A linear extension of Unscented Kalman Filter to higher-order moment-matching

This paper addresses the problem of optimal state estimation (OSE) for a wide class of nonlinear time series models. Empirical evidence suggests that the Unscented Kalman Filter (UKF), proposed by Julier and Uhlman, is a promising technique for OSE with satisfactory performance. Unscented Transformation (UT) is the central and vital operation performed in UKF. A crucial point of UT is to construct a σ-set, which consists of points with associated weights capturing the input statistics, e.g., first and second and possibly higher moments. We analyze the standard choice of σ-set and propose a novel method for generating σ-set so as to capture arbitrary higher order input statistics. This method could be considered as a linear extension of UT or UKF, and its computational complexity is the same order as that of the UKF and so EKF. The performance of the algorithm is illustrated by empirical examples. Results show an improvement in accuracy compared to traditional UKF.