  Recent advances in operations research and machine learning have revived
interest in solving complex real-world, large-size traffic control problems.
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.
