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Consistent Estimation of Revealed Route Choice Preferences for Dynamic Transit Assignment

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Jeff Hood, Principal, Hood Transportation Consulting

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Consistent Estimation of Revealed Route Choice Preferences for Dynamic Transit Assignment

  1. 1. Consistent Estimation of Route Choice Models for Dynamic Transit Assignment Portland State University Transportation Seminar Portland, Oregon February 16, 2016 Jeff Hood
  2. 2. Timed transfer Long walkFrequent service Rail comfort Crowded vehicle Short wait Quick trip
  3. 3. Travel Demand Model Dynamic Transit Assignment Vehicle Simulation Mode Choice Destination Choice Tour/Trip Generation Travel Costs Transit Trips
  4. 4. Introduction (you are here!) Limitations of path-based methods New correction for route overlap Reliability and stochastic arrivals The recursive logit model𝑃𝑃(π‘Žπ‘Ž|π‘˜π‘˜)
  5. 5. Hood Transportation ConsultingHood Transportation Consulting
  6. 6. Hood et al. (2011). Transport. Letters 3,63-75.
  7. 7. -3 -2 -1 0 1 2 3 0.00.51.01.52.0 Consistent Estimator Error ProbabilityDensity N = 1 N = 10 N = 100 N = 1000 N = INF -3 -2 -1 0 1 2 3 0.00.51.01.52.0 Inconsistent Estimator Error ProbabilityDensity N = 1 N = 10 N = 100 N = 1000 N = INF Consistent Estimator Inconsistent Estimator
  8. 8. Base Build Base Build Stochastic Sampling Path Size Link Penalty Nassir et al. (2014). Transport. Res. Rec. 2430, 170-181.
  9. 9. Hood Transportation Consulting 𝑃𝑃(π‘Žπ‘Ž|π‘˜π‘˜)
