PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 315

Note that we already saw this expression in (5.7). Recall also that the PF provides an
estimate of the filtering density in the form

f (x( n ) j y(1 : n )) ' X

M

w ( m ) ( n ) d (x( n ) x

( m ) ( n ))

(5 : 34)

m¼ 1

which when used in (5.33) yields an estimate of the predictive density in the form of a
mixture density given by

f (x( nþ 1) j y(1 : n )) ' X

M

w ( m ) ( n ) f (x( nþ 1) j x

( m ) ( n )) :

(5 : 35)

m¼ 1

If we express this PDF by using a discrete random measure, then a natural way of
expressing this measure is by {x

( m ) ( nþ 1), w ( m ) ( n )} m¼ 1 , where

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

x

and

X

M

w ( m ) ( n ) d (x( nþ 1) x

( m ) ( nþ 1)) :

f (x( nþ 1) j y(1 : n )) '

m¼ 1

In summary, we obtain an estimate of the predictive PDF f (x( nþ 1) j y(1 : n )) from
the random measure x ( n ) ¼ {x

( m ) ( n ), w ( m ) ( n )} m¼ 1 by simply (a) drawing particles

( m ) ( n )), for m ¼ 1, 2, ... , M and (b) assigning the
weights w ( m ) ( n ) to the drawn particles x

( m ) ( nþ 1) from f (x( nþ 1) j x

x

( m ) ( nþ 1). 12
Next we do the case k ¼ 2. The strategy for creating the random measure that
approximates f (x( nþ 2) j y(1 : n )) is analogous. Now

ðð f (x( nþ 2) j x( nþ 1)) f (x( nþ 1) j x( n ))

f (x( nþ 2) j y(1 : n )) ¼

f (x( n ) j y(1 : n )) d x( n ) d x( nþ 1) : (5 : 36)

Here we have two integrals, where the first involves the integration of x( n ) carried out
in the same way as above. We reiterate that we can approximate f (x( n ) j y(1 : n )) as in

12 Another approximation would be by drawing the particles from the mixture in (5.35), that is, by first
sampling the mixand and then drawing the particle from it. In that case all the drawn particles have the
same weight.