Bayesian calibration of traffic flow fundamental diagrams using Gaussian processes

TL;DR

This paper introduces a Bayesian Gaussian Process-based calibration method for traffic flow speed-density models, addressing biases in traditional least-squares approaches and providing a probabilistic framework for model parameters.

Contribution

It proposes a novel GP-based calibration method that generalizes LS and WLS, offering bias reduction and Bayesian parameter estimation for traffic models.

Findings

Significantly reduces calibration bias compared to LS.

Achieves similar results to WLS in calibration accuracy.

Enables non-parametric modeling and Bayesian inference of speed-density functions.

Abstract

Modeling the relationship between vehicle speed and density on the road is a fundamental problem in traffic flow theory. Recent research found that using the least-squares (LS) method to calibrate single-regime speed-density models is biased because of the uneven distribution of samples. This paper explains the issue of the LS method from a statistical perspective: the biased calibration is caused by the correlations/dependencies in regression residuals. Based on this explanation, we propose a new calibration method for single-regime speed-density models by modeling the covariance of residuals via a zero-mean Gaussian Process (GP). Our approach can be viewed as a generalized least-squares (GLS) method with a specific covariance structure (i.e., kernel function) and is a generalization of the existing LS and the weighted least-squares (WLS) methods. Next, we use a sparse approximation to address the scalability issue of GPs and apply a Markov chain Monte Carlo (MCMC) sampling scheme to obtain the posterior distributions of the parameters for speed-density models and the hyperparameters (i.e., length scale and variance) of the GP kernel. Finally, we calibrate six well-known single-regime speed-density models with the proposed method. Results show that the proposed GP-based methods (1) significantly reduce the biases in the LS calibration, (2) achieve a similar effect as the WLS method, (3) can be used as a non-parametric speed-density model, and (4) provide a Bayesian solution to estimate posterior distributions of parameters and speed-density functions.