  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.
