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On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models

Author

Listed:
  • Yidan Shangguan

    (Department of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong Kong)

  • Xuecheng Tian

    (Department of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong Kong)

  • Sheng Jin

    (College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China)

  • Kun Gao

    (Department of Architecture and Civil Engineering, Chalmers University of Technology, 412 96 Göteborg, Sweden)

  • Xiaosong Hu

    (State Key Laboratory of Mechanical Transmission/Automotive Collaborative Innovation Center, Chongqing University, Chongqing 400044, China)

  • Wen Yi

    (Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Hong Kong)

  • Yu Guo

    (Department of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong Kong)

  • Shuaian Wang

    (Department of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong Kong)

Abstract

In traffic flow, the relationship between speed and density exhibits decreasing monotonicity and continuity, which is characterized by various models such as the Greenshields and Greenberg models. However, some existing models, i.e., the Underwood and Northwestern models, introduce bias by incorrectly utilizing linear regression for parameter calibration. Furthermore, the lower bound of the fitting errors for all these models remains unknown. To address above issues, this study first proves the bias associated with using linear regression in handling the Underwood and Northwestern models and corrects it, resulting in a significantly lower mean squared error (MSE). Second, a quadratic programming model is developed to obtain the lower bound of the MSE for these existing models. The relative gaps between the MSEs of existing models and the lower bound indicate that the existing models still have a lot of potential for improvement.

Suggested Citation

  • Yidan Shangguan & Xuecheng Tian & Sheng Jin & Kun Gao & Xiaosong Hu & Wen Yi & Yu Guo & Shuaian Wang, 2023. "On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models," Mathematics, MDPI, vol. 11(16), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3460-:d:1213924
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    References listed on IDEAS

    as
    1. Wu, Xinkai & Liu, Henry X. & Geroliminis, Nikolas, 2011. "An empirical analysis on the arterial fundamental diagram," Transportation Research Part B: Methodological, Elsevier, vol. 45(1), pages 255-266, January.
    2. Castillo, J. M. Del & Benítez, F. G., 1995. "On the functional form of the speed-density relationship--I: General theory," Transportation Research Part B: Methodological, Elsevier, vol. 29(5), pages 373-389, October.
    3. Harold Greenberg, 1959. "An Analysis of Traffic Flow," Operations Research, INFORMS, vol. 7(1), pages 79-85, February.
    4. Leslie C. Edie, 1961. "Car-Following and Steady-State Theory for Noncongested Traffic," Operations Research, INFORMS, vol. 9(1), pages 66-76, February.
    5. G. F. Newell, 1961. "Nonlinear Effects in the Dynamics of Car Following," Operations Research, INFORMS, vol. 9(2), pages 209-229, April.
    6. Wu, Ning, 2002. "A new approach for modeling of Fundamental Diagrams," Transportation Research Part A: Policy and Practice, Elsevier, vol. 36(10), pages 867-884, December.
    7. Castillo, J. M. Del & Benítez, F. G., 1995. "On the functional form of the speed-density relationship--II: Empirical investigation," Transportation Research Part B: Methodological, Elsevier, vol. 29(5), pages 391-406, October.
    8. Zhang, Jin & Qu, Xiaobo & Wang, Shuaian, 2018. "Reproducible generation of experimental data sample for calibrating traffic flow fundamental diagram," Transportation Research Part A: Policy and Practice, Elsevier, vol. 111(C), pages 41-52.
    9. Keyvan-Ekbatani, Mehdi & Kouvelas, Anastasios & Papamichail, Ioannis & Papageorgiou, Markos, 2012. "Exploiting the fundamental diagram of urban networks for feedback-based gating," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1393-1403.
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