  10. 10. Fosgerau et al. (2014) Transport. Res. B: 56, 70-80 π‘˜π‘˜: current link π‘Žπ‘Ž: possible movement from π‘˜π‘˜ 𝑑𝑑: destination 𝑉𝑉(π‘Žπ‘Ž): expected max. utility of all paths from π‘Žπ‘Ž to 𝑑𝑑 𝐴𝐴(π‘˜π‘˜): set of all successors of π‘˜π‘˜ 𝑣𝑣(π‘Žπ‘Ž|π‘˜π‘˜): β€œinstantaneous” utility of moving from π‘Žπ‘Ž to π‘˜π‘˜
  11. 11. Recursive value equation: 𝑉𝑉 π‘˜π‘˜ = 𝐸𝐸 max π‘Žπ‘Žβˆˆπ΄π΄(π‘˜π‘˜) 𝑣𝑣 π‘Žπ‘Ž π‘˜π‘˜ + 𝑉𝑉 π‘Žπ‘Ž + πœ‡πœ‡πœ‡πœ‡(π‘Žπ‘Ž) πœ€πœ€(π‘Žπ‘Ž) i.i.d. extreme value type I implies… Logit transition probabilities: 𝑃𝑃 π‘Žπ‘Ž π‘˜π‘˜ = exp 1 πœ‡πœ‡ 𝑣𝑣 π‘Žπ‘Ž π‘˜π‘˜ +𝑉𝑉 π‘Žπ‘Ž βˆ‘ π‘Žπ‘Žβ€²βˆˆπ΄π΄(π‘˜π‘˜) exp 1 πœ‡πœ‡ 𝑣𝑣 π‘Žπ‘Žβ€² π‘˜π‘˜ +𝑉𝑉 π‘Žπ‘Žβ€² Value equation is logsum: 𝑉𝑉 π‘˜π‘˜ = πœ‡πœ‡ log οΏ½ π‘Žπ‘Žβˆˆπ΄π΄(π‘˜π‘˜) exp 1 πœ‡πœ‡ 𝑣𝑣 π‘Žπ‘Ž π‘˜π‘˜ + 𝑉𝑉 π‘Žπ‘Ž
  12. 12. If β€’ 𝐌𝐌 = matrix of exponentiated utilities exp[𝑣𝑣 π‘Žπ‘Ž π‘˜π‘˜ /πœ‡πœ‡] β€’ 𝐛𝐛 = indicator vector for the destination β€’ 𝐳𝐳 = desired vector of values exp[𝑉𝑉(π‘˜π‘˜)/πœ‡πœ‡] Then the Bellman value equation is 𝐳𝐳 = 𝐌𝐌𝐌𝐌 + 𝐛𝐛 Result is equivalent to link-additive path-based model with unrestricted choice set
  13. 13. Hood Transportation Consulting
  14. 14. dπ‘˜π‘˜0 𝐡 𝐴𝐴 𝐢 𝐷 Uncorrelated Errors Correlated Errors dπ‘˜π‘˜0 𝐡 𝐴𝐴 𝐢 𝐷 Path-Based β€œSize” Correction Path Size Prob. A 1 0.33 BC 1 0.33 BD 1 0.33 Path Size Prob. A 1 0.50 BC 0.5 0.25 BD 0.5 0.25 A B C D A B C D
  15. 15. Performs Poorly β€’ Not sensitive to extent of overlapping links β€’ Conflates route overlap and route utility β€’ Requires scaling parameter β€’ Not topologically-invariant
  16. 16. # downstream path segments # upstream path segments # paths containing π‘˜π‘˜ 𝑁𝑁 𝑑𝑑 (π‘˜π‘˜) Γ— 𝑁𝑁 𝑒𝑒 (π‘˜π‘˜) 𝑁𝑁 𝑑𝑑 (π‘˜π‘˜) = οΏ½ 𝑁𝑁 𝑑𝑑 (π‘Žπ‘Ž) π‘Žπ‘Žβˆˆπ΄π΄(π‘˜π‘˜) 𝑁𝑁 𝑒𝑒 (π‘˜π‘˜) = οΏ½ 𝑁𝑁 𝑒𝑒 (π‘Žπ‘Ž) π‘Žπ‘Žβˆˆπ΄π΄(π‘˜π‘˜) Equations have no solution in cyclic networks!
