Sensitivity-based Uncertainty Analysis
of a Combined Travel Demand Model
Chao Yang, Tongji University
Anthony Chen
Xiangdong Xu, Utah State University
S.C. Wong, University of Hong Kong
The 20th International Symposium on Transportation and Traffic Theory
July 17-19, 2013, the Netherlands
Outline
•
Introduction
•
Travel Demand Forecasting Models
•
Sensitivity Analysis
•
Uncertainty Analysis
•
Numerical Examples
•
Conclusions
2
Introduction
•
Transportation planning and project evaluation are both
based on travel demand forecasting: subject to
different types of uncertainties (Rasouli and
Timmermans, 2012)



•
Predicted socioeconomic inputs (i.e. population, employee)
Calibrated parameters (i.e. dispersion parameter, BPR)
Travel demand model itself (i.e., model structure &
assumptions)
Without considering uncertainty in travel demand
models, decision are likely to take on unnecessary risk
and forecasts may be inaccurate and misleading (Zhao
and Kockelman, 2002)
3
Introduction (cont’d)
•
Most of the existing procedures in the travel
demand forecasting are deterministic
•
Planners usually use point estimates of traffic
forecasts in practice
•
There lacks a systematic methodology to conduct
uncertainty analysis of a travel demand model
(Rasouli and Timmermans, 2012)
4
Literature Review
•
Waller et al. (2001) studied the impact of demand uncertainty
on the results of traffic assignment model
•
Zhao and Kockelman (2002) addressed the uncertainty
propagation of a sequential four-step procedure using Monte
Carlo simulation
•
Pradhan and Kockelman (2002) & Krishnamurthy and
Kockelman (2003) investigated the uncertainty propagation of
an integrated land use-transportation model over time
•
Rasouli and Timmermans (2012) reviewed the uncertainty
analysis in travel demand forecasting, including four-step
models, discrete choice models, and activity-based models
5
Typical Components of Uncertainty
Analysis of a Model
• Characterization of input/parameter uncertainty
• distribution characteristics (e.g., mean, variance) of
input/parameter uncertainty
• Uncertainty propagation
• output uncertainty resulting from input/parameter
uncertainty
• Characterization of output uncertainty
• mean, variance
• confidence level
• relationship between input/parameter & output
6
Research Objective
•
To develop a systematic and computationally efficient
network equilibrium approach for quantitative uncertainty
analysis of a combined travel demand model (CTDM)
using the analytical sensitivity-based method

Modeling multi-dimensional demands and equilibrium
flows on congested networks consistently

Less computation than the sampling-based methods

Uncertainties stemming from input data and model
parameters can be treated separately, so that the individual
and collective effects of uncertainty on the outputs can be
clearly quantified
7
Travel Demand Forecasting Models
•
Oppenheim (1995) proposed a combined travel demand
model (CTDM), which combines the travel-destinationmode-route choice based on the random utility theory
•
A viable avenue with behavioral consistency for modeling and
predicting multi-dimensional demands and equilibrium flows
8
Combined Travel Demand Model
Ni
Yes
No
Ti
j
1
J
Tij
1
m
Travel
Ti = Ni Pt|i is the travel demand in origin i
Pj|i is the probability of choosing destination j given Ti
Destination
Tij = Ni Pt|i Pj|i is the travel demand from origin i to destination j
Pm|ij is the probability of choosing mode m given Tij
M
Mode
Route
Tijm
1
Ni is the potential number of travelers in origin i
Pt|i is the probability of making a trip given Ni
Tijm = Ni Pt|i Pj|i Pm|ij is the travel demand from origin i to destination j on mode m
Pr|ijm is the probability of choosing route r given Tijm
R
r
Tijmr
Tijmr = Ni Pt|i Pj|i Pm|ij Pr|ijm is the travel demand taking route r from origin i to destination j on mode m
Pijmr  Pt|i  Pj|i  Pm|ij  Pr|ijm

e
t ( hi Wt|i )
1 e
t ( hi Wt|i )
e
 d ( hij W j|i )
e
j
 d ( hij W j|i )
e
m ( hijm Wmij| )
e
m
m ( hijm Wmij| )
e
 r gijmr
e
  r gijmr
r
9
Oppenheim’s Model (1995)
min U TDMR (Ti , Ti 0 , Tij , Tijm , Tijmr )

   ijr
am
m
1

r
1
 't

s.t.
Direct utility of route-mode-destination-travel choices
a
Tijmr  ijrm
0
T
ijmr
g am ( )d    hijmTijm   hij Tij   hi Ti
ijm
lnTijmr
ijmr
 Ti ln Ti 
i
t
i
Entropy terms of
ij Tij lnTij route, mode,
destination choices
Entropy terms of travel
and no travel choices
1

