  We present a general nonlinear Bayesian filter for high-dimensional state
estimation using the theory of reproducing kernel Hilbert space (RKHS).
Applying kernel method and the representer theorem to perform linear quadratic
estimation in a functional space, we derive a Bayesian recursive state
estimator for a general nonlinear dynamical system in the original input space.
Unlike existing nonlinear extensions of Kalman filter where the system dynamics
are assumed known, the state-space representation for the Functional Bayesian
Filter (FBF) is completely learned from measurement data in the form of an
infinite impulse response (IIR) filter or recurrent network in the RKHS, with
universal approximation property. Using positive definite kernel function
satisfying Mercer's conditions to compute and evolve information quantities,
the FBF exploits both the statistical and time-domain information about the
signal, extracts higher-order moments, and preserves the properties of
covariances without the ill effects due to conventional arithmetic operations.
This novel kernel adaptive filtering algorithm is applied to recurrent network
training, chaotic time-series estimation and cooperative filtering using
Gaussian and non-Gaussian noises, and inverse kinematics modeling. Simulation
results show FBF outperforms existing Kalman-based algorithms.
