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Ensemble Filtering for High Dimensional Non-linear State Space Models

We consider non-linear state space models in high-dimensional situations, where the two common tools for state space models both have difficulties. The Kalman filter variants are seriously biased due to non-Gaussianity and the particle filter suffers from the “curse of dimensionality”. Inspired by a regression perspective on the Kalman filter, a novel approach is developed by combining the Kalman filter and particle filter, retaining the stability of the Kalman filter in large systems as well as the accuracy of particle filters in highly non-linear systems. Its theoretical properties are justified under the Gaussian linear models as an extension of the Kalman filter. Its performance is tested and compared with other methods on a simulated chaotic system which is used widely in numerical weather forecasting.