Swarm Intelligence Techniques Applied to Nonlinear Systems State Estimation - Advances in Heuristic Signal Processing and Applications - page 222

Fig. 10.4 Illustration of a
resampling process in PF [ 3 ]

then N eff = 1(see[ 3 ]). Resam-

pling is a strategy to overcome degeneracy of samples in SIS. The idea of resam-
pling is to eliminate particles with small weights and copy those with large weights.
During this procedure, samples

N, and (ii) if severe degeneracy occurs, then N eff =

x j w j x j

.
Resampling was first proposed by Gordon, Salmond, and Smith [ 22 ], which is
illustrated in Fig. 10.4 [ 3 ], where CSW stands for the cumulative sum of particles
weight, j = 1 ¯

{

k ,

¯

k }

are replaced with samples {˜

k , 1 /N

}

w j

k , and the random variable, r j , is uniformly distributed within the
interval [0, 1]. For example, if r j =

0 . 4, then the first particle for which j = 1 ¯ w k ≥

r j is the third particle. Therefore, a particle with large weight will have a good
chance of being resampled several times.

10.3.5 State Estimation

In the final step, the posterior density at time k will be approximated as a discrete
density given by [ 9 ]:

N

p ˆ z k ≈

k δ x k − ˜ k

w j x j

x k |

1 ¯

(10.8)

j =

w j

where the normalized weights

k are updated according to Eq. ( 10.6 ) and δ is the
Dirac Delta-function. Therefore, the approximation of the posterior density can be
formulated using some statistical properties (mean, median, confidence intervals,
etc.), based on the weight of particles. For example, the states can be estimated

¯