Calibration and uncertainty quantification of macroscopic fundamental diagrams

The macroscopic fundamental diagram (MFD) provides an efficient framework for characterizing network traffic dynamics. Recent studies have underscored the importance of quantifying MFD hysteresis and explicitly distinguishing congestion loading and recovery phases for traffic control. Yet, no systematic calibration method has been proposed to accurately capture the traffic dynamics across distinct congestion phases and quantify the associated uncertainty. This study proposes a unified mathematical program for MFD calibration and uncertainty quantification that automatically distinguishes congestion loading and recovery. Specifically, the program first incorporates two conventional quantification principles as special cases and is validated using microscopic simulations based on two real-world urban networks in China, thereby demonstrating its effectiveness in characterizing MFD data scatter. Motivated by observed MFD hysteresis, we then extend this unified program to develop a novel uncertainty quantification approach that explicitly captures the distinct congestion loading and recovery dynamics. The results show that the quantified upper and lower bounds correspond to congestion loading and recovery phases, respectively, and that their distance can be interpreted as an indicator of congestion-induced capacity drop. We further examine how macroscopic factors, such as travel demand and traffic control, and microscopic factors, such as driving behavior, influence MFD hysteresis and traffic resilience. The results suggest that demand management and well-designed traffic control can alleviate hysteresis and enhance network resilience, while more homogeneous driving behavior may further improve network performance.