Kalman Filter Applications for Traffic Management

Abstract

An onÂline calibration approach for dynamic traffic assignment systems has been developed. The approach is general and flexible and makes no assumptions on the type of the DTA system, the models or the data that it can handle. Therefore, it is applicable to a wide variety of tools including simulationÂbased and analytical, as well as microscopic and macroscopic models. The objective of the onÂline calibration approach is to introduce a systematic procedure that will use the available data to steer the model parameters to values closer to the realized ones. The output of the onÂline calibration is therefore a set of parameter values that --when used as input for traffic estimation and prediction-- minimizes the discrepancy between the simulated (estimated and predicted) and the observed traffic conditions. The scope of the onÂline calibration is neither to duplicate nor to substitute for the offÂline calibration process. Instead, the two processes are complementary and synergistic in nature. The onÂline calibration problem is formulated as a stateÂspace model. StateÂspace models have been extensively studied and efficient algorithms have been developed, such as the Kalman Filter for linear models. Because of the nonÂlinear nature of the onÂline calibration formulation, modified Kalman Filter methodologies have been presented. The most straightforward extension is the Extended Kalman Filter (EKF), in which optimal quantities are approximated via first order Taylor series expansion (linearization) of the appropriate equations. The Limiting EKF is a variation of the EKF that eliminates the need to perform the most computationally intensive steps of the algorithm onÂline. The use of the Limiting EKF provides dramatic improvements in terms of computational performance. The Unscented Kalman Filter (UKF) is an alternative filter that uses a deterministic sampling approach. The computational complexity of the UKF is of the same order as that of the EKF. Empirical results suggest that joint onÂline calibration of demand and supply parameters can improve estimation and prediction accuracy of a DTA system. While the results obtained from this real network application are promising, they should be validated in further empirical studies. In particular, the scalability of the approach to larger, more complex networks needs to be investigated. The results also suggest that --in this application-- the EKF has more desirable properties than the UKF (which may be expected to have superior performance over the EKF), while the UKF seems to perform better in terms of speeds than in terms of counts. Other researchers have also encountered situations where the UKF does not outperform the EKF, e.g. LaViola, J. J., Jr. (2003) and van Rhijn et al. (2005). The Limiting EKF provides accuracy comparable to that of the best algorithm (EKF), while providing order(s) of magnitude improvement in computational performance. Furthermore, the LimEKF algorithm is that it requires a single function evaluation irrespective of the dimension of the state vector (while the computational complexity of the EKF and UKF algorithms increases proportionally with the state dimension). This property makes this an attractive algorithm for largeÂscale applications.