A traffic fundamental diagram calibrating methodology to avoid unbalanced speed–density observations

    Chunbo Zhang Affiliation
    ; Zhaoguo Huang Affiliation
    ; Yonggang Wang Affiliation


Traffic fundamental diagram is extremely important to analyse traffic flow and traffic capacity, and the central part of traffic fundamental diagram is to calibrate speed–density relationship. However, because of unbalanced speed–density observations, calibrating results using Least Square Method (LSM) with all speed–density points always lead to inaccurate effect, so this paper proposed a selecting data sample method and then LSM was used to calibrate four well-known single-regime models. Comparisons were made among the results using LSM with all speed–density points and the selecting data sample. Results indicated that the selecting data sample method proposed by this paper can calibrate the singleregime models well, and the method overcomes the inaccurate effect caused by unbalanced speed–density observations. Data from different highways validated the results. The contribution of this paper is that the proposed method can help researchers to determine more precise traffic fundamental diagram.

Keyword : traffic fundamental diagram, least square method, speed–density relationship, unbalanced speed–density observations, single-regime models, selecting data sample method

How to Cite
Zhang, C., Huang, Z., & Wang, Y. (2021). A traffic fundamental diagram calibrating methodology to avoid unbalanced speed–density observations. Transport, 36(1), 13-24.
Published in Issue
Feb 24, 2021
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Alonso, B.; Ibeas, A.; Musolino, G.; Rindone, C.; Vitetta, A. 2019. Effects of traffic control regulation on network macroscopic fundamental diagram: a statistical analysis of real data, Transportation Research Part A: Policy and Practice 126: 136–151.

Baer, N.; Boucherie, R. J.; Van Ommeren, J.-K. C. W. 2019. Threshold queueing to describe the fundamental diagram of uninterrupted traffic, Transportation Science 53(2): 585–596.

Bhouri, N.; Aron, M.; Hajsalem, H. 2019. A data-driven approach for estimating the fundamental diagram, Promet – Traffic & Transportation 31(2): 117–128.

Del Castillo, J. M.; Benitez, F. G. 1995a. On the functional form of the speed-density relationship – I: general theory, Transportation Research Part B: Methodological 29(5): 373–389.

Del Castillo, J. M.; Benitez, F. G. 1995b. On the functional form of the speed-density relationship – II: empirical investigation, Transportation Research Part B: Methodological 29(5): 391–406.

Djafarzadeh, N.; Safarpour, M.; Khataee, A. 2014. Electrochemical degradation of three reactive dyes using carbon paper cathode modified with carbon nanotubes and their simultaneous determination by partial least square method, Korean Journal of Chemical Engineering 31(5): 785–793.

Drake, J. S.; Schofer, J. L.; May, A. D. 1967. A statistical analysis of speed–density hypotheses, Highway Research Record 154: 112–117.

Edie, L. C. 1961. Car-following and steady-state theory for noncongested traffic, Operations Research 9(1): 66–76.

Fiems, D.; Prabhu, B.; De Turck, K. 2019. Travel times, rational queueing and the macroscopic fundamental diagram of traffic flow, Physica A: Statistical Mechanics and its Applications 524: 412–421.

Ghasemi, S. E.; Hatami, M.; Mehdizadeh Ahangar, G. R.; Ganji, D. D. 2014. Electrohydrodynamic flow analysis in a circular cylindrical conduit using least square method, Journal of Electrostatics 72(1): 47–52.

Greenberg, H. 1959. An analysis of traffic flow, Operations Research 7(1): 79–85.

Greenshields, B. D.; Bibbins, J. R.; Channing, W. S.; Miller, H. H. 1935. A study of traffic capacity, Highway Research Board Proceedings 14: 448–477.

Hatami, M.; Ganji, D. D. 2014. Thermal and flow analysis of microchannel heat sink (MCHS) cooled by Cu–water nanofluid using porous media approach and least square method, Energy Conversion and Management 78: 347–358.

Jeong, I.; Gu, B.-G.; Kim, J.; Nam, K.; Kim, Y. 2015. Inductance estimation of electrically excited synchronous motor via polynomial approximations by least square method, IEEE Transactions on Industry Applications 51(2): 1526–1537.

