Higher Order Kalman Filtering for Nonlinear Systems



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We seek to improve upon and generalize the Ensemble Kalman Filter (EnKF) by defining a Higher Order Kalman Filter. The Kalman filter consists of two steps: forecast and assimilation. In this thesis we develop the forecast step of our desired Higher Order Kalman Filter with the higher order unscented transform (HOUT). The HOUT is a quadrature rule that estimates the expected value of the first four moments of a distribution, i.e. the mean, covariance, skewness and kurtosis. We then discuss how to generalize the assimilation step. The original Kalman Filter can be derived in three ways: the Bayesian approach, the Minimum Mean-Square Estimate (MMSE) approach and the Closure approach. Each derivation provides a different avenue for us to derive the Higher Order Kalman Filter. In order to generalize the Bayesian approach to the first four moments, instead of using a Gaussian likelihood and prior, we use exponentials with a quartic polynomial as the exponent. In order to generalize the MMSE approach we consider deriving optimal quadratic filters. Finally we may generalize the closure approach by deriving the ordinary differential equations for the skewness and kurtosis and instead of assuming that the skewness is zero, we seek new closures for the first four moments rather than just the first two.



CP decomposition, Kalman filter, Kurtosis, Skewness, Tensors, Unscented transform