BootStrap/Sequential Importance Resampling

A simpler solution is Sequential Importance Resampling, in which the weights do not depend on previous iterations; instead, the samples 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)

SIR filters do not eliminate the degenerate case—in fact, they permanently eliminate particles with low probability—but they provide substantial computational savings and focus the search for the solution around the most probable states.

Several algorithms exist 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.