Low-Rank Unscented Kalman Filter for Freeway Traffic Estimation Problems

Abstract

Data assimilation techniques are a process of using measurements combined with model predictions to (optimally) estimate the value of a certain set of variables. These techniques belong to a group of model-based estimation approaches. In traffic science, the Kalman filter and its family, namely, the extended Kalman filter (EKF), have been widely applied to problems of traffic state estimation. The EKF is successful for computation and provides some reasonably accurate results. However, traffic systems are generally so nonlinear that the linearization used in the EKF is not always straightforward. To avoid such linearization, another family of Kalman filter, the unscented Kalman filter (UKF), has been designed to account for the nonlinear system in which a set of deterministic sample points is chosen to capture the initial probability distribution. These sample points are then propagated through the nonlinear system, and the probability density function of the actual state is approximated by the ensemble of the estimates. In the UKF, the number of sample points is determined by the dimension of the states to be estimated so that it becomes computationally expensive when the size of traffic networks is increased. Thus an effort was made to approximate a UKF in which the required sample points were reduced. Numerical experiments were carried out to assess the relevance of the proposed approach with real traffic data, and comparisons with the UKF for the level of accuracy and computational time were made.