PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 320

Table 5.7 Smoothing algorithm II

Forward PF
Run PF in the forward direction and store all the random measures
x (n) ¼ {x

(m) (n) , w (m) (n)} m ¼ 1 ,

n ¼ 1 , 2 , , N

Backward recursions
Set the smoothing weights
w

(m)
s

(N) ¼ w

(m) (N)and

(m) (N) , w (m)

s

(N)} m ¼ 1

x s (N) ¼ {x

For n=N 21, ... ,1,0

Computation of the smoothing weights w s (n)

(m) (n)by

Compute the smoothing weights of x

(n) ¼ X M

(m) (n)f(x (j) (n þ 1) j x (m) (n))

w

(m)
s

(j)
s

(n þ 1)

P l ¼ 1 w

w

j ¼ 1 w (j) (n þ 1) j x

(l) (n))
Construction of the smoothing random measure x s

(l) (n)f(x

(n)

(m) (n) , w (m s (n)} m ¼ 1

Set x s (n) ¼ {x

5.10 CONVERGENCE ISSUES

Whenever we develop algorithms for estimation, we study their performance and often
express it in some probabilistic / statistical terms. For example, when we deal with par-
ameter estimation, a benchmark for performance comparison of unbiased estimators is
the Cram´r-Rao bound. In particle filtering, where we basically rely on the Bayesian
methodology, we use the concept of posterior Cram´r-Rao bounds (PCRBs) and in
obtaining them we combine the information about the unknowns extracted from
observed data and prior information [74]. The PCRBs represent the MSEs that can
be achieved by an estimator. The computation of the PCRB in the context of sequential
signal processing is presented in [73]. Thus, often we find that particle filtering algor-
ithms are compared with PCRBs of the estimated states, where the PCRBs themselves
are computed by using Monte Carlo integrations. The difference between the MSEs
and the PCRBs provides us with quantitative information about how far the algorithm
is from optimal performance. An important issue that has not been successfully
resolved yet is the required number of particles that are needed for achieving a desired
performance. A related problem is the curse of dimensionality, that is, that for satisfac-
tory accuracy the number of particles most likely will go up steeply as the dimension of
the state increases [22].
An important type of performance issue is related to the convergence of particle
filtering methods. For example, one would like to know if the methods converge,
and if they do, in what sense and with what rate. We know that particle filtering is
based on approximations of PDFs by discrete random measures and we expect that
the approximations deteriorate as the dimension of the state space increases but