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Presented to SSAI(WA) on 8 November 2005 1 / 32
Statistical Methods in
Transportation Research
Martin Hazelton
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The Travelling Statistician Problem
Presented to SSAI(WA) on 8 November 2005 2 / 32
Perth
Palmerston North
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The Problem
Presented to SSAI(WA) on 8 November 2005 3 / 32
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Some Transport Statistics
Presented to SSAI(WA) on 8 November 2005 4 / 32
In developed countries, transportation typically accounts between
5% and 12% of GDP.
Transport accounts for around 20% of global CO2
emissions. Around 2000 Australian transport fatalities each year (90% road).
Transport sector supplied 427000 jobs in Australia in 2003.
In 2000, road traffic congestion in U.S. estimated to have caused:
3.6 billion vehicle-hours of delay
21.6 billion litres of wasted fuel
$67.5 billion in lost productivity Statisticians account for 0.1% of transportation research. (OK,
that ones a guess!)
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Outline
Presented to SSAI(WA) on 8 November 2005 5 / 32
Transportation research (and whats currently wrong with it. . .) The audience buys a car
Some problems for statisticians: Speed estimation
Estimation of trip matrices
Modelling day-to-day dynamics of traffic flow Further statistical problems
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Current State of Transport Research
Presented to SSAI(WA) on 8 November 2005 6 / 32
Variety of research fields:
Transport policy
Pyschology and transport Transport and urban planning
Transport modelling
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Current State of Transport Research
Presented to SSAI(WA) on 8 November 2005 6 / 32
Variety of research fields:
Transport policy
Pyschology and transport Transport and urban planning
Transport modelling
Most researchers transport modelling have an engineering or
applied mathematics background
Variability frequently gets ignored
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Variation and Transport Research
Presented to SSAI(WA) on 8 November 2005 7 / 32
Examples of (stochastic) variation:
Traffic counts
Vehicle speeds Route choice
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Variation and Transport Research
Presented to SSAI(WA) on 8 November 2005 7 / 32
Examples of (stochastic) variation:
Traffic counts
Vehicle speeds Route choice
Ignore variation and:
Extreme events may be missed (e.g. gridlock)
Estimates and predictions may be biased
Essential information may be lost
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Statisticians and Transport Research
Presented to SSAI(WA) on 8 November 2005 8 / 32
Variability is a statisticians trade
Statisticians have the opportunity to play an important role in
transportation research
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The Audience Buys a Car
Presented to SSAI(WA) on 8 November 2005 9 / 32
Aim: to get best fuel efficiency
Dealer 1:
Buy car that does 40 km per litre with prob.1/2 Buy car that does 10 km per litre with prob.1/2
Dealer 2:
Buy car that does 20 km per litre
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Which Dealer?
Presented to SSAI(WA) on 8 November 2005 10 / 32
The mean km per litre are:
Dealer 1:1/2 40 + 1/2 10 = 25.
Dealer 2:20. So get more km per litre on average going to Dealer 1.
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Look at Things the Other Way Up
Presented to SSAI(WA) on 8 November 2005 11 / 32
Dealer 1:
Buy car that needs 1/40 litre per km with prob.1/2
Buy car that needs 1/10 litre per km with prob.1/2 Dealer 2:
Buy car that needs 1/20 litre per km
The mean litres per km are:
Dealer 1:1/2 1/40 + 1/2 1/10 = 1/16. Dealer 2:1/20.
So need less litres per km on average from Dealer 2.
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The Importance of Variation
Presented to SSAI(WA) on 8 November 2005 12 / 32
Answers are different because generally
1/E[X]= E[1/X]
for a random variableX. Car buying example illustrates importance of accounting for
variation.
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Speed Estimation: The Problem
Presented to SSAI(WA) on 8 November 2005 13 / 32
Automated vehicle detector set in road.
Detector is occupied while vehicle passes over it.
