Functional form selection and calibration of macroscopic fundamental diagrams

Macroscopic fundamental diagram (MFD) is widely applied in network-level traffic control and management with most applications necessitating a well-calibrated MFD. With various data sources, more and more empirical MFDs are documented, while the MFD functional form is predetermined by traffic engineers based on their prior experiences. To our best, no generally accepted functional form has been identified. An automatic functional form selection method is yet to be devised. To meet this, a two-step MFD calibration framework is proposed to enable both the functional form selection and the estimation of parameters in this paper. A math program problem is first developed to identify a proper functional form from a set of candidate functions via random sampling of the measurement data. A mean-field variational Bayesian (MFVB) algorithm is then proposed to estimate the parameters of the selected MFD functions using the full measurement dataset. Both calibrations with and without the MFD dynamics are evaluated. The comparison between these calibration results highlights that the calibration considering the MFD dynamics can better characterize network traffic dynamics subject to dynamic travel demand and traffic control measures. Leveraging functional form selection and the computational advantages of the MFVB method, the two-step framework can significantly reduce the computational burden. Results using simulated data and empirical data validate the effectiveness and efficiency of the two-step framework. Furthermore, different functional forms are identified for different cities, highlighting the importance of functional form selection in the MFD calibration.