WHAT WILL THE ‘WORLD’ BE LIKE …

… … in 20 minutes’ time?in 20 minutes’ time?

… … the rest of today?the rest of today?

… … in two years’ time?in two years’ time?

… … in 20 years’ time?in 20 years’ time?

… … in 100 years’ time?in 100 years’ time?

… AND …

What does it mean to make predictions at each of What does it mean to make predictions at each of these time horizons?these time horizons?

What assumptions do we im/explicitly make when What assumptions do we im/explicitly make when extrapolating to the future?extrapolating to the future?

How relevant are past observations, and how can How relevant are past observations, and how can we make best use of them?we make best use of them?

What scale/resolution can we make predictions on What scale/resolution can we make predictions on (with ‘acceptable error’)?(with ‘acceptable error’)?

STATISTICAL INFERENCE FOR TRAFFIC NETWORK MODELS

David Watling

Institute for Transport StudiesUniversity of Leeds

d.p.watling@its.leeds.ac.uk

Open University, Milton Keynes, March 31st 2009

ACKNOWLEDGEMENTS

Joint researchers: Joint researchers: Richard Connors (Leeds)Richard Connors (Leeds)Shoichiro Nakayama (Kanazawa, Japan)Shoichiro Nakayama (Kanazawa, Japan)

Stimulating discussions with … Stimulating discussions with … Paul Timms (Leeds)Paul Timms (Leeds)Martin Hazelton (Massey, New Zealand)Martin Hazelton (Massey, New Zealand)

Financial support:Financial support:Grew out of earlier EPSRC fundingGrew out of earlier EPSRC fundingNational Japanese Visiting AwardNational Japanese Visiting Award

NETWORK MODELS : USER EQUILIBRIUM

Simplistic representation of large scale road Simplistic representation of large scale road network systems, going back to the 1950s.network systems, going back to the 1950s.

Simultaneously deal with the interaction of Simultaneously deal with the interaction of drivers’ route choices and congestion.drivers’ route choices and congestion.

Drivers play out a ‘game’ and reach a state where Drivers play out a ‘game’ and reach a state where they are satisfied.they are satisfied.

Attractive as can test/design traffic measures now, Attractive as can test/design traffic measures now, forecast impact of changes in demand (20 years?) forecast impact of changes in demand (20 years?) & test hypothetical policies (eg capacity, pricing)& test hypothetical policies (eg capacity, pricing)

In widespread use in practice, esp. at urban level.In widespread use in practice, esp. at urban level. Basic UE model since extended: intersections, Basic UE model since extended: intersections,

dynamics, departure time choice, uncertainty, etc.dynamics, departure time choice, uncertainty, etc.

EXAMPLE: UE MODEL

O D

O-D flow = 7 c1(f1) = 2 + f12

c2(f2) = 1 + f2

Generalised travel cost on route 2,typically a combination of traveltime, distance, tolls, etc.

Flow on route 2

f2

f1

EXAMPLE: UE MODEL

O D

O-D flow = 7 c1(f1) = 2 + f12

c2(f2) = 1 + f2

Generalised travel cost on route 2 Flow on route 2

f1

f2

UE solution: (f1,f2) = (2,5) when (c1,c2) = (6,6)

NOT MUCH ROOM FOR STATISTICS?

UE: a deterministic model, calibration typically UE: a deterministic model, calibration typically done by trial-and-error.done by trial-and-error.

How can we bring in statistical inference?How can we bring in statistical inference?

In late 1970s, a generalisation proposed: SUE.In late 1970s, a generalisation proposed: SUE. Assume drivers make random perceptual errors …Assume drivers make random perceptual errors …

EXAMPLE: SUE MODEL

O D

O-D flow = 7 c1(f1) = 2 + f12

c2(f2) = 1 + f2

f1

f2

e.g. SUE solution: (f1,f2) = (2.25,4.75) if 2 = 4

f1 = 7 Pr(c1(f1) + ε1 ≤ c2(f2) + ε2)f2 = 7 – f1

(ε1,ε2) ~ MVN( (0,0), 2I )

DOES THAT HELP?

Not really, as data typically traffic countsNot really, as data typically traffic counts

SUE model is still deterministic.SUE model is still deterministic.

