
-2.08
1.59
-7284.711
-6966.668
-7281.035
-7160.154
-6704.515
0.291
0.322
0.292
0.303
0.348
µ
b
Rob. std
Rob. t-test 1
Final L-L
Adj. rho bar sq.
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Numerical Results
Speed Bump Coef.
b1
+ b1
b1
b
10
5
bc
βbSB
=1
=2
=3
Scale Param. µ
b
10
+ bc
bcb
b +
+b
0
bc
+b
bc
+
b
bc
+b
bc
+b
10
b
+
5
bc
bc
PS Coef. βbPS
bc
+b
bc
5
+
b
+
bcb
0
10 50 90 130 170 210 250
Corr
βL = −1, MEPS
+bc
+bc
+bc
+bc
+bc
bc
b
+bc
0
bc
+
+
b
b
bc
+
b
bc
+b
10 50 90 130 170 210 250
10 50 90 130 170 210 250
Scale Param. µ
b
PS Coef. βbPS
βL = −1, MPCorr
S(C)
Speed Bump Coef. βbSB
10
+bcb
bc
5
bc
+bcb
+bcb
10
10
5
5
b
+bc
+b
+b
b
0 +bc
10 50 90 130 170 210 250
+bc +bc bcb
+
0
bc
+
b
bc
+b
bc
+b
bc
+b
10 50 90 130 170 210 250
0
b
+bbc +bc
b
+
bc
b
bc
bc
+
+b
+bc
b
+bcb
10 50 90 130 170 210 250
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Numerical Results
Speed Bump Coef.
b1
+ b1
b1
b
10
bc
βbSB
=1
=2
=3
Scale Param. µ
b
10 ++
++bbc
+bcb
10
bc
bc
b
PS Coef. βbPS
bbc
bc
+
5
bc
+ bc
b b
+
0 +bc b
5
bc
bc
+
+
b
b
bc
+b
b
b
0
b
bc
+b
bcb
+
+bc
5
bc
bc
+b bc
+b
+
b
+b
b
+bc
+bc
PS Coef. βbPS
10
bcb
bc
10 50 90 130 170 210 250
bc
5 ++ +
bc
+
bc
+b
Scale Param. µ
b
bc
b
0
bcb
10 50 90 130 170 210 250
10
+bcb
+bc
b
10 50 90 130 170 210 250
βL = −0.3, MPCorr
S(C)
Speed Bump Coef. βbSB
10
5
bc
+
+bcb
10 50 90 130 170 210 250
Corr
βL = −0.3, MEPS
0
b
+bc
b
b
bc
+
+bcb
b
bc
+
+
b
b
5
bc
+b +bbc
b
+bc
bc
b
0
0
10 50 90 130 170 210 250
+
+bcb
bc
+b
+bcb
10 50 90 130 170 210 250
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Conclusions
Trips
Network
Sampling of paths
Expanded Path Size
Choice sets
Description of
Path
Observations
Sampling correction
correlation
Route choice model
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Conclusions
• New paradigm for choice set generation and route
choice model estimation
• Aim: avoid bias in the econometric model
• Path generation is importance sampling of alternatives
• Sampling correction of path utilities and Path Size
attribute
• Numerical results show that true parameter values can
be retrieved
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Future Work
• Results based on real data
• Forecasting
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