  This paper presents an adaptive Kalman filter for a linear dynamic system
perturbed by an additive disturbance. The objective is to estimate both of the
state and the unknown disturbance concurrently, while learning the disturbance
as a stochastic process of the state vector. This is achieved by estimating the
state according to the extended Kalman filtering applied to the marginal
distribution of the state, and by estimating the disturbance from a backward
smoothing technique. The corresponding pair of the estimated states and
disturbances are fetched to a Gaussian process, which is constantly updated to
resemble the disturbance process. The unique feature is that all of
uncertainties in the estimated state and disturbance are accounted throughout
the learning process. The efficacy of the proposed approach is illustrated by a
numerical example.
