Fundamental Diagram Estimation Based on Random Probe Pairs on Sub-Segments

  • Xiaoli Deng School of Mathematics and Statistics, Guizhou University
  • Yao Hu School of Mathematics and Statistics, Guizhou University
  • Qian Hu School of Mathematics and Statistics, Guizhou University
Keywords: traffic fundamental diagram estimation, expectation maximisation algorithm, linear regression, confidence interval, probe pairs


A new statistical algorithm is proposed in this paper with the aim of estimating fundamental diagram (FD) in actual traffic and dividing the traffic state. Traditional methods mainly focus on sensor data, but this paper takes random probe pairs as research objects. First, a mathematical method is proposed by using probe pairs data and the jam density to determine the FD on a stationary segment. Second, we applied it to the near-stationary probe traffic state set through linear regression and expectation maximisation iterative algorithm, estimating the free flow speed and the backward wave speed and dividing the traffic state based on the 95% confidence interval of the estimated FD. Finally, simulation and empirical analyses are used to verify the new algorithm. The simulation analysis results show that the estimation error corresponding to the free flow speed and the backward wave speed are 1.0668 km/h and 0.0002 km/h respectively. The empirical analysis calculates the maximum capacity of the road and divides the traffic into three states (free flow state, breakdown state, and congested state), which demonstrates the accuracy and practicability of the research in this paper, and provides a reference for urban traffic monitoring and government decision-making.


Immers LH, Logghe S. Traffic flow theory. Transportation Research Board Special Report. 2002;62(1-2): 5-7. DOI: 10.1023/A:1011514226782

Treiber M, Kesting A. Traffic flow dynamics: Data, models and simulation. Berlin: Springer; 2013.

Yan Q, Sun Z, Gan Q, Jin WL. Automatic identifi cation of near-stationary traffic states based on the PELT changepoint detection. Transportation Research Part B: Methodological. 2018;108: 39-54. DOI: 10.1016/j.trb.2017. 12.007

Newell GF. A simplified car-following theory: A lower order model. Transportation Research Part B: Methodological. 2002;36(3): 195-205. DOI: 10.1016/S0191 2615(00)00044-8

Duret A, Buisson C, Chiabaut N. Estimating individual speed-spacing relationship and assessing ability of newell’s car-following model to reproduce trajectories. Transportation Research Board. 2008;2088: 188-179. DOI: 10.3141/2088-20

Jabari SE, Zheng J, Liu HX. A probabilistic stationary speed-density relation based on newell’s simplifi ed car-following model. Transportation Research Part B: Methodological. 2014;68: 205-223. DOI: 10.1016/j.trb.2014. 06.006

Daganzo CF. On the variational theory of traffic flow: Well-posedness, duality and applications. Networks and Heterogeneous Media (NHM). 2006;1(4): 601-619. DOI: 10.3934/nhm.2006.1.601

Seo T, et al. Calibration of fundamental diagram using trajectories of probe vehicles: Basic formulation and heuristic algorithm. Transportation Research Procedia. 2017;21: 6-17. DOI: 10.1016/j.trpro.2017.03.073

Knoop VL, Daamen W. Automatic fitting procedure for the fundamental diagram. Transportmetrica B: Transport Dynamics. 2017;5(2): 129-144. DOI: 10.1080/216805 66.2016.1256239

Wu N. A new approach for modeling of fundamental diagrams. Transportation Research Part A: Policy and Practice. 2002;36(10): 867-884. DOI: 10.1016/S0965-8564(01)00043-X

Herrera JC, et al. Evaluation of traffic data obtained via GPS-enabled mobile phones: The mobile century field experiment. Transportation Research Part C. 2010;18(4): 568-583. DOI: 10.1016/j.trc.2009.10.006

Shladover SE. Connected and automated vehicle systems: Introduction and overview. Journal of Intelligent Transportation Systems. 2018;22(3): 190-200. DOI: 10.1080/15472450.2017.1336053

Jenelius E, Koutsopoulos HN. Probe vehicle data sampled by time or space: Consistent travel time allocation and estimation. Transportation Research Part B: Methodological. 2015;71: 120-137. DOI: 10.1016/j.trb.2014.10.008

