PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 298

filtering density f (x( n ) j y(1 : n )) by the mixture density

f ( y( n ) j x( n )) X

M

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

( m ) ( n 1)) :

(5 : 23)

m¼ 1

( m ) ( n ) from this density. In
order to do so, we introduce an auxiliary variable, which is an index variable. We
denote it by k and we index it by m , so that we write it as k m . We draw it from the
set f 1, 2, ... , Mg , 9 and it denotes the particle stream which we want to update.
Thus, if we draw k m ¼ 5, we work with the 5th stream, if we have k m ¼ 11, it is the
11th stream and so on.
First we describe the basic APF method. This method makes easy the problem of
drawing x( n ) from (5.23) by using estimates of x( n ) for each stream of particles. If we
denote the estimates by x

The underlying idea behind APF is to propose samples x

( m ) ( n ), we modify (5.23) and create a proposal distribution

given by

X

M

( m ) ( n )) f ( x( n ) j x

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

( m ) ( n 1)) :

(5 : 24)

m¼ 1

( m ) ( n ) can be any value of x( n ) that is a good representative, which
means that it should be a value that can easily be computed and has high likelihood.
For example, if the state equation is

An estimate of x

x( n ) ¼ g 1 (x( n )) þ v 1 ( n )

( m ) ( n ) could be

and the noise vector v 1 ( n ) is zero mean, an estimate of x

( m ) ( n ) ¼ g 1 (x ( m ) ( n 1)) :

x

( m ) ( n ) and the new form of the proposal distribution (5.24), it is
much easier to propose new particles x

With the estimates x

( m ) ( n ). The idea has a subtle point: we use (5.24)
as a joint distribution of the auxiliary variable and the state. The implications is that
first we draw the auxiliary variable (index) k m from a multinomial distribution,
where Pr ( k m ¼ i ) / w ( i ) ( n 1) f ( y( n ) j x

( i ) ( n )). 10 The drawn index, say i , identifies

the distribution from which we draw x

( m ) ( n ), f (x( n ) j x ( k m ¼i ) ( n 1)), and so we pro-

( k m ¼i ) ( n 1)). Once the particles are
drawn, as with SIR, the last step is the computation of the weights.
Before we derive the formula for the update of the weights, we express the
proposal distribution in a form that will make the derivation easy. First, we rewrite

ceed by drawing a particle from f (x( n ) j x

9 In fact, the number of drawn auxiliary variables can be different from M [65].
10 This is basically the same procedure applied for resampling.