Applications of Koopman Operator Theory to Highway Traffic Dynamics
The ever-increasing demands on transportation systems have led to the need for a robust and universal method for the analysis and forecasting of vehicular traffic systems. Traditional methods are mainly model-based, that is, the analysis is performed by investigating a mathematical model that represents the target dynamics of a traffic system. On the other hand, contemporary efforts have focused on utilizing artificial intelligence (AI) to model or forecast vehicular traffic dynamics. Despite these large efforts, there is still no single best-performing method for the analysis and forecasting of vehicular traffic dynamics. This is due to the very well known fact that the unpredictable behaviors involved in a traffic system, like human interaction and weather, leads to a very complicated high-dimensional nonlinear dynamical system. Therefore, it is difficult to obtain a mathematical or AI model that explains all events and time evolution of vehicular traffic dynamics. Even if such a model could be attained, it would not lead to a robust and universal way of traffic analysis and forecast, due to its need of extensive parameter tuning. Thus, in contrast to the model or AI-based approach, it is necessary to develop data-driven methods that can identify dynamically important spatiotemporal structures of traffic phenomena. In this thesis, we demonstrate how the Koopman operator theory can offer a model and parameter-free, data-driven approach to accurately analyzing and forecasting traffic dynamics. The Koopman operator theory framework is a rapidly developing theory in dynamical systems that offers powerful methods for analyzing complex nonlinear systems. The effectiveness of this framework is demonstrated by an application to the Next Generation Simulation (NGSIM) data set collected by the US Federal Highway Administration and the Performance Measurement System (PeMS) data set collected by the California Department of Transportation. By obtaining a Koopman mode decomposition (KMD) of the data sets, we are able to accurately reconstruct our observed dynamics, distinguish any growing or decaying modes, and obtain a hierarchy of coherent spatiotemporal patterns that are fundamental to the observed dynamics. Furthermore, it is demonstrated how the KMD can be utilized to accurately forecast traffic dynamics by obtaining a decomposition of a subset of the data, that is then used to predict a future subset of the data.