PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 305

Figure 5.16 MSEs per sample as a function of the number of used particles.
5.6 HANDLING CONSTANT PARAMETERS
The particle filtering methodology was originally devised for the estimation of
dynamic signals rather than static parameters. The most efficient way to address the
problem is to integrate out the unknown parameters when possible, either analytically
(see next section) [20] or by Monte Carlo procedures [72]. The former methods, how-
ever, depend on the feasibility of integration that is, on the mathematical model of the
system. Generic solutions, useful for any model, are scant and limited in performance.
A common feature of most of the approaches is that they introduce artificial evolution
of the fixed parameters and thereby treat them in a similar way as the dynamic states of
the model [31, 56]. Some methods insert the use of Markov chain Monte Carlo
sampling to preserve diversity of the particles [10, 28]. A recent work [24] introduces
a special class of PFs called density-assisted PFs that approximate the filtering density
with a predefined parametric density by generalizing the concepts of Gaussian PFs and
Gaussian sum PFs. These new filters can cope with constant parameters more naturally
than previously proposed methods. In this section the problem of handling static para-
meters by PFs is reviewed under a kernel-based auxiliary PF method [56] and the
density-assisted particle filtering technique [24].
We can reformulate the state-space model introduced in Section 1.2 to explicitly
incorporate fixed parameters as
x( n ) ¼ g 1 (x( n 1), u , v 1 ( n ))
y( n ) ¼ g 2 (x( n ), u , v 2 ( n ))