Extended Ensemble Filter for High-dimensional Nonlinear State Space Models

Abstract

There are several functional forms for non-linear dynamical filters. Extended Kalman filters are algorithms that are used to estimate more accurate values of unknown quantities of internal dynamical systems from a sequence of noisy observation measured over a period of time. This filtering process becomes computationally expensive when subjected to high dimensional data which consequently has a negative impact on the filter performance in real time. This is because integration of the equation of evolution of covariances is extremely costly, especially when the dimension of the problem is huge which is the case in numerical weather prediction. This study has developed a new filter, the First order Extended Ensemble Filter (FoEEF), with a new extended innovation process to improve on the measurement and be able to estimate the state value of high dimensional data. We propose to estimate the covariances empirically, which lends the filter amenable to large dimensional models. The new filter is derived from stochastic state-space models and its performance is tested using Lorenz 63 system of ordinary differential equations and Matlab software. The performance of the newly developed filter is then compared with the performances of three other filters, that is, Bootstrap particle Filter (BPF), First order Extended Kalman Bucy Filter (FoEKBF) and Second order Extended Kalman Bucy Filter (SoEKBF). The performance of the FoEEF improves with the increase in ensemble size. Even with as low number of ensembles as 40, the FoEEF performs as good as the FoEKBF and SoEKBF. This shows, that the proposed filter can register a good performance when used in high-dimensional state-space models.