Page 139 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

128 STATE ESTIMATION

hmmtrain: Given a sequence of observations, estimate P t ( j )
and P z ( j ).
hmmviterbi: Given P t ( j ), P z ( j ) and a sequence of observa-
tions, calculate the most likely state sequence.

The function hmmtrain() implements the so-called Baum–Welch
algorithm.

4.4 MIXED STATES AND THE PARTICLE FILTER

Sections 4.2 and 4.3 focused on special cases of the general online estimation
problem. The topic of Section 4.2 was continuous state estimation, and in
particular the linear-Gaussian case or approximations of that. Section 4.3
discussed discrete state estimation. We now return to the general scheme of
Figure 4.2. The current section introduces the family of particle filters (PF). It
is a group of estimators that try to implement the general case. These
estimators use random samples to represent the probability densities, just
as in the case of Parzen estimation; see Section 5.3.1. As such, particle filters
are able to handle nonlinear, non-Gaussian systems, continuous states, dis-
crete states and even combinations. In the sequel we use probability densities
(which in the discrete case must be replaced by probability functions).

4.4.1 Importance sampling

A Monte Carlo simulation uses a set of random samples generated from
a known distribution to estimate the expectation of any function of that
(k)
distribution. More specifically, let x , k ¼ 1, ... , K be samples drawn
from a conditional probability density p(xjz). Then, the expectation of
any function g(x) can be estimated by:

K
1 X
E½gðxÞjzŠffi gx ðkÞ ð4:71Þ
K
k¼1
Under mild conditions, the right-hand side asymptotically approximates
the expectation as K increases. For instance, the conditional expectation
and covariance matrix are found by substitution of g(x) ¼ x and
T
^
^
x
x
g(x) ¼ (x x)(x x) , respectively.
In the particle filter, the set x (k) depends on the time index i.It
represents the posterior density p(x(i)jZ(i)). The samples are called the