Upload
q-fu
View
99
Download
0
Tags:
Embed Size (px)
Citation preview
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
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
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
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?
Pr( | )qr qchoice t
A conditional probability:
MOTIVATION – IN OTHERWORDS…
Pr( | )qr qchoice t
passenger q choosing route r
(in view of his/her own choice set)
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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…
Pr( | )qr qchoice t , r = 1, …, N (number of alternative routes)
would possibly offer an answer to Q1.
A conditional probability:
MOTIVATION – IN OTHERWORDS…
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…
Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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
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
Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )
Pr( )
qr q qr
q
choice t choice
tPr( | )qr qchoice t
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
Under Bayesian framework
BAYESIAN FRAMEWORK
Pr( )Pr( | )qr q qrchoice t choice∝Pr( | )qr qchoice t
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
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
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
MIXTURE DISTRIBUTION OF JOURNEYTIME
Overall observations
- passengers’ journey time t on an O-D
- there are N alternative routes on that O-D
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
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
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
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
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
THEOYSTER IN LONDON
THEOYSTER IN LONDON
EXT ENTOJT T T
Oyster Journey Time (OJT )
THEOYSTER IN LONDON
EXT ENTOJT T T
Time-stamp of EXIT
Time-stamp of ENTRY Oyster Journey Time (OJT )
(in minutes)
CASE STUDY: ONTHE LONDON UNDERGROUND
(Source: Standard Tube map, Transport for London)
CASE STUDY: ONTHE LONDON UNDERGROUND
(Source: Standard Tube map, Transport for London)
CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
(Picture edited from the Standard Tube map, Transport for London)
CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
(Picture edited from the Standard Tube map, Transport for London)
CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
(Picture edited from the Standard Tube map, Transport for London)
CASE STUDY: VICTORIA (O) - LIVERPOOL STREET (D)
Direct route(Low frequency)
(Picture edited from the Standard Tube map, Transport for London)
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)
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)
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
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
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
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
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
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
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
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
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