T
ln
T

ijm
ijm
 'd
ijm
1

 'm
1
ij
T
i0
ln Ti 0
i
Unique
Solution!
 Tijmr  Tijm , i, j, m
r
T
ijm
 Tij , i, j
m
T
ij
 Ti , i
Conservation constraints
j
Ti  Ti 0  N i , i
Ti 0 , Ti , Tij , Tijm , Tijmr  0, i, j , m, r
Oppenheim, N. (1995) Urban Travel Demand Modeling, John Wiley & Sons. 10
Sensitivity Analysis
 y( )  M ( )1 N ( ), y  (Ti , Ti 0 , Tij , Tijm , Tijmr ,  ijm , ij , i , i )







M ( )  







T2i L
0
0
I
I
0
0
0
I
0
I
T
0
I
T
0
0
T2ijmr L
T
0
0
0
0
T2i 0 L
T2ij L
T2ijm L
0
0
0
I
0
0
0
0
I

I

0

0
0
0
0
I
I
0
0
0
0
0
N ( )  Ti , L Ti 0 , L Tij , L Tijm , L Tijmr , L 0 0 0  Ni 


Follow the approach of Yang and Bell (2007), we can prove that M is invertible















T
11
Sensitivity Analysis
•
Estimated solution using the first-order Taylor
series approximation
T
y     y  0     y   
•
Matrix manipulation and differential chain rule
vam
Tijmr am
ta
va

 ijr
  ta 
  ta 
v

 i
 i
 i
 i
ijr
m
m
m
tam
 vam

TTT
  
tam 
vam 
 i
 i
m am   i

am
m
i
 vam 
TVM
  ham 

 i
m am
  i 
12
Propagation of Uncertainties
Output 1
Two possible approaches:
Probability density
of output 1
Input 2
• Sampling-based method
• high computational
effort
• non-reproducibility
Input 1
Probability density
of input 2
Probability density
of input 1
• Linear regression of
input/output
• Analytical sensitivity-based
method
13
An analytical method based on
sensitivity analysis of CTDM
Uncertainty Analysis
•
Variance-covariance matrix of outputs
Soutput   y  Sinput   y 
T
•
Confidence intervals of outputs (normality)
•
Covariance of outputs and inputs
Soutput ,input   y  Sinput
•
Given
Sensitivity
Correlation of output i & input j (critical inputs)
rij 
sij
si s j
14
Remarks

