Page 142 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

MIXED STATES AND THE PARTICLE FILTER 131

4
(a) p(x)
2
0
2
(b) q(x)
1
0
0 0.2 0.4 0.6 0.8 1
x

(c)
(d)

(e)
Figure 4.21 Representation of a probability density. (a) A density p(x). (b) The
proposal density q(x). (c) 40 samples of q(x). (d) Importance sampling of p(x) using
the 40 samples from q(x). (e) Selected samples from (d) as an equally weighted
sample representation of p(x)

4.4.3 The condensation algorithm

One of the simplest applications of importance sampling combined
with resampling by selection is in the so-called condensation algorithm
(‘conditional density optimization’). The algorithm follows the general
scheme of Figure 4.2. The prediction density p(x(i)jZ(i 1)) is used as
the proposal density q(x). So, at time i, we assume that a set x (k) is
available which is an unweighted representation of p(x(i)jZ(i 1)). We
use importance sampling to find the posterior density p(x(i)jZ(i)). For
that purpose we make the following substitutions in (4.72):

pðxÞ! pðxðiÞjZði 1ÞÞ
pðxjzÞ! pðxðiÞjzðiÞ; Zði 1ÞÞ ¼ pðxðiÞjZðiÞÞ
qðxÞ! pðxðiÞjZði 1ÞÞ
pðzjxÞ! pðzðiÞjxðiÞ; Zði 1ÞÞ ¼ pðzðiÞjxðiÞÞ

The weights w (k) that define the representation of pðxðiÞjzðiÞÞ is
norm
obtained from:
w ðkÞ ¼ pðzðiÞjx Þ ð4:77Þ
ðkÞ
Next, resampling by selection provides an unweighted representation
x (k) . The last step is the prediction. Using x (k) as a representation
selected selected