Aggregate data returned for 20 or 30 second intervals:
Vehicle count during interval,n Total time that detector is occupied during interval,y.
Goal: to estimate mean vehicle speed during interval.
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Speed Estimation: Towards a Solution
Presented to SSAI(WA) on 8 November 2005 14 / 32
0000
0000
1111
1111
Detector
Range
Effective vehicle length
Each vehicle occupies detector for time it takes to travel the
effective vehicle length. Typical effective vehicle length given by
=d+l
wheredis detector range,lis mean vehicle length.
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Speed Estimation: Simple Solution
Presented to SSAI(WA) on 8 November 2005 15 / 32
Speed=Distance
Time
Hence simple estimator of speed is
s=n
y =
y
whereyis mean occupancy of each vehicle.
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Speed Estimation: Simple Solution
Presented to SSAI(WA) on 8 November 2005 15 / 32
Speed=Distance
Time
Hence simple estimator of speed is
s=n
y =
y
whereyis mean occupancy of each vehicle. But note
s=
y=
y
= s
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Speed Estimation: Simple Solution (cont.)
Presented to SSAI(WA) on 8 November 2005 16 / 32
Simple solution gives noisy estimation through time
Fails to properly account for variation in vehicle lengths.
4 5 6 7 8 9 10 11
20
40
60
80
100
120
Time of day
Speed(mph)
Speed Estimation: A Statisticians
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Speed Estimation: A Statistician s
Approach
Presented to SSAI(WA) on 8 November 2005 17 / 32
A statistician might:
Try and apply smoothing through time.
Model individual vehicle speeds and lengths as missing data. Not so hard to do post hoc, perhaps with MCMC for model fitting.
But traffic control often needs real-time estimates.
Speed Estimation: A Statisticians
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Speed Estimation: A Statistician s
Approach
Presented to SSAI(WA) on 8 November 2005 17 / 32
A statistician might:
Try and apply smoothing through time.
Model individual vehicle speeds and lengths as missing data. Not so hard to do post hoc, perhaps with MCMC for model fitting.
But traffic control often needs real-time estimates.
Research problem: develop real-time speed estimation methods
that are computationally cheap.
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Trip Matrix Estimation: The Problem
Presented to SSAI(WA) on 8 November 2005 18 / 32
Estimate intensity of traffic flow between origins and destinations
of travel on a network.
Available data are traffic counts on certain road links over given
time intervals.
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LONDON ROAD (A6)
REGENT ROAD
LANCASTER ROAD
WATERLOOROA
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UNIVERSITYROAD
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Trip Matrix Estimation: The Crux
Presented to SSAI(WA) on 8 November 2005 19 / 32
Trip matrix easy to estimate fromroutetraffic counts.
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LONDON ROAD (A6)
REGENT ROAD
LANCASTER ROAD
WATERLO
OROAD
UNIVERSITY
ROAD
T i M i E i i Th C ( )
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Trip Matrix Estimation: The Crux (cont.)
Presented to SSAI(WA) on 8 November 2005 20 / 32
But route flows generally not uniquely determined from a single
set of link counts.
Hence good trip matrix estimation impossible from aggregate link
counts unless other information is available (e.g. informative
prior).
O1 O2
D1 D2
C
10 10
10 10
T i M t i E ti ti T d S l ti
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Trip Matrix Estimation: Towards a Solution
Presented to SSAI(WA) on 8 November 2005 21 / 32
Sequence of set of link counts provides vital additional
information.
Trip matrix parameters are identifiable for e.g. Poisson model of
trip generation.
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T i M t i E ti ti Ch ll
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Trip Matrix Estimation: Challenges
Presented to SSAI(WA) on 8 November 2005 22 / 32
What is a good stochastic model for trip generation?
Well founded statistical approaches largely restricted to
estimation of static trip matrices.
Traffic control can require dynamic updating of trip matrices.
T i M t i E ti ti Ch ll
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Trip Matrix Estimation: Challenges
Presented to SSAI(WA) on 8 November 2005 22 / 32
What is a good stochastic model for trip generation?
