IEEE Transactions on Intelligent Transportation Systems, 2013, Special Issue on "Emerging techniques for the management of uncertainty in computational traffic models" 1  Abstract—Freeway traffic state estimation is a vital component of traffic management and information systems. Macroscopic model based traffic state estimation methods are widely used in this field and have gained significant achievements. However, tests show that the inherent randomness of traffic flow and uncertainties in the initial conditions of models, model parameters as well as model structures all influence traffic state estimations. To improve the estimation accuracy, this paper presents an ensemble learning framework to appropriately combine estimation results from multiple macroscopic traffic flow models. This framework first assumes that any models existing are imperfect and have their own strengths/weaknesses. It then estimates the online traffic states in a rolling horizon scheme. This framework automatically ensembles the information from each individual estimation models, based on their performance during the selected regression horizon. In particular, we discuss three weighting algorithms: least square regression, ridge regression and lasso, which represent different presumptions of model capabilities. A field test based on real freeway measurements indicates that lasso ensemble best handles various uncertainties and improves estimation accuracy significantly. It should also be pointed out that the proposed framework is a flexible tool to assemble non-model based traffic estimation algorithms. This framework can also be extended for many other applications, including traffic flow prediction and travel time prediction. Index Terms—Traffic state estimation, multi-model ensemble, uncertainty, ridge regression, lasso. I. INTRODUCTION EAL-time freeway traffic state (flow rates, mean speeds, and densities) estimation is a key function of intelligent transportation systems. Because of time and financial cost constraints, the measurement devices (e.g. inductive loops, video sensors, radar detectors) in most freeways only deliver Manuscript received July 1st, 2013, revised October 30th, 2013, December 30th, 2013. This work was supported in part by National Science Foundation CAREER Award Project 1150925 "Reliability as an Emergent Property of Transportation Networks", U.S. Federal Highway Administration Exploratory Advanced Research Program, National Natural Science Foundation of China 51278280, the Chinese National Science and Technology Support Program 2013BAG18B00. L. Li is with Department of Automation, Tsinghua University, Beijing, China 100084. X. Chen is with Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742, USA (corresponding author, e-mail: xmchen@umd.edu). L. Zhang is with Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20742, USA. real-time traffic information at a few locations. As a result, the spatial resolution of available real-time traffic measurements is typically insufficient for the direct implementation of appropriate control actions. How to estimate traffic states of road segments between detectors is an important issue for traffic management and control, and has thus received an increased amount of research interest. Due to significant space inhomogeneity and time-varying characteristics of traffic flow, we need to design proper traffic state estimators to recover real-time traffic flow states for the whole freeway based on limited traffic measurements. Related studies had attracted considerable attention in the past few decades [1]. We may roughly categorize the existing approaches into two types: traffic flow model based approaches and non-model based approaches. Every traffic flow model based approach explicitly adopts one specific traffic flow model (macroscopic models are the most widely used). In such models, vehicular traffic is often treated as a continuum and described by aggregated fluid-like quantities such as density, flow rate and speed. These quantities are point-wise defined at all points x along the roadway (in the direction of the traffic) and are assumed to evolve with time t . Usually, this evolution process is described by a set of equations consisting of a flow conservation equation and speed momentum equation, both in the form of a partial differential equation (PDE), with certain initial and boundary conditions. In traffic estimation problems, we usually discretize these models in both space and time, and then apply finite-difference approximations to replace partial derivatives. This trick turns the continuous PDEs into discrete difference equations which can be numerically solved [2]-[3]. Based on these difference equations, we can depict the dependence relations between nearby (space and time) traffic flow states, then estimate the states that are not directly measured. The frequently used models include the first order Lighthill-Whitham-Richards (LWR) model [4]-[15] and a special second order model proposed by Papageorgiou et al. [16]-[20]. In comparison, non-model based approaches do not presume a fixed physical traffic flow model to describe system dynamics. They attempt to establish a mapping relation between the historical values of traffic flow states and their future evolutions. Either regression techniques [21]-[23] or artificial intelligence modeling skills [24]-[25] can be applied to discover the mapping relation. On the basis of the reconstructed system dynamics, we can then estimate unknown traffic flow Multi-Model Ensemble for Freeway Traffic State Estimations Li Li, Senior Member, IEEE, Xiqun (Michael) Chen, Member, IEEE, Lei Zhang R