PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 300

The last step amounts to computing the weights of these particles by

f ( y( n ) j x

( m ) ( n ))

w ( m ) ( n ) /

( k m ) ( n )) :

(5 : 26)

f ( y( n ) j x

The algorithm is summarized in Table 5.4.
So, what do we gain by computing estimates of x( n ) and implementing the drawing
of particles by auxiliary variables? By using the estimates of x

( m ) ( n ), we look ahead to
how good the particle streams may be. Rather than resampling from samples obtained
from the prior, we first resample by using the latest measurement and then propagate
from the surviving streams. Thereby, at the end of the recursion instead of having
particles propagated without the use of y( n ), we have particles moved in directions
preferred by y( n ). With SIR, the data y( n ) affect the direction of particle propagation
later than they do with APF.

Table 5.4 Auxiliary particle filter
Initialization
For m = 1,2, ... ,M
Sample

x(0) (m)

f(x(0))

1
M
Recursions
For n = 1,2, ...
For m = 1,2, ... , M
Estimation of next particles

(m) (0) ¼

w

(m) (n) ¼ E(x(n) j x

(m) (n 1))

Compute x

Sample the indexes of the streams that survive

k m

Sample k m = i with probability

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

w

Sample the new particles for time instant n

(k m ) (n 1))
Computation of weights
Evaluate the weights w

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

x

(m) (n) ¼ f ( y ( n ) j x

(m) (n))

(k m ) (n))

f(y(n) j x

(m) (n)
P j ¼ 1 w

w

(m) (n) ¼

Normalize the weights w

(j) (n)