A Data-driven Approach for Estimating the Fundamental Diagram

Keywords: non-analytical, calibration, empirical data, shortest-path algorithm, convex quadratic program, safety constraint, critical density function

Abstract

The fundamental diagram links average speed to density or traffic flow. An analytic form of this diagram, with its comprehensive and predictive power, is required in a number of problems. This paper argues, however, that, in some assessment studies, such a form is an unnecessary constraint resulting in a loss of accuracy. A non-analytical fundamental diagram which best fits the empirical data and respects the relationships between traffic variables is developed in this paper. In order to obtain an unbiased fundamental diagram, separating congested and non-congested observations is necessary. When defining congestion in parallel with a safety constraint, the density separating congestion and non-congestion appears as a decreasing function of the flow and not as a single critical density value. This function is here identified and used. Two calibration techniques – a shortest path algorithm and a quadratic optimization with linear constraints – are presented, tested, compared and validated.

Author Biographies

Neila Bhouri, IFSTTAR/COSYS/GRETIA
researcher
Maurice Aron, IFSTTAR/COSYS/GRETIA
Researcher
Habib Hajsalem, IFSTTAR/COSYS/GRETIA
Senior Researcher

References

[1] Gerlough DL, Huber MJ. Traffic Flow Theory: a Monograph. Transportation Research Board. Special Report 165, 1975.
[2] Transportation Research Circular E-C 149. 75 Years of the Fundamental Diagram for traffic flow theory, Greenshields Symposium. TRB Traffic Flow, Theory and Characteristics Committee; 2011.
[3] Dhingra SL, Gull I. Traffic Flow Theory Historical Research Perspectives. In: Transportation Research Circular E-C 149. 75 Years of the Fundamental Diagram for traffic flow theory. Greenshields Symposium. TRB Traffic flow, Theory and Characteristics Committee; 2011. p. 45-62.
[4] Pipes LA. Car-Following models and the Fundamental Diagram of road traffic. Transportation Research. Part B: Methodological. 1967;1(1): 21-29.
[5] Li J, Chen QY, Ni D, Wang H. Analysis of LWR model with Fundamental Diagram subject to uncertainty. In: Transport. Research Circular E-C 149. 75 Years of the Fundamental Diagram for Traffic Flow Theory. Greenshields Symposium. TRB Traffic flow Theory and Characteristics Committee; 2011. p. 74-83.
[6] Lu Y, Wong SC, Zhang M, Shu CW, Chen W. Explicit construction of entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a piecewise quadratic flow-density relationship. Transportation Research Part B: Methodological. 2008;42(4): 355-372.
[7] Aron M, Seidowsky R, Cohen S. Ex-ante assessment of a speed limit reducing operation – A data-driven approach. In: Yannis G, Cohen S. (Eds.) Traffic Safety. Wiley; 2016. p. 177-197.
[8] Cohen S, Christoforou Z. Travel time estimation between loop detectors and FCD: A compatibility study on the Lille network, France. Transportation. Research Procedia. 2015;10: 245-255. Available from: https://ac.els-cdn.com/S2352146515002616/1-s2.0-S2352146515002616-main.pdf?_tid=cbda321a-d945-11e7-9781-00000aab0f02&acdnat=1512428125_be15da67407e1b228fff263a8ed88619 [Accessed 5th Dec 2017].
[9] Neumann T, Böhnke PL, Touko Tcheumadjeu LC. Dynamic representation of the fundamental diagram via Bayesian networks for estimating traffic flows from probe vehicle data. In: 16th IEEE ITSC, October 6-9, 2013, The Hague, The Netherlands. IEEE; 2013. p. 1870-1875.
[10] Underwood RT. Speed, volume and density relationships: Quality and theory of traffic flow. New Haven, Connecticut: Yale Bureau of Highway Traffic; 1961. p. 141-188.
[11] Coifman, B. Revisiting the empirical fundamental relationship. Transportation Research Part B: Methodological. 2014;68: 173-184.
[12] Lighthill MJ, Whitham GB. On kinematic waves. II. A theory of traffic flow on long crowded roads. In: Proceedings of the Royal Society of London,UK. Mathematical and Physical Sciences, series A. 1955;229(1178): 317-345.
[13] Richards P. Shock waves on the highway. Operations Research. 1956;4(1): 42-51.
[14] Payne HJ. Models of freeway traffic and control. In: Bekey GA. (ed.) Simulation Councils Proceedings. Mathematical Models Public Systems. 1971;1(1): 51-61.
[15] Lebacque JP, Khoshyaran MM. A variational formulation for higher order macroscopic traffic flow models of the GSOM family. Transportation Research Part B: Methodological. 2013;57: 245-265.
[16] Greenshields BD. A study in highway capacity. In: Proceedings of the Highway Research Board. Washington DC.USA. Vol. 14; 1935. p. 448-477.
[17] Greenberg H. An analysis of traffic flow. Operations Research. 1959;7(1): 79-85.
[18] Edie LC. Car-Following and steady state theory for non congested traffic. Tunnel traffic capacity study. Port of New York Authority, New York, USA. Report number: 6, 1960.
[19] May AD. Traffic flow fundamental. Englewood Cliffs: Prentice Hall; 1990.
[20] Treiterer J, Myers JA. The hysteresis phenomenon in traffic flow. In: Proceedings of the 6th Int. Symp. on Transportation and Traffic Theory, 26-28 August 1974, Sydney, Australia. Vol. 6; 1974. p.13-38.
[21] Zhang HM. A mathematical theory of traffic hysteresis. Transportation Research Part B: Methodological. 1999;33(1): 1-23.
[22] Castillo J, Benítez F. On the functional form of the speed-density relationship I: General theory. Transportation Research Part B: Methodological. 1995;29(5): 373-389.
[23] Li J, Zhang HM. Fundamental Diagram of traffic flow. New identification scheme and further evidence from empirical data. Transportation Research Records, Journal of the Transportation Research Board. 2011;2260: 50-59.
[24] Kerner BS. Three-phase theory of city traffic: Moving synchronized flow patterns in under-saturated city traffic at signals. Physica A: Statistical Mechanics and its Applications. 2014;397: 76-110.
[25] Daganzo CF, Geroliminis N. An analytical approximation of the macroscopic fundamental diagram. Transportation Research Part B: Methodological. 2008;42(9): 771-781.
[26] Ji Y, Xu M, Li J, van Zuylen HJ. Determining the Macroscopic Fundamental Diagram from Mixed and Partial Traffic Data. Promet – Traffic & Transportation. 2018;30(3): 267-279. Available from: https://traffic.fpz.hr/index.php/PROMTT/article/view/2406/ [Accessed 29 August 2018].
[27] Lu XY, Varaiya P, Horowitz R, Skabardonis A. Fundamental Diagram modeling and analysis based NGSIM data. In: 12th IFAC Symposium on Control in Transportation Systems, 2009, Redondo Beach, California, USA.
[28] Coifman B. Jam occupancy and other lingering problems with empirical fundamental relationships. Transportation Research Records, Journal of the Transportation Research Board. 2014;2422: 104-112.
[29] Goldfarb D, Idnani A. A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming. 1983;27: 1-33.
Published
2019-04-01
How to Cite
1.
Bhouri N, Aron M, Hajsalem H. A Data-driven Approach for Estimating the Fundamental Diagram. Promet - Traffic & Transportation [Internet]. 1Apr.2019 [cited 22Apr.2019];31(2):117-28. Available from: http://traffic.fpz.hr/index.php/PROMTT/article/view/2849
Section
Articles