PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 317

A naive approach to these two problems would be to conduct particle filtering in the
usual way and store all the random measures, and for obtaining approximations of the
smoothing densities use the set of particles at time instant n and the weights of these
streams at time instants n þ k or N , respectively. In other words, we use

X

M

w ( m ) ( nþk ) d (x( n ) x

( m ) ( n ))

f (x( n ) j y(1 : nþk )) '

m¼ 1

and

X

M

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

( m ) ( n )) :

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

m¼ 1

These approximations may be good if k or N 2 n are small numbers. For even mod-
erate values of k or N 2 n , the approximations may become quite inaccurate,
especially when resampling is implemented at each time instant. Namely, with resam-
pling, we deplete the number of surviving streams k or N 2 n lags later, which may
make the representation at time instant n rather poor.
In the sequel, we show two approaches for improved smoothing. They are both
based on backward smooth recursions, where the first recursion uses only the set of
weights generated during the forward filtering pass [52] and the second one generates
an additional set of weights during the backward filtering pass [25]. Here we address
the fixed-interval smoothing problem because modifications for fixed-lag smoothing
are straightforward.
First we write the joint a posteriori PDF of the states as follows

f (x(0 : N ) j y(1 : N )) ¼ f (x( N ) j y(1 : N )) Y

N 1

f (x( n ) j x( nþ 1), y(1 : N )) :

n¼ 0

We note that

f (x( n ) j x( nþ 1), y(1 : N )) ¼ f (x( n ) j x( nþ 1), y(1 : n ))
/ f (x( nþ 1) j x( n )) f (x( n ) j y(1 : n )) :

From the above two expressions, we obtain the recursion that takes us from the
smoothing PDF f (x( nþ 1: N ) j y(1 : N )) to f (x( n : N ) j y(1 : N )), that is,

f (x( n : N ) j y(1 : N )) ¼ f (x( nþ 1: N ) j y(1 : N )) f (x( n ) j x( nþ 1), y(1 : N ))
¼ f (x( nþ 1: N ) j y(1 : N )) f (x( n ) j x( nþ 1), y(1 : n ))
/ f (x( nþ 1: N ) j y(1 : N )) f (x( nþ 1) j x( n )) f (x( n ) j y(1 : n )) :
(5 : 37)

If we approximate f (x( n þ 1: N ) j y(1 : N )) by the smoothing random measure
x s ( nþ 1) ¼ {x

( m ) ( nþ 1), w ( m )
s

( nþ 1)} m¼ 1 , where the weights w ( m ) ( nþ 1) are the

s