Jiang, Z.; Huang, Y.-X. 2009. Parametric calibration of speed–density relationships in mesoscopic traffic simulator with data mining, Information Sciences 179(12): 2002–2013.

Kim, B.; Lee, T.; Ouarda, T. B. M. J. 2014. Total least square method applied to rating curves, Hydrological Processes 28(13): 4057–4066.

Knoop, V. L.; Daamen, W. 2017. Automatic fitting procedure for the fundamental diagram, Transportmetrica B: Transport Dynamics 5(2): 129–144.

Lam, W. H. K.; Tam, M. L.; Cao, X.; Li, X. 2013. Modeling the effects of rainfall intensity on traffic speed, flow, and density relationships for urban roads, Journal of Transportation Engineering 139(7): 758–770.

Maghrour Zefreh, M.; Torok, A. 2020. Distribution of traffic speed in different traffic conditions: an empirical study in Budapest, Transport 35(1): 68–86.

Newell, G. F. 1961. Nonlinear effects in the dynamics of car following, Operations Research 9(2): 209–229.

Poole, A.; Kotsialos, A. 2016. Second order macroscopic traffic flow model validation using automatic differentiation with resilient backpropagation and particle swarm optimization algorithms, Transportation Research Part C: Emerging Technologies 71: 356–381.

Qu, X.; Wang, S.; Zhang, J. 2015. On the fundamental diagram for freeway traffic: a novel calibration approach for single-regime models, Transportation Research Part B: Methodological 73: 91–102.

Qu, X.; Zhang, J.; Wang, S. 2017. On the stochastic fundamental diagram for freeway traffic: Model development, analytical properties, validation, and extensive applications, Transportation Research Part B: Methodological 104: 256–271.

Sun, L.; Zhou. J. 2005. Development of multiregime speed–density relationships by cluster analysis, Transportation Research Record: Journal of the Transportation Research Board 1934: 64–71.

Wang, H.; Li, J.; Chen, Q.-Y.; Ni, D. 2011. Logistic modeling of the equilibrium speed–density relationship, Transportation Research Part A: Policy and Practice 45(6): 554–566.

Wang, H.; Ni, D.; Chen, Q.-Y.; Li, J. 2013. Stochastic modeling of the equilibrium speed–density relationship, Journal of Advanced Transportation 47(1): 126–150.

Washington, S.; Karlaftis, M.; Mannering, F.; Anastasopoulos, P. 2020. Statistical and Econometric Methods for Transportation Data Analysis. Chapman and Hall/CRC. 496 p.

Yin, G.; Zhang, Y.; Fan, H.; Ren, G.; Li, Z. 2015. One-step calibration of magnetic gradient tensor system with nonlinear least square method, Sensors and Actuators A: Physical 229: 77–85.

Zhang, C.; Guo, X.; Xi, Z. 2017. Determination of observation weight to calibrate freeway traffic fundamental diagram using weighted least square method (WLSM), Promet – Traffic & Transportation 29(2): 203–212.

Zhang, P.; Yue, H.; Wang, P.; Shao C.; Zhang, X. 2019. Modeling the enveloping macroscopic fundamental diagram based on the traffic assignment with deterministic user equilibrium, IEEE Access 7: 69776–69794.

Zheng, H.; Liang, Z.-F.; Li, M.-S.; Li, K. 2015. Optimization of parameters for LCL filter of least square method based threephase PWM converter, Journal of Electrical Engineering and Technology 10(4): 1626–1634.

Zhong, R.; Chen, C.; Chow, A. H. F.; Pan, T.; Yuan, F.; He, Z. 2016. Automatic calibration of fundamental diagram for first‐order macroscopic freeway traffic models, Journal of Advanced Transportation 50(3): 363–385.

Zhu, W.-X.; Li, S. 2019. Study on discrete boundary-feedbackcontrol strategy for traffic flow based on macroscopic fundamental diagram, Physica A: Statistical Mechanics and its Applications 523: 1237–1247.