Description
A.2 (c) (textbook). You can take parts (a) and (b) as a given (no need for re-derivation)
since you did those parts in HW 12.
A. Consider the following special case of a nonlinear state-space model:
xk+1 = fk (xk) + wk ,
zk = Hkxk + vk ,
where Hk is a matrix and wk and vk are mutually independent zero-mean Gaussian noise
processes with covariance matrices Qk and Rk (as usual). As discussed in the reading
Arulampalam et al. (2002) and in the online class handout
Kalman_particle_MCMC_handout.pdf, the optimal importance density for particle
filtering under the independent noise assumption is 1 ( ) , kk k − p xx z . For the model, above, it is
known that that 1 ( ) , kk k − p xx z corresponds to a N(ak, Σk) distribution with ak being a mean vector
(not relevant here) and Σk = Qk−1 − Qk−1Hk
T
Sk
−1
HkQk−1 and Sk = HkQk−1Hk
T + Rk. Derive this
formula for Σk. (Hint: The matrix inversion lemma [sometimes called the
Sherman−Morrison−Woodbury formula] may be useful: 1 ( ) + − C AUB = −1 C − 1 1 11 1 ( ) + − − −− − C A BC A U BC , given that the indicated inverses are assumed to exist.)
Note: The main reading for particle filtering in the class, Arulampalam et al. (2002), almost gives
the result to be proved in expression (60) in the reading, but the paper does not show how the
expression is derived (simply stating “one obtains….” before (60)). You are to prove the result
using the given model form, first principles, and Bayes formula, as needed. You may need to
prove the result (58) in Arulampalam et al. (2002) as part of the derivation ((58) is not to be
taken as a “given”).
B. Consider the following linear state-space model of a three-state dynamical system with scalar
measurements:
xk+1 = �
0.5 0.5 0
0 0.5 0.5
0 0 0.5
� xk + wk
zk = [1 1 1] xk + vk
where wk and vk are mutually independent zero-mean Gaussian noise processes with covariance
matrices Qk = I3 (3×3 identity matrix) and Rk = 1 for all k. Further, assume that the initial state
has x0 = [0 0 0]T and P0 = I3. Generate one simulated set of measurements {z1,z2,…,z200} and use
both the Kalman filter and the particle filter with resampling to estimate the state vectors
corresponding to the 200 measurements. Using N = 10 or 1000 particles, calculate the relative
errors for each three components between the two estimates according to the following:
(a) For each of the three components in the state vector, using N = 10 and N = 1000, plot the true
state, Kalman filter estimate and the particle filter estimate over the time range [0,200] (i.e.,
produce three plots with three paths on each plot for each N, so a total of 6 plots). All studies use
the same set of 200 measurements.
(b) Compare and comment briefly on the observed accuracy of the particle filter estimates
relative to the Kalman filter estimates across the time range [0,200] for the two values of N. That
is, for each of the three components, take the (Euclidean) norm of the difference between the
200-dimensional vectors of particle filter estimates and Kalman filter estimates, and divide by
the norm of the 200-dimensional vector of Kalman filter estimates.
Note: For problem B, you may use particle filter code posted at the course website. You should
use your formerly developed (HW12) Kalman filter code as well.
