Page 143 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

132 STATE ESTIMATION

for p(x(i)jZ(i)), the representation of p(x(i þ 1)jZ(i)) is found by
generating one new sample x (k) for each sample x (k) using
selected
p(x(i þ 1)jx (k) ) as the density to draw from. The algorithm is as
selected
follows:
Algorithm 4.4: The condensation algorithm

1. Initialization
. Set i ¼ 0
(k)
. Draw K samples x , k ¼ 1, ... , K, from the prior probability
density p(x(0))

2. Update using importance sampling:
(k)
. Set the importance weights equal to: w (k) ¼ p(z(i)jx )
. Calculate the normalized importance weights: w (k) ¼
norm
(k)
w = P w (k)
3. Resample by selection:
k
. Calculate the cumulative weights w (k) ¼ P w (j)
norm
cum
. for k ¼ 1, ... , K: j¼1
. Generate a random number r uniformly distributed in [0, 1]
. Find the smallest j such that w (j) r (k)
cum
. Set x (k) ¼ x ( j)
selected
4. Predict:

. Set i ¼ i þ 1
. for k ¼ 1, ... , K:
(k)
. Draw sample x , from the density p(x(i)jx(i 1) ¼ x (k) )
selected
5. Go to 2

After step 2, the posterior density is available in terms of the samples x (k)
and the weights w (k) . The MMSE estimate and the associated error
norm
covariance matrix can be obtained from (4.75). For insance, the MMSE
is obtained by substitution of g(x) ¼ x. Since we have a representation of
the posterior density, estimates associated with other criteria can be
obtained as well.
The calculation of the importance weights in step 2 involves the
(k)
conditional density p(z(i)jx ). In the case of the nonlinear measurement
functions of the type z(i) ¼ h(x(i)) þ v(i), it all boils down to calculating
(k)
the density of the measurement noise for v(i) ¼ z(i) h(x ). For