PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 325

5.13 EXERCISES

5.1 Consider the following dynamic system

x ( n ) ¼ x ( n 1) þ v 1 ( n )
y ( n ) ¼ x ( n ) þ v 2 ( n )

where x( n ) and y ( n ) are scalars and v 1 ( n ) and v 2 ( n ) are independent Gaussian
noises of parameters m 1 and s 1 , and m 2 and s 2 , respectively. This system
evolves for n ¼ 0 : 500 s .
(a) If the prior proposal distribution is chosen for generation of particles, derive
the weight update expression.
(b) If the posterior proposal distribution is chosen for generation of particles,
derive the weight update expression.
(c) Program a particle filter in M ATLAB based on Table 5.3 for the previous
system and using as proposal distribution the prior distribution.
(d) Extend the previous program to plot the evolution of the weights. Justify the
degeneracy of the weights as time evolves.
(e) Implement a particle filter that uses the posterior proposal distribution and
compare its performance with the previous one in terms of mean square
error (MSE). The MSE is defined as

J X

J

1

( x j ( n ) x j ( n )) 2

MSE( n ) ¼

j¼ 1

where J denotes the number of simulation runs, and x ( n ) and x j ( n ) are the
true and estimated (in the j -th run) values of the state at time instant n ,
respectively.

5.2 For the model from the previous example, implement the SIR PF with two poss-
ible resampling schemes: one that performs the resampling at each time step, and
another one where the resampling is carried out depending on the effective par-
ticle size measure. Compare the performances of the filters for different numbers
of particles M and different thresholds of effective particle sizes.

5.3 A model for univariate nonstationary growth can be written as

x ( n 1)
1 þx 2 ( n 1) þg cos(1 : 2( n 1)) þ v 1 ( n )

x ( n ) ¼ ax ( n 1) þb

x 2 ( n )
20 þ v 2 ( n ), n ¼ 1, ... , N

y ( n ) ¼

where x(0) ¼ 0.1, a¼ 0.5, b¼ 25, g¼ 8, N ¼ 500, v 1 ( n ) N (0, 1) and
v 2 ( n ) N (0, 1). The term in the process equation that is independent of x ( n )