### A Bayesian Approach to Traffic Estimation in Stochastic User

```The 20th International Symposium on Transportation and Traffic Theory
Noordwijk, the Netherlands, 17 – 19, July, 2013
A Bayesian approach to traffic estimation in
stochastic user equilibrium networks
Chong WEI
Beijing Jiaotong University
Yasuo ASAKURA
Tokyo Institute of Technology
1
Purpose
Estimating
Traffic flows
on congested
networks
Path Flows
O-D Matrix
2
Background
• Likelihood-based methods
- Frequentist:
Watling (1994), Lo et al. (1996),
Hazelton (2000), Parry & Hazelton (2012)
- Bayesians:
Maher (1983), Castillo et al. (2008),
Hazelton (2008), Li (2009),
Yamamoto et al. (2009), Perrakis et al. (2012)
3
Background
• On congested networks: Bi-level model
Bi - level
Likelihood
equilibrium constraint
4
Background
• On congested networks: Single level model
Bayesian
Likelihood
equilibrium constraint
5
Highlights
• Use a likelihood to present the estimation
problem along with equilibrium constraint
• Exactly write down the posterior distribution
of traffic flows conditional on both link count
data and equilibrium constraint through a
Bayesian framework
• Develop a sampling-based algorithm to obtain
the characteristics of traffic flows from the
posterior distribution
6
Primary problem
• On a congested network, estimating  based on
∗ and .
: vector of route flows;
∗ : vector of observed link counts;
: pre-specified O-D matrix ;
• equilibrium constraint:
the network is in Stochastic User Equilibrium.
7
Representation
• Bi-level approach:
min ( ,  ∗ )
s.t.  and
• Our approach:
(| ∗ , , )
. . denotes a conditional probability density;
∗ , ,  are the given conditions;
∗ , ,
→ E(), Var(), E(), Var  .
8
Decomposition
∗ , ,

′ ℎ
(|)

,  ∗ , )
ℎ
( ∗ |)   , )

9
Equilibrium constraint
•  and  (see Hazelton et al. 1998):
⟺  ∀ ∈
: user  displays Stochastic User Behaviour
i.e., user  selects the route that he or she
perceives to have maximum utility;
: set of users on the networks;
• The equilibrium constraint can be obtained as:
| ,  = ∀∈ ( |, )
10
An illustrative example
• Two-route network
110
O
200
A
? (90)
D
Proposed model
Equilibrium model
91.81
105.15
True value = 90
True value = 90
11
Path flow estimation problem
• The representation of the problem:
|,  ∗
here,  is no longer a given condition.
• Using Bayes’ theorem
∗
(,

| )()
∗
|,  =
(,  ∗ )
• The constant term
,  ∗ =
12
Path flow estimation problem
• The posterior distribution
|,  ∗ ∝ (,  ∗ | )()/(,  ∗ )
!
∗
,  , )
Likelihood
∀∈
∀∈  !
∙
Prior probability: the
principle of indifference
13
Prior knowledge of O-D matrix
• Dirichlet distribution
=
Γ
∀∈
∀∈ Γ

∀∈
−1
: the relative magnitude of the demand of
the O–D pair in the total demand across the network
• Do estimation with prior knowledge
|,  ∗ ,  ∝     |,  ∗
∝     ,  ∗ )()
14
Estimation
• Sampling-based algorithm
|,  ∗
E()
E()
Var()
Var()
15
Blocked sampler
0
(1) Specify initial samples 10 , … , ||
for 1 , … , || ,
set  ← 1 and  ← 1.
(2) For the O–D pair :
draw  using the Metropolis–Hastings (M–H) algorithm.
(3) If  < || then  ←  + 1, and go to step (1);
otherwise, go to step (3).
(4) If  <  then  ←  + 1,  ← 1, and go to step (1);
otherwise, stop the iteration.
16
Test network
60 O-D pairs
17
Test network
“observed” flow on link , ∗ may
be different from the “true” flow,
# due to observational errors, so
that inconsistencies can arise in
illustrative purposes, we created
the “observed” flow, ∗ by
drawing a sample from the
Poisson distribution as
∗ ~Poisson(# ).
we created  by introducing
Poisson-perturbed errors to the
true O–D matrix
18
19
O-D estimates without prior knowledge
20
21
95% Bayesian confidence interval
22
O-D estimates with prior knowledge
23
Conclusions
• A likelihood-based statistical model that can take
into account data constraint and equilibrium
constraint through a single level structure.
• Therefore, the proposed method does not find an
equilibrium solution in each iteration.
• The proposed model uses observed link counts as
input but does not require consistency among the
observations.
24
Conclusions
• The probability distribution of traffic flows can be
obtained by the proposed model.
• No special requirements for route choice models.
The National Basic Research Program of
China (No. 2012CB725403)
25
Questions？
Chong WEI
[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */
Yasuo ASAKURA
[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */
26
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