  17. 17. Probability-scaled downstream path segments Probability-scaled upstream path segments (𝐹𝐹(π‘Žπ‘Ž) is uncorrected link flow) 𝑁𝑁� 𝑑𝑑 (π‘˜π‘˜) = 𝑁𝑁� 𝑒𝑒 (π‘˜π‘˜) = οΏ½ 𝑃𝑃(π‘Žπ‘Ž|π‘˜π‘˜) maxπ‘Žπ‘Žβ€² ∈𝐴𝐴 𝑃𝑃(π‘Žπ‘Ž|π‘˜π‘˜) π‘Žπ‘Žβˆˆπ΄π΄ 𝑁𝑁� 𝑑𝑑 (π‘Žπ‘Ž) οΏ½ 𝐹𝐹(π‘Žπ‘Ž)𝑃𝑃(π‘˜π‘˜|π‘Žπ‘Ž) maxπ‘Žπ‘Žβ€² ∈𝐴𝐴 𝐹𝐹(π‘Žπ‘Žβ€² ) 𝑃𝑃(π‘Žπ‘Žβ€² ) π‘Žπ‘Žβˆˆπ΄π΄ 𝑁𝑁� 𝑒𝑒 (π‘Žπ‘Ž)
  18. 18. D Prob: 0.33 Scaled Path Count: 3 Prob: 0.67 Scaled Path Count: 2 S.P.C. = 0.33 0.67 Γ— 3 + 0.67 0.67 Γ— 2 = 1.5 + 2.0 = 3.5
  19. 19. 𝐿𝐿𝐿𝐿(π‘˜π‘˜) = π‘šπ‘š(π‘˜π‘˜) �𝑀𝑀 π‘˜π‘˜ log �𝑁𝑁 𝑑𝑑 (π‘˜π‘˜) �𝑁𝑁 𝑒𝑒 (π‘˜π‘˜) π‘šπ‘š π‘˜π‘˜ : measure of link extent �𝑀𝑀 π‘˜π‘˜ : expected measure of all paths containing π‘˜π‘˜
  20. 20. dπ‘˜π‘˜0 𝐡 𝐴𝐴 𝐢 𝐷 Link Measure π’Žπ’Ž(𝒂𝒂) Travel Time A 1 1 B 𝑑𝑑 𝑑𝑑 C 1 βˆ’ 𝑑𝑑 1 βˆ’ 𝑑𝑑 D 1 βˆ’ 𝑑𝑑 1 βˆ’ 𝑑𝑑 A B C D
  21. 21. 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0 0.5 1 FlowonLinkB Proportion of Travel Time on Link B LS proposed LS unscaled LS flow Nested Logit
  22. 22. dπ‘˜π‘˜0 𝐡 𝐴𝐴 𝐢 𝐷 Link Measure π’Žπ’Ž(𝒂𝒂) Travel Time A 1 1 B 0.5 0.5 C 0.5 0.5 D 𝑑𝑑 𝑑𝑑 A B C D
  23. 23. 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.0 0.5 1.0 FlowonUpstreamLinkB Travel Time on Downstream Link D LS proposed LS unscaled LS flow Nested Logit
  24. 24. Hood Transportation Consulting
  25. 25. Poor schedule adherence reduces boarding probability for multiple reasons β€’ Direct disutility of excess wait times β€’ Missed connections β€’ Lateness in arrival sequence β€œReliability” term in utility function will not work Problem requires dynamic choice probabilities
  26. 26. -5 0 5 10 15 0.00.20.40.60.81.0 ArrivalProbabi dwe𝑏𝑏𝑏 arrπ‘˜π‘˜depπ‘Ÿπ‘Ÿπ‘Ÿ dwe𝑔𝑔𝑔 arrπ‘Ÿπ‘Ÿ2 arr𝑏𝑏2 arrπ‘Ÿπ‘Ÿπ‘Ÿ arr𝑔𝑔𝑔 = ? 𝐴𝐴 π΄π΄βˆ— 𝐴𝐴 𝒕𝒕 (min.)
