  Sequential Bayesian filters in non-linear dynamic systems require the
recursive estimation of the predictive and posterior distributions. This paper
introduces a Bayesian filter called the adaptive kernel Kalman filter (AKKF).
With this filter, the arbitrary predictive and posterior distributions of
hidden states are approximated using the empirical 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
dynamical system model. Specifically, particles are generated and updated in
the data space, while the corresponding kernel weight mean vector and
covariance matrix associated with the feature mappings of the particles 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 particle numbers, by comparing with the
unscented Kalman filter (UKF), particle filter (PF) and Gaussian particle
filter (GPF). For example, compared with the GPF, the proposed approach
provides around 5% logarithmic mean square error (LMSE) tracking performance
improvement in the bearing-only tracking (BOT) system when using 50 particles.
