The compact gaussian mixture optimal transport ensemble kalman filter and its applications to partially linear systems

In this paper, we revisit the Gaussian mixture optimal transport Ensemble Kalman filter (GM-OT-EnKF) proposed in Li and Luo (2024) to make it more efficient in some partially linear problems. We first show the equivalence between the OT-EnKF with the classical Kalman filter (KF) in the linear system with Gaussian initial distribution. Next, we compare the Gaussian sum filter (GSF) with the GM-OT-EnKF in the setting of linear system with GM initial distribution. The updating of the components’ weights in GM-OT-EnKF is finer than those in the GSF, due to the flexibility induced by the particles. These observations in the linear system suggest two adaptions to the original GM-OT-EnKF corresponding to partially linear systems. The one is when the state’s equation is linear, the EM algorithm is unnecessary in every cycles; the other one is when the observation is linear, the posterior mean and covariance matrix should be updated explicitly, rather than the empirical ones. The GM-OT-EnKF with either one of the two adaptions above is called the compact GM-OT-EnKF in this paper. The efficiency and accuracy of our proposed algorithm have been numerically verified in the estimation of the states in the Lorenz 63 system and the prediction of the remaining useful life of the lithium-ion batteries.