Bayesian Estimation and Tracking: A Practical Guide

17.1 The Gaussian Particle Filter

Let us return once again to the fundamental equations of Bayes estimation, remembering that it is a two-step process, with the filtering step given by the estimation of the posterior (filtering or update) density

(17.1)

and the predictive step encompassing the estimation of the predictive (or prior, because it uses all of the prior measurements) density

(17.2)

The fundamental concept of the GPF is to make the assumption that both the posterior and prior densities are Gaussian. That is, let

(17.3)

(17.4)

Assume that at time t ₀, prior to any observations, we have prior information about the initial density and that

(17.5)

To initialize the GPF, we first draw samples from and propagate them forward in time using the dynamic equation

(17.6)

Then we compute ...