For non-separable link cost with
asymmetric interaction, CTDM can be
formulated as VI, and sensitivity analysis
for VI could be adopted
 Sampling-based methods and sensitivity
based analytical method is a tradeoff
between information richness and
computational burden
15
Numerical Results
4
2
4
O-D pair
1
5
1
(1, 4)
3
6
(1, 5)
2
3
7
5
Route
1
2
3
4
5
6
Link sequences
1-4
1-3-6
2-6
1-5
1-3-7
2-7
• 2 modes: car (c) and transit (t)
• # of potential travelers: N1=200
• Attractiveness: h1=5.0, h14=3.5, h15=3.8, h14c=3.5, h14t=3.6, h15c=3.8, h15t=3.4
• Parameters associated with route, mode, destination and travel choices
r  2.0, m  1.0, d  0.5, t  0.2
16
Selected Outputs for Analysis
4
2
T1: production from zone 1
T10: number of non-travelers from zone 1
4
O-D p
1
5
1
(1, 4
3
6
(1, 5
2
T14: O-D demand from zone 1 to zone 4
3
7
5
T14c, T14t: O-D demands from zone 1 to zone 4 by car and transit
T14c1, T14c2, T14c3: flows on three routes b/t O-D (1, 4) using car
v1c, v1t: flows on link 1 in car and transit networks
TTT: total travel time (TTT)
TVM: total vehicle miles (TVM) traveled
17
Multi-Dimensional Equilibrium Solution
travel
Wt|i
Pt|i
(1,4)
W j|i
-5.02
0.48
Pj|i
Wm|ij
Pm|ij
1
2
car
transit
-9.66
0.32
-9.01
0.68
3
1
2
3
not travel
Travel choice
-0.05
0.73
(1,5)
Destination choice
-5.15
0.52
transit
car
-9.95
0.37
4
Mode choice
-9.01
0.63
5
6
4
5
6 Route choice
gijmr 10.15 10.53 10.06 9.46 9.72 9.52 10.38 10.86 10.40 9.46 9.72 9.53
Pr|ijm 0.38 0.18 0.45 0.40 0.24 0.36 0.43 0.16 0.41 0.40 0.24 0.36
Choice probability and expected received utility
18
Multi-Dimensional Equilibrium Solution
Consistent with the tree
Ni 200
structure (i.e., traveler’s
travel
not travel
expected received utility
Ti 145.83
at the corresponding
choice stage)
(1,5)
(1,4)
Tij
Tijm
1
2
69.83
car
transit
22.36
47.47
3
1
2
3
Travel choice
Destination choice
76.00
transit
car
27.97
4
Mode choice
48.02
5
6
4
5
6 Route choice
Tijmr
8.46
3.94
9.97 19.15 11.44 16.88 11.98 4.53 11.47 19.41 11.56 17.05
Multi-dimensional equilibrium demand
19
Sensitivity Analysis Results
• Conservation
• Significance
Derivatives of outputs with respect to inputs
N1
C1c
C2c
C3c
C4c
C5c
C6c
C7c
T1
0.676
0.085
0.043
0.000
0.007
0.022
0.035
0.051
T10
0.324
-0.085
-0.043
0.000
-0.007
-0.022
-0.035
-0.051
T14
0.334
0.070
0.034
-0.001
0.046
-0.144
0.171
-0.222
T14c
0.058
0.341
0.172
-0.002
0.088
-0.155
0.352
-0.190
T14t
0.276
-0.271
-0.138
0.001
-0.042
0.010
-0.181
-0.032
T14c1
0.031
0.295
0.047
-0.011
0.122
-0.130
-0.198
-0.076
T14c2
-0.003
0.154
-0.078
0.013
-0.030
-0.064
0.215
0.039
T14c3
0.031
-0.108
0.203
-0.004
-0.004
0.040
0.335
-0.154
v1c
0.046
0.844
-0.093
0.009
0.035
0.104
0.033
0.059
v1t
0.306
-0.280
-0.144
0.002
-0.023
-0.071
-0.116
-0.165
TTT
7.462
-0.169
-0.260
0.046
-0.054
-0.103
-0.078
0.053
TVM
6.158
0.669
0.443
0.016
0.062
0.196
0.348
0.481
link capacities in car network
20
Sensitivity Analysis Results
Derivatives of outputs with respect to parameters
h1
h14
h14c
h14t
βt
βd
βm
βr
αc
γc
αt
γt
T1
7.346 3.627 0.630 2.997 181.851 -20.381 -3.860 -1.809 -8.207 -0.006 -44.079 -1.781
T10
-7.346 -3.627 -0.630 -2.997 -181.851 20.381 3.860 1.809 8.207 0.006 44.079 1.781
T14
3.627 16.138 3.769 12.369 89.788 -14.927 -1.781 -0.898 -0.608 0.048 -18.625 -0.740
T14c
0.630 3.769 7.386 -3.616 15.590
T14t
2.997 12.369 -3.616 15.985 74.198 -12.007 2.264 -0.953 23.182 -0.009 -46.448 -1.862
