Data-driven uncertainty quantification in macroscopic traffic flow models

Abstract

We propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from cross-sectional data. We consider both a simple first order model consisting in the mass conservation equation and its second order version including a speed evolution equation. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error in the macroscopic variables (flow, speed, density) for both models, showing that second order models globally perform better in reconstructing traffic quantities of interest.

Appendix A: Least square approach

A commonly used approach to calibrate the optimal parameters is the minimization of a least square cost function taking both the real data and simulated data into account (see, e.g. [45, 53, 56]). Accordingly, the calibration is based on the minimization of the following cost function

$$ \begin{array}{@{}rcl@{}} C(\theta) = \sum \limits_{(x,t) \in (X, T)} \left|y^{F}(x,t) - y^{M}(x,t,\theta)\right|^{2}. \end{array} $$

Thus, the optimal parameter 𝜃^∗ is given by

$$ \begin{array}{@{}rcl@{}} \theta^{*} = \underset{\theta \in {\Theta}}{\text{argmin}} C(\theta). \end{array} $$

The bounds for the three-dimensional parameter space Θ are those defined in the right columns of Table 1. The optimization results for the calibration parameters are summarized in Table 7.

Table 7 Optimization results for the least squares approach

Comparing the errors reported in Table 8 with those of Tables 3 and 6, we conclude that both the optimization and the Bayesian approaches greatly outperform this basic calibration procedure, thus evidencing the benefit of introducing a bias term.

Table 8 Time-space error results for the least squares approach. In bold, the lowest flow, speed, density and total errors per data scenario