Adaptive Kernel Kalman Filter

Sequential Bayesian filters in non-linear dynamic systems require the recursive estimation of the predictive and posterior probability density functions (pdfs). This paper introduces a Bayesian filter called the adaptive kernel Kalman filter (AKKF). The AKKF approximates the arbitrary predictive and posterior pdfs of hidden states using the kernel mean embeddings (KMEs) in reproducing kernel Hilbert spaces (RKHSs). In parallel with the KMEs, some particles in the data space are used to capture the properties of the dynamic system model. Specifically, particles are generated and updated in the data space. Moreover, the corresponding kernel weight means vector and covariance matrix associated with the particles’ kernel feature mappings are predicted and updated in the RKHSs based on the kernel Kalman rule (KKR). Simulation results are presented to confirm the improved performance of our approach with significantly reduced numbers of particles by comparing with the unscented Kalman filter (UKF), particle filter (PF), and Gaussian particle filter (GPF). For example, compared with the GPF, the AKKF provides around 50% logarithmic mean square error (LMSE) tracking performance improvement in the bearing-only tracking (BOT) system when using 50 particles.