On the equivalence of the extended Kalman smoother and the expectation maximisation algorithm for polynomial signal models

Abstract

The iterated extended Kalman smoother (IEKS) is shown to be equivalent to one iteration of the expectation maximisation (EM)-based SAGE algorithm for the class of nonlinear signal models containing polynomial dynamics. Thus the IEKS is a maximum a posteriori (MAP) state sequence estimator for this class of systems. The iterated extended Kalman filter (IEKF) can be thought of as a heuristic, online version of a SAGE algorithm, derived via EM formalism rather than via linearisation around approximate conditional mean state estimates. We apply the polynomial SAGE algorithm to the discrete time, cubic sensor problem and show that it outperforms the standard extended Kalman smoother.