Direct, prediction- and smoothing-based Kalman and particle filter algorithms

We address the recursive computation of the filtering probability density function (pdf) P-n vertical bar n in a hidden Markov chain (HMC) model. We first observe that the classical path p(n-1 vertical bar n-1)-> p(n vertical bar n-1)-> p(n vertical bar n) is not the only possible one that enables to compute p(n vertical bar n) recursively, and we explore the direct, prediction-based (P-based) and smoothing-based (S-based) recursive loops for computing P-n vertical bar n. We next propose a common methodology for computing these equations in practice. Since each path can be decomposed into an updating step and a propagation step, in the linear Gaussian case these two steps are implemented by Gaussian transforms, and in the general case by elementary simulation techniques. By proceeding this way we routinely obtain in parallel, for each filtering path, one set of Kalman filter (KF) equations and one generic sequential Monte Carlo (SMC) algorithm. Finally we classify in a common framework four KF (two of which are original), which themselves can be associated to four generic SMC algorithms (two of which are original). We finally compare our algorithms via simulations. S-based filters behave better than P-based ones, and within each class filters better results are obtained when updating precedes propagation. (C) 2011 Elsevier B.V. All rights reserved.