PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 310

Gaussian of the bias vector by

m ( n 1) ¼ X

M

w ( m ) ( n 1) u

( m ) ( n 1)

m¼ 1

S ( n 1) ¼ X

M

w ( m ) ( n 1)( u

( m ) ( n 1) m ( n 1))( u ( m ) ( n 1) m ( n 1)) ` :

m¼ 1

( m ) ( n )
N ( m ( n 1), S ( n 1)) : Once we have the particles of the biases, we proceed by
using the favorite PF. For example, if it is the APF, first we project the dynamic
particles as in the previous example, that is

Then we draw the particles of u for the next time step, that is, u

( m ) ( n ) ¼ Ax

( m ) ( n ) :

x

This is followed by resampling of the streams whose weights are given by
w ( k m ) ( n 1) N ( g 2 ( x

( k m ) ( n ), S v ), and where the Gaussian is computed
at y( n ). Once the indexes of the streams for propagation are known, we draw the
particles of the dynamic variables, x

( k m ) ( n )) þu

( m ) ( n ). Next, the new weights of the streams

are computed by

( k m ) ( n ))

( m ) ( n ), u

f ( y( n ) j x

w ( m ) ( n ) /

( k m ) ( n )) :

( k m ) ( n ), u

f ( y( n ) j x

5.7 RAO-BLACKWELLIZATION

In many practical problems, the considered dynamic nonlinear system may have some
states that are conditionally linear given the nonlinear states of the system. When the
applied methodology is particle filtering, this conditional linearity can be exploited
using the concept of Rao-Blackwellization [20, 25]. Rao-Blackwellization is
a statistical procedure that is used for reducing variance of estimates obtained by
Monte Carlo sampling methods [18], and by employing it, we can have improved
filtering of the unknown states.
The main idea consists of tracking the linear states differently from the nonlinear
states by treating the linear parameters as nuisance parameters and marginalizing
them out of the estimation problem. This strategy allows for more accurate estimates
of the unknowns because the dimension of the space that is explored with particles is
reduced and therefore it is much better searched. At every time instant the particles of
the nonlinear states are propagated randomly, and once they are known, the problem
is linear in the rest of the states. Therefore, one can find their 'optimal' values by
employing Kalman filtering and associate them with the sampled nonlinear states.
Some recent applications of Rao-Blackwellized PFs include tracking of maneuvering