a bayesian modelling framework for individual passenger’s probabilistic route choice: a case study...

3 CASE STUDY

(edited from the Standard Tube map, Transport for London)

UNDERGROUND STATIONS: VICTORIA (Orig) – LIVERPOOL STREET (Destn)

The Oyster in London & test network

Oyster Journey Time,EXT ENTOJT T T

time-stamp of EXIT (end)

time-stamp of ENTRY (start)

(minutes)

Frequency distribution of OJT in AM peak (07:00-10:00)(35,992 valid observations, 26/06/2011 – 31/03/2012)

Suppose that cr (t; θr), for all r (r = 1, 2), is

» Gaussian distribution» Lognormal distribution

Oyster Journey Time (minutes)

Freq

uenc

y

Stats (in minutes)——————— min. max. mean med. stdev.

: : : : :

174226.3429264.9664

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Indirect route(high service frequency)

A Bayesian modelling framework for individual passenger’s probabilistic route choice: A case study on the London Underground

Qian Fu, Ronghui Liu and Stephane Hess, Institute for Transport Studies (ITS), University of Leeds, UK Email: [email protected]; Tel: +44 (0)113 343 1790; 34-40 University Road, Leeds LS2 9JT, UK

Presentation #14-5328, session 775The 93rd Transportation Research Board Annual Meeting

Washington D.C., 12-16 January 2014


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0.1Oyster data (AM Peak)Est. Lognorm mixtureRoute1 (Victoria - Central)Route2 (Circle)


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0.1Oyster data (A whole weekday)Est. Lognorm mixtureRoute1 (Victoria - Central)Route2 (Circle)


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providing knowledge for revealing passenger-flow distributions and traffic congestion, etc.; and

assisting public-transport managers in delivering a more effective transit service, especially during rush hours

small samples of individual passengers’ route choices

data collection

» costly, hence a/ and small sample size; and» lack of accuracy or even loss of essential information, e.g. travel time

To understand passengers’ route choice behaviour

1 MOTIVATION & QUESTIONS

Issues of data availability for model estimation

from automatic fare collection system implemented on local public transport, e.g. Oyster in London, SmarTrip in Washington D.C., Octopus in Hong Kong, and SPTC in Shanghai, to name but a few

times-tamps of the start and the end of a journey

» a sufficiently large sample of the smart-card users’ journey time between O-D stations; BUT …

their detailed itineraries?

» each individual’s actual route choice?

( | ), 1,..., Nqr qchoice t r Prob (Number of alternative routes)

Given only the observed journey time, would it be possible to tell the most likely (or even the actual) route choice that the passenger made?

observed journey time of passenger q

the passenger q choosing route r(from his/her own route-choice set)

ANS1: A conditional probability

Would there be a link that potentially relates a passenger’s route choice to his/her journey time observed from the smartcard data?

Q1

RESEARCH QUESTIONS POSSIBLE ANSWERS&

Q2

time-stamp ofend

time-stamp ofstart

Smart-card data

Given a pair of O-D stations:

2 METHOD

Mixture distribution of the journey time

For simplicity, assume that all the passengers consider an identical route-choice set that contains all the N alternative routes

Pr( ) Pr( )qr r rchoice choice

In accordance with Bayesian framework,

1Pr( ) Pr( ) Pr( | ) ( ; , )q r rrt choice chocie m

N

t t

Pr( | ) Pr( | ) ( ; )q qr r r rt choice t choice c t

Apply Expectation-Maximization (EM) algorithm(Dempster, Laird & Rubin, 1977)

1Pr( | ) 1qr qrchoice t

N

1Pr( ) 1qrrchoice

N

1Pr( ) Pr( )Pr( | )q qr q qrrt choice t choice

N

Pr( | ) Pr( )Pr( | )qr q qr q qrchoice t choice t choice

under BAYESIAN FRAMEWORK

Pr( )Pr( | )Pr( | )

Pr( )qr q qr

qr qq

choice t choicechoice t

t

If ∃r* ∊ arg max Pr(choiceqr | tq),

For all r = 1, 2, …, N

Pr(choiceq1 | tq)

Pr(choiceq2 | tq)

…

Pr(choiceqN | tq)Route r* could then be deemed as the most likely route passenger q might have chosen.

Based on the law of total probability,

N component distributions, cr (t; θr),

where r = 1, …, N

Consider, on the given O-D, the overall observations of all the passengers’ journey time, t

There are supposed to be N sub-populations of the observedjourney times, in conformity with the N alternative routes, r

A mixture distribution of journey time,

m (t; Ω, Θ)

1( ; , ) ( ; ), r r rrm t c t

N

11rr

Nwhere

How frequently is route rused? It should be learnt, a priori, from history data.

The prior probability

The likelihood that the observed journey time would be tq given the evidence that route r was actually chosen by the passenger q

The likelihood function

Q2… ANSWERING

Estimated results(AM Peak)

K-means clustering

Gaussian mixture

Lognormal mixture

Route Label Route1 Route2 Route1 Route2 Route1 Route2

Est. Mean (min) 23.12 31.72 22.02 28.75 21.78 28.69

Est. Stdev. (min) 2.39 3.18 1.83 4.51 1.78 4.43

Est. Mixing probability 62.55% 37.45% 35.77% 64.23% 34.02% 65.98%

Naive inference passenger prop. 62.55% 37.45% 42.60% 57.40% 35.36% 64.64%

Final inference passenger prop. 62.55% 37.45% 35.50% 64.50% 34.04% 65.96%

Indirect route(Victoria Line – Central Line)

Direct route(Circle Line only)

22.50 28.24


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Posterior probabilities of an individual choosing an alternative route conditional on his/her OJT

Estimated results & route matching

Validation – Passenger-flow proportions (on a weekday):

(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 RODSGaussianmixture

Lognormal mixture

RODSGaussian 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%

The average journey time of each of the alternative routes was calculated by aggregating the average travel time of every journey segment

Estimation & Results

Posterior probabilities

Survey results (AM Peak)

Avg. journey time (min):

(Data source: Transport for London)

PDFs of the estimated mixture distributions

4 FURTHER WORK

To further involve the timetable

» passengers’ arrival time at origin station (time of ENTRY) » train’s scheduled departure time from platform

To explicitly specify/identify each individual’s perceived choice set

Journey time distribution of each alternative route

The next steps …

Applying to other comparable public transport networks with smartcard data

Understanding route choice behaviour:

» model estimation using the posterior probability estimates in the absence of actual route choices

Potential applications

ACKNOWLEDGEMENT

The authors appreciate funding support from China Scholarship Council -University of Leeds Scholarship, and would also like to express their gratitude to the staff of Transport for London for their continued support.

a bayesian modelling framework for individual passenger’s probabilistic route choice: a case study...

Data & Analytics