Page 133 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

122 STATE ESTIMATION

Minimizing the error probabilities of all individually estimated states
does not imply that the sequence of states is estimated with minimal
error probability. It might even occur that a sequence of ‘individually
best’ estimated states contains a forbidden transition, i.e. a transition for
which P t (x(i)jx(i 1)) ¼ 0. In order to circumvent this, we need a criter-
ion that involves all states jointly.
This section discusses two offline estimation methods. One is an
‘individually best’ solution. The other is an ‘overall best’ solution.

Individually most likely states
Here, the strategy is to determine the posterior probability P(x(i)jZ(I)), and
x
then to determine the MAP estimate: ^ x(ijI) ¼ arg max P(x(i)jZ(I)). As said
before, this method minimizes the error probabilities of the individual states.
As such, it maximizes the expected number of correctly estimated states.
Section 4.3.1 discussed the forward algorithm, a recursive algorithm
for the calculation of the probability P(x(i), Z(i)). We now introduce the
backward algorithm which calculates the probability P(z(i þ 1), .. . ,
z(I)jx(i)). During each recursion step of the algorithm, the probability
P(z(j), .. . , z(I)jx(j 1)) is derived from P(z(j þ 1), ... , Z(I)jx(j)). The
recursion proceeds as follows:

PðzðjÞ; .. . ; zðIÞjxðj 1ÞÞ
K
X ð4:63Þ
¼ P t ðxðjÞjxðj 1ÞÞP z ðzðjÞjxðjÞÞPðzðj þ 1Þ; .. . ; ZðIÞjxðjÞÞ
xðjÞ¼1

The algorithm starts with j ¼ I, and proceeds backwards in time, i.e.
I, I 1, I 2, .. . until finally j ¼ i þ 1. In the first step, the expression
P(z(I þ 1)jx(I)) appears. Since that probability does not exist (because
z(I þ 1) is not available), it should be replaced by 1 to have the proper
initialization.
The availability of the forward and backward probabilities suffices for
the calculation of the posterior probability:

PðxðiÞ; ZðIÞÞ
PðxðiÞjZðIÞÞ ¼
PðZðIÞÞ
Pðzði þ 1Þ; ... ; zðIÞjxðiÞ; ZðiÞÞPðxðiