Incorporating Perceived Travel Time Reliability Into Transportation Planning and Simulation Models Using Information Entropy as the Measure Jiangbo Gabriel Yu R. Jayakrishnan jiangby@uci.edu rjayakri@uci.edu • Measuring perceived travel time uncertainty using current consensus of category-based perception and (quantum) information entropy in cognitive science in order to set up theoretical foundations for modeling and analysis on uncertainty. • Offering models with more flexibility especially comparing with variance-based measures. • linearly embedding uncertainty into general cost function, which is critical in considering perceived uncertainty in the time-of- departure, mode choice, and traffic assignment modeling. • Associating the input and output with economic/econometric concepts such as willingness-to-pay, time budget, time choice of departure, etc. • Demonstrating the importance of considering the initial condition and order of interventions versus taking it as a model’s shortcoming. • Cognitive Science: offers theoretical support in modeling perception and behavior, category-based perception, Misperception • Information Theory: Entropy is the natural candidate for measuring system uncertainty. • Bayesian Update: Travelers/Shippers’ perception is influenced by the historical events/information as well as the current observations (or traveler information with various degrees of beliefs). It has been considered a shortcomings that a model’s results change with different initial conditions, but this is suggested to reconsider. • Quantum Cognitive Model: Used in the field of cognitive science, finance, and decision making for modeling misperception (“fail of total probability theorem”). More general case: Quantum Markov Chain Model under Hilbert Space • Transportation Networks: Path-based traffic assignment • Economics/Econometrics: The methods proposed can be relatively easy to associate with Value of Time, Travel Time Budget, Willingness to Pay, and so on. • Examples: When traveler information systems are improved, the actual travel time distribution might not vary, while the perceived entropy (uncertainty) for travelers/shippers might be reduced or increased – traveler information becomes easier to incorporate into the models ; Due to misperception, the entropy reduction might be systematically under- /over-estimated. • Stochastic Segmentation of Perceptive Categories 1 , 2 ,.., , 1 , 2 ,…, | = , 1 , 2 ,…, >0, 0≤ 1 ≤ 2 ≤⋯≤ 0, ℎ • Information Entropy Measure =− ∀∈ ∙log where, is an event in the sample space . A continuous case makes the measure no more realistic due to the categorical perception nature of human. Function is the (perceived) probability of event to happen. • Combining Perceived Uncertainty with Stochastic Segmentation of Categories − ℝ ×1 ∀∈ | ∙log | is the probability density for the market section ∈. This also applies to discrete market segmentation by replacing integral with summation. • Bayesian Update for Sensed and Perceived Travel Time Distribution Θ|t Θ 1 ,Θ 2 ,Θ 3 … , ξ… ∝(Θ 1 ,Θ 2 ,Θ 3 ,…;)∙ Θ (Θ 1 ,Θ 2 ,Θ 3 ,..| ξ) Travel time distribution can be selected from gamma family (for non-negative support) and distribution for the hyper-parameters can be the corresponding conjugate priors in order to simplify the computation. • Misperception Using Quantum-like Cognitive Model in Hilbert Space Mental State under quantum superposition:| >= ∀ ∈ < | >∙ For example, >=>+ ℎ > + > Pr | 0 = < 1 >< 1 0 >+< 2 >< 2 0 > 2 =|< 1 >< 1 0 >| 2 +|< 2 >< 2 0 >| 2 +2(< 1 >< 1 0 >) ∗ < 2 >< 2 0 >∙cos() =− ∙ is the Von Neumam entropy, and = >< as the density matrix. X 0 1 , 1 2 , 2 1 , 1 2 , 2 Initial Mental State, |X0> Sensed (Actual) Uncertainty, |> Perceived Uncertainty with Interference |> 1 1 2 2 1 1 2 2 A B In order to test the methods and algorithms presented in this paper, a 24-link 12-node unimodal network (generally aggregated from that of Southern California Association of Government region) is used. The coefficients for converting entropy to general cost are estimated through minimizing the weighted differences between the actual observed flow data and the modeled one. Due to non- linear space, here only selected three coefficients to demonstrate. Algorithm Description: Step 0: Perform path-based traffic assignment in different scenarios in the scenario set to obtain the flow variation list (need to