PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 319

The second algorithm is based on an identity from [46], which connects the
smoothing density f (x( n ) j y(1 : N )) with f (x( nþ 1) j y (1 : N )) through an integral

and which is given by

f (x( n ) j y(1 : N )) ¼ f (x( n ) j y(1 : n ))

ð f ( x ( n þ 1) j x ( n )) f ( x ( n þ 1) j y (1 : N ))
f (x( nþ 1) j y(1 : n ))

d x( nþ 1) : (5 : 38)

The identity can readily be proved by noting that

ð f (x( n ), x( nþ 1) j y(1 : N )) d x( nþ 1)

f (x( n ) j y(1 : N )) ¼ (5 : 39)

and

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

f ( x ( n þ 1) j x ( n )) f ( x ( n ) j y (1 : n ))
f (x( nþ 1) j y(1 : n ))

: (5 : 40)

We use the following approximations

X

M

w ( m )
s

( m ) ( n ))

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

m¼ 1

X

M

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

( m ) ( n ))

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

m¼ 1

and

ð f (x( nþ 1) j x( n )) f (x( n ) j y(1 : n )) d x( n )

f (x( nþ 1) j y(1 : n )) ¼

X

M

w ( m ) ( n ) f (x( nþ 1) j x

( m ) ( n ))

m¼ 1

to obtain

( n ) ¼ X

M

w ( m ) ( n ) f (x

( j ) ( nþ 1) j x ( m ) ( n ))

w ( m )
s

w ( j )
s

( nþ 1) P l¼ 1 w ( l ) ( n ) f (x

( l ) ( n )) : ( j ) ( nþ 1) j x

j¼ 1

The algorithm is summarized in Table 5.7. The second algorithm is more complex
than the first.