BootStrap/Sequential Importance Resampling

A simpler solution is Sequential Importance Resampling, in which the weights do not depend on previous iterations; instead, the particles change following a resampling phase.

The resampling phase consists of generating a new set of particles $x'$ by resampling $N_s$ times from a discrete approximation of $p(\mathbf{x}_k \vert \mathbf{z}_k)$ given by

$$p( \mathbf{x}_k \vert \mathbf{z}_k ) \approx \sum_{i=1}^{N_s} w_{k,i} \delta (\mathbf{x}_k - \mathbf{x}_{k,i} )$$ (3.48)

where

$$w_{k,i} \propto p(\mathbf{z}_k \vert \mathbf{x}_k)$$ (3.49)

has been defined.

SIR filters do not avoid the degenerate case (in fact, they permanently eliminate unlikely particles), but they provide substantial computational savings and concentrate the search for a solution around the most probable states.

Several algorithms are available for performing resampling. A non-exhaustive list is: Simple Random Resampling, Roulette Wheel / Fitness proportionate selection, Stochastic universal sampling, Multinomial Resampling, Residual Resampling, Stratified Resampling, Systematic Resampling.