PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 291

function of the state, we can sample from the optimal proposal distribution even if
the function in the state equation is nonlinear.

A popular choice for the importance function is the prior

( m ) (0 : n 1), y(1 : n )) ¼ f (x( n ) j x ( m ) ( n 1))

p (x( n ) j x

which yields importance weights proportional to the likelihood

w ( m ) ( n ) / w ( m ) ( n 1) f ( y( n ) j x

( m ) ( n )) :

The main advantage of this choice is the ease in the computation of the weights, which
amounts to obtaining the likelihood function. However, the generation of the particles
is implemented without the use of observations, and therefore not all of the available
information is used to explore the state space. This may lead in some practical cases to
poor estimation results. Strategies to improve this performance consist of the inclusion
of a prediction step like that of the auxiliary PF [65] (see Subsection 5.5.2) or the use of
a hybrid importance function if possible [37]. We refer to an importance function as a
hybrid importance function if part of the state is proposed from a prior and the remain-
ing state from the optimal importance function.

B EXAMPLE 5.6
Consider a state space whose parameters can be divided in two groups x( n ) ¼
f x 1 ( n )x 2 ( n ) g and where sampling can be carried out from f (x 2 ( n ) j x

( m )
1 ( n 1),

( m ) ( m ) ( m ) ( m )

1 ( n 1), y( n )). A hybrid proposal
that combines the prior and the posterior importance functions is given by

2 ( n 1)) and f (x 1 ( n ) j x 2 ( n ), x 2 ( n 1), x

x

( m )
1

( m )
2

( m ) ( n 1), y( n )) ¼ f (x 2 ( n ) j x

p (x( n ) j x ( n 1), x ( n 1))

( m )
2

( m )
2

( m )
1

f (x 1 ( n ) j x ( n ), x ( n 1), x ( n 1), y( n ))

( m )
2 ( n 1)). The update of the
weights can readily be obtained from the general expression given by (5.21).

( m )
2

( m )
1

where x ( n ) is a sample from f (x 2 ( n ) j x ( n 1), x

5.4.2 Resampling

In particle filtering the discrete random measure degenerates quickly and only few
particles are assigned meaningful weights. This degradation leads to a deteriorated
functioning of particle filtering. Figure 5.10 illustrates three consecutive time instants
of the operation of a PF which does not use resampling. Particles are represented by
circles and their weights are reflected by the corresponding diameters. Initially, all
the particles have the same weights, that is, all the diameters are equal, and at each
time step the particles are propagated and assigned weights (note that the true posterior
at each time instant is depicted in the figure.) As time evolves, all the particles except