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A Bayesian modelling framework for individual passengers’ probabilistic route choices:
A case study on the London Underground
THE 46TH ANNUAL UTSG CONFERENCE, NEWCASTLE, 6-8 JANUARY 2014
QIAN FUPhD student
Institute for Transport Studies (ITS)University of Leeds
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Motivation
Methodology- Bayesian framework
- finite mixture distribution
Case study- a pair of O-D stations on the London Underground
Conclusions - future research
- potential applications
CONTENTS
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To understand passengers’ route choice behaviour(e.g. route choice models … )
- individual’s route choice for estimation of a route choice model
- data availability?
High cost; small sample size; and lack of accuracy
Smart-card data on local public transport(e.g. Oyster in London, Octopus in Hong Kong, SPTC in Shanghai …)
- entry time and exit time of a journey → individual’s journey time
- detailed itinerary ? → each individual’s actual route choice?
MOTIVATION
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Q1: Would there be a link that potentially relates a passenger’s route choice to his/her journey timeobserved from the smartcard data?
If such a ‘link’ exists…
Q2: Given only the observed journey time, would it be possible to tell the most probable (or even the actual)route choice that the passenger made?
MOTIVATION –QUESTIONS?
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Pr( | )qr qchoice t
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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Pr( | )qr qchoice t
passenger q choosing route r
(in view of his/her own choice set)
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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Pr( | )qr qchoice t
passenger q choosing route r
(in view of his/her own choice set)
observed journey time of the passenger q
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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Pr( | )qr qchoice t , r = 1, …, N (number of alternative routes)
would possibly offer an answer to Q1.
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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Pr( | )qr qchoice t , r = 1, …, N (number of alternative routes)
Under Bayesian framework
A posterior probability of a passenger’s route choice,
conditional on an observation of the passenger’s journey time
would possibly offer an answer to Q1.
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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Under Bayesian framework
For all r = 1, 2, …, N
Pr(choiceq1 | tq)
Pr(choiceq2 | tq)
…
Pr(choiceqN | tq)
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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Under Bayesian framework
For all r = 1, 2, …, N
Pr(choiceq1 | tq)
Pr(choiceq2 | tq)
…
Pr(choiceqN | tq)
maxr Pr(choiceqr | tq)
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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1Pr( ) Pr( )Pr( | )q qr q qrr
t choice t choice
N
According to the law of total probability,
Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )qr q qrchoice t choice∝Pr( | )qr qchoice t
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The prior probability
Under Bayesian framework
How frequently is route r used?
It should be learnt, a priori, from
history data
BAYESIAN FRAMEWORK
Pr( )Pr( | )qr q qrchoice t choice∝Pr( | )qr qchoice t
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The prior probability
The likelihood function
Under Bayesian framework
The likelihood that the observed
journey time would be tq given the
evidence that route r was actually
chosen by the passenger q
How frequently is route r used?
