The fundamental diagram serves as the foundation of traffic flow modeling for almost a century. With the increasing availability of road sensor data, deterministic parametric models have proved inadequate in describing the variability of real-world data, especially in congested area of the density-flow diagram. In this paper we estimate the stochastic density-flow relation introducing a nonparametric method called convex quantile regression. The proposed method does not depend on any prior functional form assumptions, but thanks to the concavity constraints, the estimated function satisfies the theoretical properties of the density-flow curve. The second contribution is to develop the new convex quantile regression with bags (CQRb) approach to facilitate practical implementation of CQR to the real-world data. We illustrate the CQRb estimation process using the road sensor data from Finland in years 2016-2018. Our third contribution is to demonstrate the excellent out-of-sample predictive power of the proposed CQRb method in comparison to the standard parametric deterministic approach.

翻译：基本图理论作为交通流建模的基础已持续近一个世纪。随着道路传感器数据的日益普及，确定性参数模型已难以描述真实数据的变异性，尤其是在密度-流量图的拥堵区域。本文通过引入称为凸分位数回归的非参数方法，对随机密度-流量关系进行估计。该方法不依赖于任何先验函数形式假设，但借助凹性约束，所估计函数满足密度-流量曲线的理论性质。第二个贡献是开发了基于袋装法的凸分位数回归（CQRb）新方法，以促进CQR在实际数据中的应用。我们使用2016-2018年芬兰道路传感器数据展示了CQRb的估计过程。第三个贡献是证明相较于标准参数确定性方法，所提出的CQRb方法具有优异的样本外预测能力。