A Data-driven Approach for Estimating the Fundamental Diagram

Bhouri, Neila; Aron, Maurice; Hajsalem, Habib · 2019 · Crossref

DOI: 10.7307/ptt.v31i2.2849

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Summary

This paper addresses the limitations of using analytical forms for the fundamental diagram (FD), which links traffic speed, density, and flow. While analytical FDs are necessary for certain theoretical models, the authors argue that they impose unnecessary constraints in assessment studies and macroscopic simulations, leading to a loss of accuracy. Specifically, analytical curves often fail to capture empirical data variability and perform poorly at high speeds due to flat branches near free-flow conditions. The study proposes a data-driven, non-analytical approach that constructs an FD strictly fitting empirical data while respecting the physical constraint that speed must decrease as density increases. The methodology utilizes six-minute aggregated data records of average speed and flow collected from a dual-loop detector on the A1 motorway in France (2009–2010). After filtering irrelevant data, 58,000 observations were used. The core innovation is the definition of congestion not by a single critical density value, but by a density threshold that is a decreasing function of flow, derived from safety constraints regarding stopping distances. Two calibration techniques are developed to separate congested and non-congested observations: a Shortest Path Algorithm (SPA) and a Quadratic Optimization with Linear Constraints (QOLC). The SPA minimizes the residual error between observed and modeled speeds across flow classes, enforcing monotonicity constraints where non-congested speed decreases with flow and congested speed increases with flow. The QOLC approach directly optimizes a decreasing speed-density relationship. The results demonstrate that the proposed non-analytical FD provides a more accurate representation of empirical traffic data than traditional analytical models. By identifying the density threshold as a function of flow rather than a constant, the method better accounts for phenomena such as capacity drop and driver adaptation to time headways. The validation shows that the calibrated curves respect the theoretical requirement of decreasing speed with increasing density while capturing the scatter inherent in real-world data. The SPA and QOLC methods are compared, showing that both effectively generate unbiased FDs that align closely with the observed speed-flow and speed-density relationships. The significance of this work lies in its application to traffic assessment and simulation, where high-fidelity representation of traffic states is crucial. By abandoning rigid analytical forms, the approach allows for more precise estimation of state variables, particularly at high speeds where analytical models often fail. This data-driven framework improves the reliability of ex-ante assessments for traffic management measures, such as speed limit reductions, by ensuring that simulated traffic behavior closely mirrors empirical observations. The study concludes that utilizing empirical FDs with appropriate physical constraints enhances model accuracy without the computational burden of complex stochastic models.

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• traffic density