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W HAT W ILL T HE ‘W ORLD ’ B E L IKE …

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?. … A ND …. What does it mean to make predictions at each of these time horizons?

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W HAT W ILL T HE ‘W ORLD ’ B E L IKE …

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  1. WHAT WILL THE ‘WORLD’BE LIKE … … in 20 minutes’ time? … the rest of today? … in two years’ time? … in 20 years’ time? … in 100 years’ time?

  2. … AND … 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. STATISTICAL INFERENCE FORTRAFFIC NETWORK MODELS David Watling Institute for Transport Studies University of Leeds d.p.watling@its.leeds.ac.uk Open University, Milton Keynes, March 31st 2009

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

  5. NETWORK MODELS : USER EQUILIBRIUM • 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. EXAMPLE: UE MODEL c1(f1) = 2 + f12 O-D flow = 7 f1 O D f2 c2(f2) = 1 + f2 Generalised travel cost on route 2, typically a combination of travel time, distance, tolls, etc. Flow on route 2

  7. EXAMPLE: UE MODEL c1(f1) = 2 + f12 O-D flow = 7 f1 O D f2 c2(f2) = 1 + f2 Generalised travel cost on route 2 Flow on route 2 UE solution: (f1,f2) = (2,5) when (c1,c2) = (6,6)

  8. NOT MUCH ROOM FOR STATISTICS? • 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. EXAMPLE: SUE MODEL c1(f1) = 2 + f12 O-D flow = 7 f1 O D f2 c2(f2) = 1 + f2 f1 = 7 Pr(c1(f1) + ε1 ≤ c2(f2) + ε2) f2 = 7 – f1 (ε1,ε2) ~ MVN( (0,0), 2I ) e.g. SUE solution: (f1,f2) = (2.25,4.75) if 2 = 4

  10. DOES THAT HELP? Not really, as data typically traffic counts  SUE model is still deterministic. Some possible remedies: • ‘Generalised’ SUE model • Markov process model of day-to-day dynamics

  11. EXAMPLE: ‘GENERALISED’ SUE h1(f1) = E[c1(F1)] = E[ 2 + F12 ] O-D flow = 7 F1 O D F2 h2(f2) = E[c2(F2)] = E[ 1 + 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

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

  13. STATISTICAL INFERENCE Maximise log-likelihood L(, f | X ) subject to constraint: f = Φ(f ; ) .

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

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

  16. CASE STUDY EXAMPLE Nodes 140 Links 472 ODs 1383

  17. CASE STUDY EXAMPLE In Kanazawa example, estimated parameter of logit-based GSUE model, where: pr = Pr(E[cr(F)] + εr ≤ E[cs(F)] + εs s) where εi ~ Gumbel, i.i.d. MLE: 0.169 (99% C.I.: 0.163 to 0.175) LSE: 2.080 .

  18. ALTERNATIVE TO GSUE:DAY-TO-DAY DYNAMIC MARKOV PROCESS Initialisation: day k = 0 Decision model Increment day: k = k + 1 Traffic loading Memory filter

  19. 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)) )

  20. 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(k1), … , F(km+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

  21. 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

  22. Origin Multi-scale network example • Mode/route choice: • 240 OD movements • 188 road links • Focus on one movement • Mode: Car vs car+train Train City Centre

  23. Origin Aim: Replace entire road network with a single link. Qu: What is the “cost function” for this link? To Station Via Road Network Road demand changes for all OD movements Train Equilibrium route choice changes for network Link flows/times change OD travel times/costs change for all OD movements City Centre

  24. Origin Original cost function on “rail” link. Approx OD cost on road network link, derived by analytical sensitivity analysis of SUE problem. Train City Centre

  25. Multi-scale Predictions of Commuter Flows Road Rail

  26. 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

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