Non-Gaussian Filtering by Local Linear Approximation

We propose a new approach to nonlinear non-Gaussian filtering, by explicitly mimicking the optimal filter using local linear approximation to the underlying dynamical system. At each time step, instead of directly using the forecast ensemble as the forecast distribution as in the ensemble Kalman filter, we find a suitable linear transformation that maps the previous updated ensemble to the current forecast ensemble, with small additive Gaussian noises. Once this linear transformation and the noise distribution are estimated, the optimal filtering can be approximated by a standard Kalman filtering. Such a local linear approximation shall have good robustness against the highly nonlinear dynamics. It can also make use of temporal smoothness by borrowing information from recent time steps using techniques such as weighted local averaging. Spatial smoothness can be taken into account by regularizing the estimated linear transformation and noise distribution. The method will be illustrated on several classical models of nonlinear dynamical systems.