Some possible remedies:Some possible remedies:

• ‘‘Generalised’ SUE modelGeneralised’ SUE model

• Markov process model of day-to-day dynamicsMarkov process model of day-to-day dynamics

EXAMPLE: ‘GENERALISED’ SUE

O D

O-D flow = 7 h1(f1) = E[c1(F1)] = E[ 2 + F12 ]

h2(f2) = E[c2(F2)] = E[ 1 + F2 ]

F1

F2

Fi ~ Poisson(fi) (i =1,2; independent)

f1 = 7 Pr(h1(f1) + ε1 ≤ h2(f2) + ε2)f2 = 7 – f1 (ε1,ε2) ~ MVN( (0,0), 2I )

e.g. GSUE solution: (f1,f2) = (1.94,5.06) if 2 = 4

STATISTICAL INFERENCE

As an example, suppose:As an example, suppose:

• Observe flow vector Observe flow vector XX(i)(i) on a sample of links on a sample of links

• … … over several days {over several days {XX(1)(1),…,,…,XX(n)(n)} = } = XX

• Dependent within-day, but i.i.d. over daysDependent within-day, but i.i.d. over days

• GSUE model fitted to determine GSUE model fitted to determine

STATISTICAL INFERENCE

Maximise log-likelihood L(Maximise log-likelihood L(, f | , f | X X ))

subject to constraint: f = subject to constraint: f = ΦΦ(f ; (f ; ) .) .

STATISTICAL INFERENCE

Maximise log-likelihood L(Maximise log-likelihood L(, f | , f | X X ))

subject to constraint: f = subject to constraint: f = ΦΦ(f ; (f ; ) .) .i.e. f is a GSUEsolution given

internally determined from

determined bymax likelihood

STATISTICAL INFERENCE

Maximise log-likelihood L(Maximise log-likelihood L(, f | , f | X X ))

subject to constraint: f = subject to constraint: f = ΦΦ(f ; (f ; ) .) .

• Complex constraint Complex constraint unusual ML problem unusual ML problem

• Efficient gradient-based algorithm developed Efficient gradient-based algorithm developed for general networks, solve using sensitivity for general networks, solve using sensitivity analysis on implicit mathematical program.analysis on implicit mathematical program.

CASE STUDY EXAMPLE

J apan Railway

Kanazawa Station

0 1km 2km0 1km 2km

Fig. 2 Kanazawa Road Network

Nodes 140Links 472ODs 1383

CASE STUDY EXAMPLE

In Kanazawa example, estimated parameter In Kanazawa example, estimated parameter of logit-based GSUE model, where:of logit-based GSUE model, where:

pprr = Pr( = Pr(E[cE[crr((FF)] + )] + εεrr ≤ ≤ E[cE[css((FF)] + )] + εεss s)s)

where where εεii ~ Gumbel, i.i.d. ~ Gumbel, i.i.d.

MLE: 0.169 (99% C.I.: 0.163 to 0.175)MLE: 0.169 (99% C.I.: 0.163 to 0.175)

LSE:LSE: 2.080 . 2.080 .

ALTERNATIVE TO GSUE:DAY-TO-DAY DYNAMIC MARKOV PROCESS

Decision model

Initialisation: day k = 0

Memory filter

Traffic loadingIncrement day:

k = k + 1

DAY-TO-DAY DYNAMIC MARKOV PROCESS

U(k) = w1 C(k) + w2 C(k–1) + … + wm C(k–m+1)

C(k) = c(F(k))

Fi(k) | U(k–1) ~ Multinomial( di , pi(U(k–1)) )

EQUILIBRIUM, LIMITS & SCALE

m-dependent Markov process, with discrete network link flows as state variables

‘Equilibrium’ now relates to equilibrium joint probability distribution of m-sequences of network flows, {F(k), F(k1), … , F(km+1)} .

p(.) has infinite support existence & uniqueness of equilibrium distribution (Cascetta, 1989).

Eq. dist. of SP → Nor( SUE, (SUE, w1, w2, …) ) Multi-scale theory: can apply to individual level. Future: a sounder theoretical basis for inference?

d

INFERENCE VS. SCALE

GSUE or Markov Process addresses problem of stochasticity in network models, paving way for statistical inference.

Markov Process approach also gives scaleable theory: can apply to individual decisions as much as macro-level.

But data may not be at decision-maker (traveller) level: can we also infer what network resolution appropriate to data?

To do so, need scaleable network problems …d

City Centre

Origin

Train

Mode/route choice:

• 240 OD movements• 188 road links• Focus on one movement• Mode: Car vs car+train

Multi-scale network example

Train

City Centre

Origin

Via Road Network

To Station

Aim: Replace entire road network with a single link.

Qu: What is the “cost function” for this link?

Road demand changesfor all OD movements

Equilibrium route choicechanges for network

Link flows/times change

OD travel times/costs changefor all OD movements

Train

City Centre

Origin

Original cost function on “rail” link.

Approx OD cost on road network link, derived by analytical sensitivity analysis of SUE problem.

Multi-scale Predictions of Commuter Flows

Road

Rail

CONCLUSIONS & FUTURE

Traditional UE/SUE models for medium-term planning are deterministic, therefore do not support statistical inference

GSUE or Markov Process address problem of stochasticity in network models, paving way for statistical inference.

Theory can also be made scaleable at both the decision-maker and network levels.

Putting these tools together provides powerful methods for using new data sources, themselves at different scales.

Systematic theory also allows judgement of statistical quality of the models derived.

d