Page 141 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

130 STATE ESTIMATION

as K increases. One of the requirements is that q(x) overlaps the support
of p(x).
Usually, the shorter notation for the unnormalized importance
(k)
weights w (k) ¼ w(x ) is used. The so-called normalized importance
weights are w (k) ¼ w (k) P w (k): . With that, expression (4.74) simpli-
norm
fies to:
K
X
E½gðxÞjzŠffi w ðkÞ gx ðkÞ ð4:75Þ
norm
k¼1

4.4.2 Resampling by selection

Importance sampling provides us with samples x (k) and weights w (k) .
norm
Taken together, they represent the density p(xjz). However, we can trans-
form this representation to a new set of samples with equal weights. The
procedure to do that is selection. The purpose is to delete samples with low
weights, and to retain multiple copies of samples with high weights. The
number of samples does not change by this; K is kept constant. The various
members from the PF family may differ in the way they select the samples.
However, an often used method is to draw the samples with replacement
according to a multinomial distribution with probabilities w (k) .
norm
Such a procedure is easily accomplished by calculation of the cumu-
lative weights:

k
X
w ðkÞ ¼ w ð jÞ ð4:76Þ
cum
norm
j¼1
We generate K random numbers r (k) with k ¼ 1, .. . , K. These numbers
must be uniformly distributed between 0 and 1. Then, the k-th sample
x (k) in the new set is a copy of the j-th sample x (j) where j is the
selected
(k)
smallest integer for which w (j) r .
cum
Figure 4.21 is an illustration. The figure shows a density p(x) and a
proposal density q(x). Samples x (k) from q(x) can represent p(x) if they
(k)
(k)

are provided with weights w (k) / p(x ) q(x ). These weights are
norm
visualized in Figure 4.21(d) by the radii of the circles. Resampling by
selection gives an unweighted representation of p(x). In Figure 4.21(e),
multiple copies of one sample are depicted as a pile. The height of the
pile stands for the multiplicity of the copy.