Page 140 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

MIXED STATES AND THE PARTICLE FILTER 129

particles. The density can be estimated from the particles by some
kernel-based method, for instance, the Parzen estimator to be discussed
in Section 5.3.1.
A problem in the particle filter is that we do not know the posterior
density beforehand. The solution for that is to get the samples from some
other density, say q(x), called the proposal density. The various members
of the PF family differ (among other things) in their choice of this
density. The expectation of g(x) w.r.t. p(xjz) becomes:
Z
E½gðxÞjzŠ¼ gðxÞpðxjzÞdx
Z
pðxjzÞ
¼ gðxÞ qðxÞdx
qðxÞ
Z
pðzjxÞpðxÞ
¼ gðxÞ qðxÞdx ð4:72Þ
pðzÞqðxÞ
Z
wðxÞ pðzjxÞpðxÞ
¼ gðxÞ qðxÞdx with: wðxÞ¼
pðzÞ qðxÞ
1 Z
¼ gðxÞwðxÞqðxÞdx
pðzÞ
The factor 1/p(z) is a normalizing constant. It can be eliminated as
follows:

Z
pðzÞ¼ pðzjxÞpðxÞdx

Z
qðxÞ
¼ pðzjxÞpðxÞ dx ð4:73Þ
qðxÞ
Z
¼ wðxÞqðxÞdx

Using (4.72) and (4.73) we can estimate E[g(x)jz] by means of a set of
samples drawn from q(x):
K
P
ðkÞ ðkÞ
wðx Þgðx Þ
k¼1
E½gðxÞjzŠffi ð4:74Þ
K
P
ðkÞ
wðx Þ
k¼1
Being the ratio of two estimates, E[g(x)jz] is a biased estimate. However,
under mild conditions, E[g(x)jz] is asymptotically unbiased and consistent