estimate the frequency for each scenario happening) so that any path-based travel cost distribution can be reconstructed later if need. (If need, each OD pair and user class has corresponding perceived prior travel time distribution for future update to obtain posterior distribution. To simplify the computation in this case study, we assumed “noninformative” prior to match the frequentists’ results). Step 1: Perform again the path-based traffic assignment in different scenarios in the scenario set, yet during which, add ∙ obtained from previous step onto the path cost to obtain the total cost (if need, the entropy should be calculated based on posterior distribution of the travel time rather than just the sample itself obtained in this step) before determine the shortest one in the path set K from r to s. Step 2: Compare both the link flows and variations with the previous assignment results to determine if it converges. If does, stop, or else, continue by going back to Step 1. Results Different coefficients for entropy-cost conversion might lead to change of astringency. CONCLUSION & FUTURE RESEARCH SELECTED REFERENCE Lyman, K., & Bertini, R. L. Using travel time reliability measures to improve regional transportation planning and operations. Transportation Research Record: Journal of the Transportation Research Board, 2046(1), 1-10. 2008 Anderson, J. A., Silverstein, J. W., Ritz, S. A., & Jones, R. S. Distinctive features, categorical perception, and probability learning: some applications of a neural model. Psychological Review, 84(5), 413. 1977 Harnad, S. Psychophysical and cognitive aspects of categorical perception: A critical overview. Categorical perception: The groundwork of cognition, 1-52. 1987. Busemeyer, J. R., Wang, Z., & Lambert-Mogiliansky, A. Empirical comparison of Markov and quantum models of decision making. Journal of Mathematical Psychology, 53(5), 423-433. 2009. Jayakrishnan, R., Tsai, W. T., Prashker, J. N., & Rajadhyaksha, S. A faster path-based algorithm for traffic assignment. University of California Transportation Center. 1994. PDF Perceived Travel Time Distribution of the categoriy bar (2-category case) Perceived travel time distribution PDF Perceived Travel Time _1 _2 _3 Category bars realization (4-category case) Models & Simulators (Scenario Manager) (Base) Scenarios/Alternatives Surveys and Detector Data Calibrate Traffic Patterns (Validate) Feedback Identify Institutions and Public 0 500 1000 1500 2000 2500 3000 3500 4000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Root Mean Square Error Iteration Theta1 Theta2 Theta3 This paper presented a scheme to incorporate travel time reliability in transport network planning model, using Shannon’s information entropy formulation and Von Neumann’s quantum information entropy formulation. Future work expected includes economic interpretation, coefficients estimation methods, quantum Bayesian, choice of prior and initial conditions, path- based solution uniqueness issues, test of different transformations of entropy-converted cost, incorporating into dynamic and activity-based models, and so on. METHODOLOGY OBJECTIVES BACKGROUND CASE TEST ON PATH-BASED TRAFFIC ASSIGNMENT T o Base, P=0.3 S1, P=0.25 S2, P=0.2 S3, P=0.15 S4, P=0.1 Weighted Average Weighted Variance Applying Assignment Model Without Considering Entropy Corridor 1, NB 99.0 106.3419 105.9213 106.4085 105.8283 106.3827 106.1771 0.077525 Corridor 1, SB 99.0 102.1285 101.9883 102.2552 102.2073 102.1794 102.1357 0.010456 Corridor 2, EB 60.0 61.8293 61.7735 61.6944 61.7091 61.7632 61.7632 0.003347 Corridor 2, WB 60.0 64.00508 64.11324 63.94818 63.9434 64.21629 64.03261 0.013822 Applying Assignment Model Considering Entropy (Theta3) Corridor 1, NB 99.0 107.1907 106.6498 107.3826 106.8813 107.0536 107.0337 0.079345 Corridor 1, SB 99.0 101.9435 101.7353 102.1017 101.9245 102.0056 101.9264 0.018143 Corridor 2, EB 60.0 61.09683 61.11214 61.08301 61.04685 61.06227 61.08694 0.000685 Corridor 2, WB 60.0 71.71383 71.56257 71.83199 71.55277 71.62671 71.66677 0.013637 Difference (%) Corridor 1, NB -- 0.79818 0.687775 0.915434 0.995008 0.630648 0.806765 2.34763 Corridor 1, SB -- -0.18114 -0.24807 -0.15011 -0.27669 -0.17009 -0.20492 73.5176 Corridor 2, EB -- -1.18466 -1.07062 -0.991 -1.07318 -1.13487 -1.09492 -79.5339 Corridor 2, WB -- 12.04397 11.61902 12.32844 11.90016 11.53978 11.9223 -1.33845