Second order Extended Ensemble Kalman Filter with Stochastically Perturbed Innovation

Abstract

Studies have shown several forms of non-linear dynamic filters. However, Extended Kalman filters have proved to provide more accurate values of the state of dynamic systems over period of time. Though, the results of estimation by use extended Kalman filters are accurate, there is involvement of computation of high dimension covariance matrix that are very expensive. Although Bayesian methods offer a robust and accurate approach, they are often hindered by the computational complexity involved in computing high-dimensional matrices. This study introduces a new filter, the Second Order Extended Ensemble Filter with pertubed innovation (SoEEFPI), designed to numerically address the inversion of high-dimensional covariance matrices and then stochastically perturbing the innovation. The SoEEFPI is derived from the numerical expansion of the expected values of non-linear terms in the stochastically perturbed Kushner-Stratonovich equation, utilizing a second- order Taylor series expansion. Validation of the SoEEFPI is conducted on a three-dimensional stochastic Lorenz 63 model, with simulations performed using MATLAB software. In the validation process , SoEEKFPII is compared with First Order Extended Ensemble Filter (FoEEF), First Order Extended Kalman Bucy Filter (FoEKBF), Second order Extended Ensemble Filter (SoEEF), Bootstrap Particle Filter, and Second Order Extended Kalman Bucy Filter (SoEKBF). Results indicated that SoEEFPI outperformed the other filters (KBF, FoEEF, SoEEF) across all three variables of the Lorenz 63 model: x₁, x₂ and x₃. While SoEKBF exhibited the lowest root mean square error (RMSE), its computational cost is significantly higher due to the integration of high-dimensional covariance, making SoEEFPI a more desirable option since its covariance computation is performed empirically.