Chiabaut N, et al. Fundamental diagram estimation through passing rate measurements in congestion. IEEE Transactions on Intelligent Transportation Systems. 2009;10(2): 355-359. DOI: 10.1109/TITS. 2009.2018963

Coifman B. Revisiting the empirical fundamental relationship. Transportation Research Part B: Methodological. 2014;68: 173-184. DOI: 10.1016/j.trb.2014.06.005

Coifman B. Empirical flow-density and speed-spacing relationships: Evidence of vehicle length dependency. Transportation Research Part B: Methodological. 2015;78: 54-65. DOI: 10.101 6/j.trb.2015.04.006

Seo T, Kusakabe T, Asakura Y. Traffic state estimation with the advanced probe vehicles using data assimilation. IEEE International Conference on Intelligent Transportation Systems, 15-18 Sep. 2015, Gran Canaria, Spain; 2015. p. 824-830. DOI: 10.1109/ITSC.2015.139

Seo T, Kusakabe T. Traffic state estimation using satellite remote sensing and probe vehicles. Journal of Traffic Engineering (JSTE). 2019;5(2): 1-10.

Herrera JC, Bayen AM. Incorporation of lagrangian measurements in freeway traffic state estimation. Transportation Research Part B: Methodological. 2010;44(4): 460-481. DOI: 10.1016/j.trb.2009. 10.005

Zhang N, Yang X, Ma W. Empirical approximation for the stochastic fundamental diagram of traffic flow on signalized intersection. Journal of Advanced Transportation. 2018;2018: 1-17. DOI: 10.1155/2018/4603614

Rempe F, et al. A phase-based smoothing method for accurate traffic speed estimation with floating car data. Transportation Research. 2017;85: 644-663. DOI: 10.1016/j.trc.2017.10.015

Seo T, et al. Fundamental diagram estimation by using trajectories of probe vehicles. Transportation Research Part B: Methodological. 2019;122: 40-56. DOI: 10.10 16/j.trb.2019.02.005

Li L, Chen X. Vehicle headway modeling and its inferences in macroscopic /microscopic traffic flow theory: A survey. Transportation Research Part C: Emerging Technologies. 2017;76: 170-188. DOI: 10.1016/j.trc.2017.01. 007

Xu T, et al. Fundamental diagram model of considering reaction time in environment of intelligent connected vehicles. Journal of Highway and Transportation Research and Development. 2020;37(08): 108-117. DOI: 10.3969/j.issn.1002-0268.2020.08.014. Chinese.

Wang Q, et al. Effects of adaptive cruise control and cooperative adaptive cruise control on traffic flow. China J. Highw. Transp. 2019;32(06): 188-197+205. DOI: 10.19721/ j.cnki.1001-7372.2019.06.019. Chinese.

Sunderrajan A, Viswanathan V, Cai W, Knoll A. Traffic state estimation using floating car data. Procedia Computer Science. 2016;80: 2008-2018. DOI: 10.1016/j.procs.2016.05.521

Lighthill MJ, Whitham GB. On kinematic waves II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society A. 1955;229(1178): 317-345. DOI: 10.1098/rspa.1955.0089

Richards PI. Shock waves on the highway. Operations Research. 1956;4(1): 42-51. DOI: 10.1287/opre.4.1.42

Edie L. Discussion of traffic stream measurements and defi nitions. Proceedings of 2nd International Symposium on the Theory of Traffic Flow, 25-27 June 1963, London, UK; 1963. p. 139-154.

Karush W. Minima of functions of several variables with inequalities as side conditions. In: Giorgi G, Kjeldsen T. (eds.) Traces and Emergence of Nonlinear Programming. Basel: Birkhäuser; 2014. p. 217-245. DOI: 10.1007/978-3-0348-0439-4_10.

Kuhn HW, Tucker AW. Nonlinear programming. In: Neyman J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability; 1951. p. 481- 492.

How to Cite
Deng X, Hu Y, Hu Q. Fundamental Diagram Estimation Based on Random Probe Pairs on Sub-Segments. Promet [Internet]. 2021Oct.8 [cited 2022Aug.17];33(5):717-30. Available from: