Prediction-Based Particle Filter with dependent noises

Bayesian Filtering in nonlinear stochastic dynamical systems has been addressed for a long time. Among other solutions, Particle Filtering (PF) algorithms propagate in time a Monte Carlo (MC) approximation of the a posteriori filtering measure. This paper presents an algorithm for particle prediction that takes into account a variant of the traditional Hidden Markov Model where noises (process noise and measurement noise) are not independent. We present a comparison between three different algorithms working over the mentioned model. The first algorithm takes samples from p(x
k
|x
k−1
z
k−1
) as it is the optimal importance function for prediction. The second algorithm takes samples form the optimal importance function p(x
k
|x
k−1
z
k−1
z
k
) for the filtering case. This last approach has the drawback that filtering samplers are often difficult (or even imposible) to compute or to sample from. The third Algorithm bypass this difficulty by rather considering the prediction problem as in Algorithm 1 but obtains the MC filtering estimate as a byproduct of the prediction process in an indirect approximation. Different simulations were realized with a well known semi-non linear example in order to compare the estimation error and the performance of the three methods. This idea was previously developed but for independent noises. We show here how it can be taken to a context with dependent noises in a natural way obtaining little higher errors than the direct filtering method but better performance in terms of runtime. Results confirm the validity of Prediction-Based Particle filtering method in a dependent noises enviroment.