A Semiparametric Framework for Stochastic Fundamental Diagram Modeling
Authors: Pengnan Chi, Xiaoliang Ma, Magnus Jansson, Magnus Nordenvaad
Organizations: KTH Royal Institute of Technology, Stockholm, Sweden · Swedish Transport Administration, Borlänge, Sweden
Abstract
The stochastic fundamental diagram (SFD) provides a probabilistic description of the relationship between traffic density and flow or speed, enabling uncertainty-aware traffic modeling. However, existing stochastic models frequently struggle to accommodate rigorous physical constraints while retaining sufficient flexibility to capture complex nonlinear patterns. To address this, we propose a novel semiparametric SFD modeling framework by leveraging specially designed functional forms. These functions intrinsically satisfy physical constraints defined on the moments of the conditional flow distribution given traffic density while incorporating neural-network-based structures to capture complex empirical patterns. We derive a system of moment-matching equations to convert physical constraints into the parameterization of the conditional distribution, proving that a unique solution exists for the location-scale family of distributions, thereby guaranteeing model well-posedness. Furthermore, we demonstrate that the framework can be extended to non-location-scale distributions, including those requiring additional boundary constraints. Empirical evaluations on a real-world dataset reveal that our approach consistently outperforms representative baselines, delivering superior probabilistic accuracy and robust uncertainty quantification, particularly in congested regimes. Overall, the proposed framework provides a theoretically grounded and flexible foundation for stochastic traffic flow modeling.
Traffic forecasting is a challenging spatio-temporal modeling task and a critical component of urban transportation management. Current studies mainly focus on deterministic predictions, with limited considerations on the uncertainty and stochasticity in traffic dynamics. Therefore, this paper proposes an elegant yet universal approach that transforms existing models into probabilistic predictors by replacing only the final output layer with a novel Gaussian Mixture Model (GMM) layer. The modified model requires no changes to the training pipeline and can be trained using only the Negative Log-Likelihood (NLL) loss, without any auxiliary or regularization terms. Experiments on multiple traffic datasets show that our approach generalizes from classic to modern model architectures while preserving deterministic performance. Furthermore, we propose a systematic evaluation procedure based on cumulative distributions and confidence intervals, and demonstrate that our approach is considerably more accurate and informative than unimodal or deterministic baselines. Finally, a more detailed study on a real-world dense urban traffic network is presented to examine the impact of data quality on uncertainty quantification and to show the robustness of our approach under imperfect data conditions. Code available at https://github.com/Weijiang-Xiong/OpenSkyTraffic
Car-following behavior is fundamental to traffic flow theory, yet traditional models often fail to capture the stochasticity of naturalistic driving. This paper proposes an empirical probabilistic sampling approach to car-following modeling that bypasses conventional parametric assumptions. Under this approach, we introduce the Markov Chain Car-Following (MC-CF) model, which represents state transitions as a Markov process and predicts behavior by randomly sampling accelerations from empirical distributions within discretized state bins. Evaluation on the Waymo Open Motion Dataset (WOMD) demonstrates that MC-CF variants significantly outperform all physics-based baselines (IDM, Gipps, FVDM, and SIDM) across both one-step and open-loop trajectory prediction metrics, and remain competitive with modern data-driven baselines including neural network and Gaussian mixture model approaches. Zero-shot generalization on the Naturalistic Phoenix (PHX) dataset further confirms cross-domain transferability. Finally, microscopic ring road simulations validate the framework's scalability: by incrementally integrating unconstrained free-flow trajectories and high-speed freeway data (TGSIM) alongside a conservative inference strategy, the model substantially reduces collisions across most tested scenarios and successfully reproduces naturalistic and stochastic shockwave propagation, though crashes persist under severe shockwave conditions. Overall, the proposed MC-CF model provides a robust and scalable foundation for simulating population-level stochastic traffic behavior that requires no behavioral parameter calibration, making it well-suited for the data-rich future of intelligent transportation.
Disordered traffic flow is characterized by weak or non-existent lane discipline in the presence of strong vehicle heterogeneity and continuous lateral interactions, challenging traditional lane-based modeling assumptions. This study presents an empirical study of macroscopic and microscopic aspects of disordered traffic using high-resolution UAV trajectory data collected on an urban arterial. A two-dimensional extension of Edie's framework is applied to quantify aggregate traffic variables and produce a two-dimensional fundamental diagram, revealing that traffic states cannot be adequately represented using one-dimensional formulations and highlighting the persistent role of lateral redistribution. The propagation of congestion is estimated directly from the spatiotemporal speed fields, demonstrating the emergence of coherent stop-and-go waves and showing a similar dynamics as conventional lane-based flow, in spite of the heterogeneous vehicle interactions. At the microscopic level, steady-state follower-leader identification is used to examine desired time gaps and minimum lateral spacing, vehicle dimension distributions, and kinematic characteristics, revealing pronounced inter-class heterogeneity that explains disordered traffic behavior. The study provides an empirical framework linking vehicle-level interactions and aggregate traffic dynamics and establishes a data-driven basis for the calibration and validation of traffic models for disordered mixed traffic systems.