It should be learnt, a priori, from
history data
BAYESIAN FRAMEWORK
Pr( )Pr( | )qr q qrchoice t choice∝Pr( | )qr qchoice t
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1Pr( | ) 1qr qr
choice t
N
Under Bayesian framework
1Pr( ) 1qrr
choice
N
1Pr( ) Pr( )Pr( | )q qr q qrr
t choice t choice
N
BAYESIAN FRAMEWORK
Pr( )Pr( | )qr q qrchoice t choice∝Pr( | )qr qchoice t
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MIXTURE DISTRIBUTION OF JOURNEYTIME
Overall observations
- passengers’ journey time t on an O-D
- there are N alternative routes on that O-D
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MIXTURE DISTRIBUTION OF JOURNEYTIME
N sub-populations of journey time observations
- a sub-population: all passengers who chose the same route
- a component distribution cr (t; θr) where r = 1, …, N
Mixture distribution of journey time m (t; Ω, Θ)
- a finite mixture distribution of journey time t
Overall observations
- passengers’ journey time t on an O-D
- there are N alternative routes on that O-D
- a weighted sum of all the N component distributions
by mixing probabilities ωr
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MIXTURE DISTRIBUTION OF JOURNEYTIME
N sub-populations of journey time observations
- a sub-population: all passengers who chose the same route
- a component distribution cr (t; θr) where r = 1, …, N
Mixture distribution of journey time m (t; Ω, Θ)
- a finite mixture distribution of journey time t
Overall observations
- passengers’ journey time t on an O-D
- there are N alternative routes on that O-D
1( ; , ) ( ; ),r r rr
m t c t
N
11rr
N
where
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For simplicity, assuming all passengers consider an identical
choice set that contains all the N alternative routes on the O-D,
Pr( ) Pr( )qr r rchoice choice
Pr( | ) Pr( | ) ( ; )q qr r r rt choice t choice c t
MIXTURE DISTRIBUTION OF JOURNEYTIME
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For simplicity, assuming all passengers consider an identical
choice set that contains all the N alternative routes on the O-D,
Pr( ) Pr( )qr r rchoice choice
In accordance with Bayesian framework,
1Pr( ) Pr( ) Pr( | ) ( ; , )q r rr
t choice chocie m
N
t t
Pr( | ) Pr( | ) ( ; )q qr r r rt choice t choice c t
MIXTURE DISTRIBUTION OF JOURNEYTIME
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For simplicity, assuming all passengers consider an identical
choice set that contains all the N alternative routes on the O-D,
Pr( ) Pr( )qr r rchoice choice
In accordance with Bayesian framework,
1Pr( ) Pr( ) Pr( | ) ( ; , )q r rr
t choice chocie m
N
t t
Pr( | ) Pr( | ) ( ; )q qr r r rt choice t choice c t
Expectation-Maximization (EM) algorithm
(Dempster, Laird & Rubin, 1977)
MIXTURE DISTRIBUTION OF JOURNEYTIME
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THEOYSTER IN LONDON
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THEOYSTER IN LONDON
EXT ENTOJT T T
Oyster Journey Time (OJT )
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THEOYSTER IN LONDON
EXT ENTOJT T T
Time-stamp of EXIT
Time-stamp of ENTRY Oyster Journey Time (OJT )
(in minutes)
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CASE STUDY: ONTHE LONDON UNDERGROUND
(Source: Standard Tube map, Transport for London)
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CASE STUDY: ONTHE LONDON UNDERGROUND
(Source: Standard Tube map, Transport for London)
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CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
(Picture edited from the Standard Tube map, Transport for London)
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CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
(Picture edited from the Standard Tube map, Transport for London)
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CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
(Picture edited from the Standard Tube map, Transport for London)
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CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
Direct route(Low frequency)
(Picture edited from the Standard Tube map, Transport for London)
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CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
Direct route(Low frequency)
Indirect route(High frequency)
(Picture edited from the Standard Tube map, Transport for London)
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O-D: JOURNEYTIME DISTRIBUTION
Frequency distribution of OJT in AM peak (07:00-10:00), 26/06/2011 – 31/03/2012
(35,992 valid observations)
from Victoria station (origin) to Liverpool Street station (destination)