T14c1
0.335 2.323 4.403 -2.081 8.295
-1.661 -2.232 0.456 -6.953 -0.131 14.822 0.598
T14c2
-0.038 0.296 0.336 -0.041 -0.935
-0.002
T14c3
0.332 1.151 2.646 -1.495 8.230
-1.257 -1.926 0.275 -9.371 0.442 14.638 0.591
v1c
0.496 0.41 1.995 -1.586 12.283
-1.433 -2.546 -0.477 -37.325 -0.681 21.767 0.881
v1t
3.322 1.45 -2.104 3.556 82.261
-9.155
TTT
80.944 36.860 18.608 18.261 2004.097 -223.570 -63.076 -23.585 22.114 -0.323 405.862 16.340
-2.920 -4.046 0.054 -23.790 0.057 27.822 1.122
0.112 -0.677 -7.465 -0.254 -1.637 -0.066
1.075 -1.565 27.112 0.017 -136.973 -7.393
TVM 66.810 32.996 5.818 27.188 1654.130 -185.397 -35.218 -16.253 -73.453 0.087 -382.486 -15.074
attractiveness
choices
link cost functions
21
Estimated and Exact Solutions for
Perturbed Input and Parameter
Estimate the equilibrium solution without the need to resolve the CTDM
δN1=10 (i.e., 200×5%)
δβt=0.01 (i.e., 0.2×5%)
Solution Unperturbed
Difference
Difference
variable solution
Exact Estimated
Exact Estimated
(Exact-Estimated)
(Exact-Estimated)
T1
145.827
152.575 152.585
-0.010
147.621 147.645
-0.024
T10
54.173
57.425
57.415
0.010
52.379
52.355
0.024
T14
69.829
73.182
73.165
0.016
70.720
70.726
-0.007
T14c
22.358
22.918
22.938
-0.019
22.509
22.514
-0.005
T14t
47.470
50.263
50.228
0.035
48.211
48.212
-0.002
T14c1
8.455
8.755
8.763
-0.008
8.535
8.538
-0.003
T14c2
3.936
3.899
3.901
-0.002
3.926
3.926
0.000
T14c3
9.967
10.264
10.273
-0.009
10.048
10.050
-0.002
v1c
28.896
29.343
29.356
-0.014
29.017
29.019
-0.003
v1t
61.560
64.607
64.620
-0.012
62.371
62.382
-0.011
TTT
1432.011 1506.850 1506.632
0.218
1451.807 1452.052
-0.245
TVM
1323.514 1384.904 1385.094
-0.191
1339.834 1340.056
-0.222
N1 and βt have a large derivative value
22
Uncertainty from Inputs
Solution
variable
Mean
SD
CoV
T1
145.83
40.56
T10
54.17
T14
Coefficient of variation
(CoV) of inputs = 0.30
90% confidence interval
5%
95%
0.28
79.11
212.54
19.47
0.36
22.15
86.19
69.83
20.08
0.29
36.80
102.86
T14c
22.36
4.92
0.22
14.27
30.45
T14t
47.47
16.72
0.35
19.96
74.98
T14c1
8.46
3.16
0.37
3.26
13.65
T14c2
3.94
1.67
0.42
1.19
6.69
T14c3
9.97
3.02
0.30
5.00
14.94
v1c
28.90
6.96
0.24
17.44
40.35
v1t
61.56
18.54
0.30
31.07
92.05
TTT
1432.01
447.73
0.31
695.49
2168.53
TVM
1323.51
369.54
0.28
715.62
1931.41
23
Correlation of Outputs with Inputs
Identify critical inputs relative to output uncertainty by the correlation of
inputs and outputs
Correlation
T1
T10
T14
T14c
T14t
T14c1
T14c2
T14c3
v1c
v1t
TTT
TVM
N1
1.000
0.999
0.997
0.707
0.989
0.585
-0.125
0.608
0.396
0.990
1.000
1.000
C1c
0.016
-0.033
0.026
0.520
-0.122
0.699
0.692
-0.267
0.909
-0.113
-0.003
0.014
C2c
0.008
-0.017
0.013
0.262
-0.062
0.112
-0.351
0.503
-0.100
-0.058
-0.004
0.009
C3c
0.000
0.000
0.000
-0.002
0.000
-0.016
0.034
-0.006
0.006
0.000
0.000
0.000
C4c
0.001
-0.002
0.010
0.081
-0.011
0.173
-0.080
-0.006
0.023
-0.006
-0.001
0.001
C5c
0.002
-0.005
-0.032
-0.141
0.003
-0.185
-0.173
0.059
0.067
-0.017
-0.001
0.002
C6c
0.004
-0.008
0.038
0.322
-0.049
-0.282
0.578
0.500
0.021
-0.028
-0.001
0.004
C7c
0.006
-0.012
-0.050
-0.174
-0.009
-0.108
0.105
-0.229
0.038
-0.040
0.001
0.006
24
Uncertainty from Parameters
Solution
variable
Mean
SD
CoV
T1
145.83
16.72
T10
54.17
T14
Coefficient of variation
(CoV) of paramters=
0.30
90% confidence interval
5%
95%
0.11
118.32
173.34
16.72
0.31
26.66