  27. 27. 𝑃𝑃 π‘Žπ‘ŽοΏ½π‘˜π‘˜, ̅𝐴𝐴, π‘Žπ‘Ž ∈ π΄π΄βˆ— = 𝑒𝑒 1 πœ‡πœ‡ 𝛽𝛽dwe 𝐸𝐸 dweπ‘Žπ‘Ž +𝑣𝑣 π‘Žπ‘Ž π‘˜π‘˜ +𝑉𝑉 π‘Žπ‘Ž βˆ‘π‘Žπ‘Žβ€²βˆˆπ΄π΄βˆ— π‘˜π‘˜ 𝑒𝑒 1 πœ‡πœ‡ 𝛽𝛽dwe 𝐸𝐸 dweπ‘Žπ‘Žβ€² +𝑣𝑣 π‘Žπ‘Žβ€² π‘˜π‘˜ +𝑉𝑉 π‘Žπ‘Žβ€² + 𝑒𝑒 1 πœ‡πœ‡ 𝐸𝐸 π‘ˆπ‘ˆwaitοΏ½π‘˜π‘˜, ̅𝐴𝐴 Depends on expected utility of waiting 𝐸𝐸 π‘ˆπ‘ˆwaitοΏ½π‘˜π‘˜, ̅𝐴𝐴
  28. 28. Ξ¦π‘˜π‘˜ 𝑑𝑑� ̅𝐴𝐴 = 𝑃𝑃 arrπ‘˜π‘˜ 𝑑𝑑� ̅𝐴𝐴 Recursive formula depending on β€’ Conditional distribution of arrπ‘˜π‘˜ given ̅𝐴𝐴 β€’ Conditional probability that next arrival is π‘Žπ‘Žπ‘–π‘– β€’ Expected utility of waiting for π‘Žπ‘Žπ‘–π‘– 𝑃𝑃 min π‘Žπ‘Žπ‘—π‘—βˆˆ ̅𝐴𝐴 arrπ‘Žπ‘Žπ‘—π‘— = arrπ‘Žπ‘Žπ‘–π‘– 𝐸𝐸 𝑀𝑀(π‘Žπ‘Žπ‘–π‘–|π‘˜π‘˜, arrπ‘˜π‘˜) = οΏ½ 0 ∞ 𝑀𝑀(π‘Žπ‘Žπ‘–π‘–|π‘˜π‘˜, 𝑑𝑑+ )π‘‘π‘‘οΏ½Ξ¦π‘Žπ‘Ž+ 𝑑𝑑�𝐴𝐴+
  29. 29. 𝑃𝑃 π‘Žπ‘Ž|wait( ̅𝐴𝐴) = οΏ½ π‘Žπ‘Žπ‘–π‘–βˆˆ ̅𝐴𝐴 𝑃𝑃 min π‘Žπ‘Žπ‘—π‘—βˆˆ ̅𝐴𝐴 arrπ‘Žπ‘Žπ‘—π‘— = arrπ‘Žπ‘Žπ‘–π‘– Γ— 𝛿𝛿 π‘Žπ‘Ž = π‘Žπ‘Žπ‘–π‘– 𝑃𝑃 board ̅𝐴𝐴{π‘Žπ‘Žπ‘–π‘–} + 𝛿𝛿 π‘Žπ‘Ž β‰  π‘Žπ‘Žπ‘–π‘– 𝑃𝑃 wait ̅𝐴𝐴{π‘Žπ‘Žπ‘–π‘–} 𝑃𝑃 π‘Žπ‘Ž|wait ̅𝐴𝐴{π‘Žπ‘Žπ‘–π‘–}
  30. 30. 𝑃𝑃 π‘Žπ‘Ž|π‘˜π‘˜ = οΏ½ 𝑖𝑖=0 𝐴𝐴 π‘˜π‘˜ οΏ½ 𝑗𝑗=0 𝑖𝑖 οΏ½ 𝐴𝐴∈𝐢𝐢 𝐴𝐴 π‘˜π‘˜ ,𝑖𝑖 οΏ½ Μ…π΄π΄βˆˆπΆπΆ 𝐴𝐴 π‘˜π‘˜ 𝐴𝐴,𝑗𝑗 𝑃𝑃 𝐴𝐴, ̅𝐴𝐴, π΄π΄βˆ— Γ— 𝛿𝛿 π‘Žπ‘Ž ∈ π΄π΄βˆ— Γ— 𝑃𝑃 π‘Žπ‘ŽοΏ½π‘˜π‘˜, ̅𝐴𝐴, π‘Žπ‘Ž ∈ π΄π΄βˆ— + 𝛿𝛿 π‘Žπ‘Ž ∈ ̅𝐴𝐴 1 βˆ’ οΏ½ π‘Žπ‘Žβ€²βˆˆπ΄π΄βˆ— 𝑃𝑃 π‘Žπ‘ŽοΏ½π‘˜π‘˜, ̅𝐴𝐴, π‘Žπ‘Ž ∈ π΄π΄βˆ— Γ— 𝑃𝑃 π‘Žπ‘Ž|wait ( ̅𝐴𝐴) If delays are independent exponentials, there is an (excruciating) closed-form solution
  31. 31. Hood Transportation Consulting Kathryn Coffel David Ory Lisa Zorn Shimon Israel Suzanne Childress Stefan Coe Joe Castiglione Drew Cooper Elizabeth Sall Alireza Khani Emma Frejinger Tien Mai Contact: jeff@hoodconsulting.net

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