Aggregated Kalman filter: efficient switching linear dynamic estimation via merging priors

A switching linear dynamic system (SLDS) represents a system wherein the state evolution model randomly switches between several operation modes over the observation interval. Bayesian, recursive state estimation in an SLDS is performed by running a bank of standard Kalman filters (KF) at each time step in the form of switching KF (SKF). In this paper we develop the aggregated Kalman filter (AKF), which produces a computationally efficient approximation to the SKF estimator by merging the prior conditional distributions calculated in the prediction step of SKF in terms of Kullback-Leibrer (KL) divergence, and subsequently performs the update step for a single Gaussian distribution. Considering some practical approximations, the AKF is shown to be the optimal estimator in the mean squared error (MSE) sense for the class of linear filters given the posterior probability of the modes, i.e., it is the projection of the optimal estimator onto the space of linear estimators. Quantifying the induced mean squared error by AKF projection is in general only possible via numerical integration, but its closed form is obtained by approximating operating modes’ conditional probabilities with deterministic quantities. The quantification of this approximation error provides insight into the impact of parameters of the SLDS model on the induced approximation error due to aggregation, which we then verify through experiments. We show that AKF provides significant computational savings over the SKF while maintaining near identical performance.