1. W HAT W ILL T HE ‘W ORLD ’ B E L IKE … … in 20 minutes’ time? … the rest of today? … in two years’ time? … in 20 years’ time? … in 100 years’ time?

2. … A ND … What does it mean to make predictions at each of these time horizons? What assumptions do we im/explicitly make when extrapolating to the future? How relevant are past observations, and how can we make best use of them? What scale/resolution can we make predictions on (with ‘acceptable error’)?

3. S TATISTICAL I NFERENCE F OR T RAFFIC N ETWORK M ODELS David Watling Institute for Transport Studies University of Leeds d.p.watling@its.leeds.ac.uk Open University, Milton Keynes, March 31 st 2009

4. A CKNOWLEDGEMENTS Joint researchers: Joint researchers: Richard Connors (Leeds) Shoichiro Nakayama (Kanazawa, Japan) Stimulating discussions with … Stimulating discussions with … Paul Timms (Leeds) Martin Hazelton (Massey, New Zealand) Financial support: Financial support: Grew out of earlier EPSRC funding National Japanese Visiting Award

5. N ETWORK M ODELS : U SER E QUILIBRIUM  Simplistic representation of large scale road network systems, going back to the 1950s.  Simultaneously deal with the interaction of drivers’ route choices and congestion.  Drivers play out a ‘game’ and reach a state where they are satisfied.  Attractive as can test/design traffic measures now, forecast impact of changes in demand (20 years?) & test hypothetical policies (eg capacity, pricing)  In widespread use in practice, esp. at urban level.  Basic UE model since extended: intersections, dynamics, departure time choice, uncertainty, etc.

6. E XAMPLE: UE M ODEL OD O-D flow = 7 c 1 (f 1 ) = 2 + f 1 2 c 2 (f 2 ) = 1 + f 2 Generalised travel cost on route 2, typically a combination of travel time, distance, tolls, etc. Flow on route 2 f2f2 f1f1

7. E XAMPLE: UE M ODEL OD O-D flow = 7 c 1 (f 1 ) = 2 + f 1 2 c 2 (f 2 ) = 1 + f 2 Generalised travel cost on route 2Flow on route 2 f1f1 f2f2 UE solution: (f 1,f 2 ) = (2,5) when (c 1,c 2 ) = (6,6)

8. N OT M UCH R OOM F OR S TATISTICS ?  UE: a deterministic model, calibration typically done by trial-and-error.  How can we bring in statistical inference?  In late 1970s, a generalisation proposed: SUE.  Assume drivers make random perceptual errors …

9. E XAMPLE: SUE M ODEL OD O-D flow = 7 c 1 (f 1 ) = 2 + f 1 2 c 2 (f 2 ) = 1 + f 2 f1f1 f2f2 e.g. SUE solution: (f 1,f 2 ) = (2.25,4.75) if 2 = 4 f 1 = 7 Pr(c 1 (f 1 ) + ε 1 ≤ c 2 (f 2 ) + ε 2 ) f 2 = 7 – f 1 (ε 1,ε 2 ) ~ MVN( (0,0), 2 I )

10. D OES T HAT H ELP ? Not really, as data typically traffic counts  SUE model is still deterministic. Some possible remedies: ‘Generalised’ SUE model ‘Generalised’ SUE model Markov process model of day-to-day dynamics Markov process model of day-to-day dynamics

11. E XAMPLE: ‘G ENERALISED ’ SUE OD O-D flow = 7 h 1 (f 1 ) = E[c 1 (F 1 )] = E[ 2 + F 1 2 ] h 2 (f 2 ) = E[c 2 (F 2 )] = E[ 1 + F 2 ] F1F1 F2F2 F i ~ Poisson(f i ) (i =1,2; independent) f 1 = 7 Pr(h 1 (f 1 ) + ε 1 ≤ h 2 (f 2 ) + ε 2 ) f 2 = 7 – f 1 (ε 1,ε 2 ) ~ MVN( (0,0), 2 I ) e.g. GSUE solution: (f 1,f 2 ) = (1.94,5.06) if 2 = 4

12. S TATISTICAL I NFERENCE As an example, suppose: Observe flow vector X (i) on a sample of links Observe flow vector X (i) on a sample of links … over several days {X (1),…,X (n) } = X … over several days {X (1),…,X (n) } = X Dependent within-day, but i.i.d. over days Dependent within-day, but i.i.d. over days GSUE model fitted to determine GSUE model fitted to determine

13. S TATISTICAL I NFERENCE Maximise log-likelihood L(, f | X ) subject to constraint: f = Φ(f ; ).

14. S TATISTICAL I NFERENCE Maximise log-likelihood L(, f | X ) subject to constraint: f = Φ(f ; ). i.e. f is a GSUE solution given internally determined from determined by max likelihood

15. S TATISTICAL I NFERENCE Maximise log-likelihood L(, f | X ) subject to constraint: f = Φ(f ; ). Complex constraint  unusual ML problem Complex constraint  unusual ML problem Efficient gradient-based algorithm developed for general networks, solve using sensitivity analysis on implicit mathematical program. Efficient gradient-based algorithm developed for general networks, solve using sensitivity analysis on implicit mathematical program.

16. C ASE S TUDY E XAMPLE Nodes 140 Links 472 ODs 1383

17. C ASE S TUDY E XAMPLE In Kanazawa example, estimated parameter of logit-based GSUE model, where: p r = Pr( E[c r (F)] + ε r ≤ E[c s (F)] + ε s  s) where ε i ~ Gumbel, i.i.d. MLE: 0.169 (99% C.I.: 0.163 to 0.175) LSE: 2.080.

18. A LTERNATIVE T O GSUE: D AY-TO- D AY D YNAMIC M ARKOV P ROCESS Decision model Initialisation: day k = 0 Memory filter Traffic loading Increment day: k = k + 1

19. D AY-TO- D AY D YNAMIC M ARKOV P ROCESS U (k) = w 1 C (k) + w 2 C (k–1) + … + w m C (k–m+1) C (k) = c(F (k) ) F i (k) | U (k–1) ~ Multinomial( d i, p i (U (k–1) ) )

20. E QUILIBRIUM, L IMITS & S CALE 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 (k  1), …, F (k  m+1) }. p(.) has infinite support  existence & uniqueness of equilibrium distribution (Cascetta, 1989). Eq. dist. of SP → Nor( SUE,  (SUE, w 1, w 2, …) ) Multi-scale theory: can apply to individual level. Future: a sounder theoretical basis for inference? d

21. I NFERENCE VS. S CALE 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

22. 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

23. 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 changes for all OD movements Equilibrium route choice changes for network Link flows/times change OD travel times/costs change for all OD movements

24. 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.

25. Multi-scale Predictions of Commuter Flows Road Rail

26. C ONCLUSIONS & F UTURE 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