Well founded statistical approaches largely restricted to
estimation of static trip matrices.
Traffic control can require dynamic updating of trip matrices.
Research problem: develop sound method of dynamic estimationfor trip matrices.
Modelling and Inference for Day-to-Day
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g y y
System Dynamics
Presented to SSAI(WA) on 8 November 2005 23 / 32
Aim: to model time series of day-to-day traffic counts on network.
Model can be:
Descriptive: used to understand properties of traffic system.
Predictive
Example applications:
estimate flow patterns after road closure;
assess effect of proposed bypass.
Elements of Day to Day Model
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Elements of Day-to-Day Model
Presented to SSAI(WA) on 8 November 2005 24 / 32
Travel demand model
trip matrix;
may be adjusted for seasonality, day of week etc. Traffic assignment model
describes how given demand will distributed itself over the
network;
based on aggregation of route choices for all travellers.
Modelling Traveller Route Choice
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Modelling Traveller Route Choice
Presented to SSAI(WA) on 8 November 2005 25 / 32
Route choice modelling pivotal.
Natural to define model at microscopic (individual traveller) level.
Travellers will select cheap routes:
minimum travel time
short distance
easy drive (Random) variation in perceptions of costs between travellers.
Properties of model required at macroscopic level (aggregate
traffic flows).
Markov Models
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Markov Models
Presented to SSAI(WA) on 8 November 2005 26 / 32
Travellers base route choice on combination of travel costs
experienced over finite history.
Inertia may suggest behaviour change only in case of clearly
superior alternative.
Pro:model properties relatively well understood.
Con:no attempt to represent contemporaneous traveller
interaction.
Inference for Behavioural Parameters
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Inference for Behavioural Parameters
Presented to SSAI(WA) on 8 November 2005 27 / 32
Parameters can represent:
sensitivity to cost differences
length of memory weighting of historical costs
Inference may be based on:
traffic counts (very indirect information) stated preference surveys (believable?)
travel diaries
Inference is far from straightforward in all cases.
Inference from Traffic Counts
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Inference from Traffic Counts
Presented to SSAI(WA) on 8 November 2005 28 / 32
Route flow evolve as Markov process.
Observed link flows linear combination of route flows, plus
measurement error.
Hidden Markov model technology applicable, but:
Model has peculiar structure;
Practical problems may have huge size (e.g. thousands ofroad links, even more routes, for urban network model).
Inference from Traffic Counts
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Inference from Traffic Counts
Presented to SSAI(WA) on 8 November 2005 28 / 32
Route flow evolve as Markov process.
Observed link flows linear combination of route flows, plus
measurement error.
Hidden Markov model technology applicable, but:
Model has peculiar structure;
Practical problems may have huge size (e.g. thousands of
road links, even more routes, for urban network model).
Research problem: develop methodology for inference for route
choice models from readily available data.
Image Analysis in Transportation
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Research
Presented to SSAI(WA) on 8 November 2005 29 / 32
Applications include:
number plate identification;
automatic evaluation of road surface cracks.
Environmental Statistics in Transportation
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Environmental Statistics in Transportation
Presented to SSAI(WA) on 8 November 2005 30 / 32
Spatial statistics applied to: modelling road vehicle emissions in
space and time;
modelling health impacts (both glob-ally and locally);
modelling impact of road accidents
with hazardous substances.
Further Statistical Problems in
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Transportation
Presented to SSAI(WA) on 8 November 2005 31 / 32
Survey design and analysis
Random utility modelling and inference
Stochastic operations research and system optimization
Road accident modelling
Summary
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Summary
Presented to SSAI(WA) on 8 November 2005 32 / 32
Transportation research likely to receive increasing attention as
fuel prices and environmental costs spiral.
Transportation science traditionally dominated by deterministic
models and methods.
Lots of opportunities for statisticians to make major contributions.