A Bayesian modelling framework for individual passenger’s probabilistic route choices: a case...

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

Questions?

Qian FuPhD student

tsqf@leeds.ac.uk

Institute for Transport Studies

University of Leeds