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O-D:MIXTURE DISTRIBUTION OF JOURNEYTIME
Suppose that cr (t ; θr), for all r (r = 1, 2), is
- Gaussian distribution- Lognormal distribution
The two mixture distributions estimated by the EM algorithm
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Gaussian mixture
Route Label Route1 Route2
Est. Mean (min) 22.02 28.75
Est. Standard deviation (min) 1.83 4.51
Est. Mixing probability 35.77% 64.23%
Naive inference of
passenger-flow proportion 42.60% 57.40%
Final inference of
passenger-flow proportion 35.50% 64.50%
ESTIMATED RESULT
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Gaussian mixture
Route Label Route1 Route2
Est. Mean (min) 22.02 28.75
Est. Standard deviation (min) 1.83 4.51
Est. Mixing probability 35.77% 64.23%
Naive inference of
passenger-flow proportion 42.60% 57.40%
Final inference of
passenger-flow proportion 35.50% 64.50%
Direct route(low frequency)
28.24
SURVEY RESULT
Average journey time (min):
(Survey data source: Transport for London)
ESTIMATED RESULT
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Gaussian mixture
Route Label Route1 Route2
Est. Mean (min) 22.02 28.75
Est. Standard deviation (min) 1.83 4.51
Est. Mixing probability 35.77% 64.23%
Naive inference of
passenger-flow proportion 42.60% 57.40%
Final inference of
passenger-flow proportion 35.50% 64.50%
Indirect route(high frequency)
22.50
SURVEY RESULT
Average journey time (min):
(Survey data source: Transport for London)
ESTIMATED RESULT
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Gaussian mixture
Route Label Route1 Route2
Est. Mean (min) 22.02 28.75
Est. Standard deviation (min) 1.83 4.51
Est. Mixing probability 35.77% 64.23%
Naive inference of
passenger-flow proportion 42.60% 57.40%
Final inference of
passenger-flow proportion 35.50% 64.50%
Direct route(low frequency)
28.24
Indirect route(high frequency)
22.50
SURVEY RESULT
Average journey time (min):
(Survey data source: Transport for London)
ESTIMATED RESULT
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Gaussian mixture
Route Label Route1 Route2
Est. Mean (min) 22.02 28.75
Est. Standard deviation (min) 1.83 4.51
Est. Mixing probability 35.77% 64.23%
Naive inference of
passenger-flow proportion 42.60% 57.40%
Final inference of
passenger-flow proportion 35.50% 64.50%
Direct route(low frequency)
28.24
Indirect route(high frequency)
22.50
SURVEY RESULT
Average journey time (min):
(Survey data source: Transport for London)
Lognormal mixture
Route1 Route2
21.78 28.69
1.78 4.43
34.02% 65.98%
35.36% 64.64%
34.04% 65.96%
ESTIMATED RESULT
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Oyster Journey Time (minutes)
Pro
bab
ilit
y D
en
sity
Lognormal Mixture
15 20 25 30 35 40 45 500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Oyster data (AM Peak)
Est. Lognorm mixture
Route1 (Victoria - Central)
Route2 (Circle)
Oyster Journey Time (minutes)
Pro
bab
ilit
y D
en
sity
15 20 25 30 35 40 45 500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Oyster data (AM Peak)
Est. Gaussian mixture
Route1 (Victoria - Central)
Route2 (Circle)
Estimated Gaussian mixture Estimated Lognormal mixture
Estimated PDFs of OJT in AM peak (07:00-10:00), 26/06/2011 – 31/03/2012
(35,992 valid observations)
from Victoria station (origin) to Liverpool Street station (destination)
O-D:MIXTURE DISTRIBUTION OF JOURNEYTIME
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Passenger-flow proportions on a weekday(from Victoria to Liverpool Street)
(Survey data source: Rolling Origin and Destination Survey (RODS), Transport for London)
Direct route(Circle Line only)
Indirect route(Victoria Line – Central Line)
Time-band RODSGaussian
mixtureLognormal
mixtureRODS
Gaussian mixture
Lognormal mixture
AM Peak
(07:00-10:00)51.89% 64.50% 65.96% 48.11% 35.50% 34.04%
PM Peak
(16:00-19:00)62.28% 64.20% 71.50% 37.72% 35.80% 28.50%
A whole day
(05:34-00:30)61.06% 61.02% 66.52% 38.94% 38.98% 33.48%
CASE STUDY – VALIDATION
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FUTURE RESEARCH& APPLICATIONS
Future research- timetable
- other component distributions
- perceived route choice set
Potential applications- applying to other similar public transport networks with the use of
smart-card data
- understanding route choice behaviour:providing knowledge for revealing passenger-flow distributions and traffic
congestion; and assisting public-transport managers in delivering a more
effective transit service, especially during rush hours
• model estimation using the posterior probability estimates in the
absence of actual route choices