PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 292

Figure 5.10 A pictorial description of particle filtering without resampling.

for very few are assigned negligible weights. In the last step there is only one particle
with significant weight.
The mechanism of resampling as well as the use of good importance function can
reduce this degeneracy. A measure of this degeneracy is the effective particle size
defined by [54]

M
1 þ Var( w ( m ) ( n ))

M eff ¼

f (x( n ))
p (x( n )) is the true particle weight. This metric can be estimated as

where w ( m ) ( n ) ¼

1
P m¼ 1 ( w ( m ) ( n )) 2

M eff ¼

with w ( m ) ( n ) being the normalized weight corresponding to the m -th particle at time
instant n . If the effective particle size is below a predefined threshold, resampling is
carried out. Clearly, when all the particles have the same weights, the variance of
the weights is zero and the particle size is equal to the number of particles, M . The
other extreme occurs when all the particles except one have negligible weights, and
the particle size is equal to one.
Resampling eliminates particles with small weights and replicates particles with
large weights. In general, if the random measure at time instant n is x ( n ), it proceeds
as follows.