81.68
69.83
23.36
0.33
31.40
108.26
T14c
22.36
9.82
0.44
6.21
38.50
T14t
47.47
22.98
0.48
9.66
85.28
T14c1
8.46
5.81
0.69
0.00*
18.02
T14c2
3.94
0.78
0.20
2.66
5.21
T14c3
9.97
3.65
0.37
3.96
15.97
v1c
28.90
3.64
0.13
22.91
34.88
v1t
61.56
10.06
0.16
45.01
78.11
TTT
1432.01
182.49
0.13
1131.81
1732.21
TVM
1323.51
152.03
0.11
1073.42
1573.60
25
Benefit of Improving Parameter Estimation
Coefficient of Variation (CoV) of Outputs
1.2
Parameter CoV = 0.1
Parameter CoV = 0.3
Parameter CoV = 0.5
1
0.8
0.6
0.4
0.2
0
T1
T10
T14
T14c
T14t
T14c1 T14c2 T14c3
v1c
v1t
TTT TVM
26
Correlation of Outputs with Parameters
h1
T1
0.659
T10 -0.659
T14 0.233
T14c 0.096
T14t 0.196
T14c1 0.086
T14c2 -0.073
T14c3 0.137
v1c 0.204
v1t
0.495
TTT 0.665
TVM 0.659
h14
0.228
-0.228
0.725
0.403
0.565
0.420
0.399
0.331
0.118
0.151
0.212
0.228
h14c
0.040
-0.040
0.169
0.790
-0.165
0.795
0.454
0.761
0.576
-0.220
0.107
0.040
h14t
0.194
-0.194
0.572
-0.398
0.751
-0.387
-0.056
-0.442
-0.471
0.382
0.108
0.193
βt
0.652
-0.652
0.231
0.095
0.194
0.086
-0.072
0.135
0.203
0.490
0.659
0.653
βd
-0.183
0.183
-0.096
-0.045
-0.078
-0.043
0.000
-0.052
-0.059
-0.136
-0.184
-0.183
βm
-0.069
0.069
-0.023
-0.124
0.030
-0.115
0.043
-0.158
-0.210
0.032
-0.104
-0.069
βr
-0.065
0.065
-0.023
0.003
-0.025
0.047
-0.523
0.045
-0.079
-0.093
-0.078
-0.064
αc
-0.022
0.022
-0.001
-0.109
0.045
-0.054
-0.432
-0.115
-0.462
0.121
0.005
-0.022
γc
0.000
0.000
0.002
0.007
0.000
-0.027
-0.393
0.145
-0.225
0.002
-0.002
0.001
αt
-0.047
0.047
-0.014
0.051
-0.036
0.046
-0.038
0.072
0.108
-0.245
0.040
-0.045
γt
-0.064
0.064
-0.019
0.069
-0.049
0.062
-0.051
0.097
0.145
-0.441
0.054
-0.059
27
Uncertainty from
Both Input and Parameter Uncertainty
Solution
variable
T1
T10
T14
T14c
T14t
T14c1
T14c2
T14c3
v1c
v1t
TTT
TVM
Mean
SD
CoV
145.83
54.17
69.83
22.36
47.47
8.46
3.94
9.97
28.90
61.56
1432.01
1323.51
43.87
25.66
30.81
10.98
28.42
6.62
1.84
4.74
7.86
21.09
483.49
399.59
0.30
0.47
0.44
0.49
0.60
0.78
0.47
0.48
0.27
0.34
0.34
0.30
90% confidence interval
5%
95%
73.66
217.99
11.96
96.39
19.15
120.51
4.30
40.42
0.71
94.23
0.00*
19.34
0.90
6.97
2.17
17.76
15.97
41.82
26.86
96.26
636.66
2227.36
666.19
1980.84
Outputs uncertainty (SD and CoV) from both inputs and parameters
uncertainty is not simply the sum of individual uncertainties
28
Output Uncertainty
at Each Travel Choice Step
Average Coefficient of Variation (CoV)
0.5
0.4
0.3
0.2
0.1
0
Travel Demand
O-D Demand
O-D Mode
equilibrium nature of traffic assignment
Link Flow
29
Concluding Remarks
•
•
•
•
Proposed a systematic analytical sensitivity-based approach
for the uncertainty analysis of a CTDM
Required significantly less computational efforts than the
sampling-based methods
Quantified the individual & collective effects of input and
parameter uncertainties on outputs
Can estimate the possible benefits of improving the
parameter accuracy
30
Thank You!
Acknowledgements
The authors are grateful to three anonymous referees and
especially to Prof. Hai Yang for valuable comments on the
sensitivity analysis formulation.
This research was supported by the Oriental Scholar
Professorship Program sponsored by the Shanghai
Ministry of Education in China to Tongji University,
National Natural Science Foundation of China (71171147),
Fundamental Research Funds for the Central Universities,
and the China Scholarship Council.
31