Swarm Intelligence Techniques Applied to Nonlinear Systems State Estimation - Advances in Heuristic Signal Processing and Applications - page 223

Fig. 10.5 Particle impoverishment due to narrowness of the maximum-likelihood region [ 14 ]

based on the average position of particles as follows:

N

1
N

x k .

x k = ˆ

1 ˜

(10.9)

j =

It can also be shown that as N →∞ the above approximation approaches the true

posterior density [ 9 ].

10.3.6 Limitations

Some versions of PF adapt the SIS algorithm to calculate the posterior distribution
using the importance sampling density such as Sampling Importance Resampling
(SIR) filter [ 22 , 23 ] and Auxiliary Sampling Importance Resampling (ASIR) fil-
ter [ 24 ]. Since PF algorithms are suboptimal estimators, they have some accuracy
problems. In the following subsections, these problems are discussed in detail.

10.3.6.1 Particle Impoverishment

Particle impoverishment happens when the likelihood is so narrow that the overlap-
ping region of likelihood and prior distribution is quite small [ 3 , 25 ] and no particle
lies within the region of likelihood probability. Thus, many particles are wasted in
the low likelihood region, as depicted in Fig. 10.5 , and few particles are located in
the high likelihood region. Therefore, the weights of most particles become rela-
tively small and their efficiency is decreased; the result of this is the degradation of
the estimation accuracy.