Illustrative Examples and Possible Explanation for an Unexpected Behaviour of the Particle Filter

The particle filter (PF) approximates the posterior distribution of the states in filtering problems, and it is well-known that it converges to the true posterior when the number of particles tends to infinity. It would be natural to assume that measures such as mean square error (MSE) decreases monotonically as the number of particles increases. This is, however, not always true. We present a simple two-dimensional linear Gaussian system where the MSE grows initially before it starts to decrease to eventually reach the optimal filter performance, which in this case is provided by the Kalman filter (KF). Other indicators such as the efficient number of particles and trace of the particle covariance show a similar strange behavior. Inspired by this, we derive a condition for what we term projected instability, which means that the particle in the standard SIR PF that gives the best prediction actually increases the state estimation error. For linear systems, this gives an explicit condition in terms of the state space matrices when this situation occurs. Monte Carlo simulations of a large number of random linear systems indicate that everything works as expected as long as the system does not have a projected instability, otherwise the particle filter can perform badly or even diverage.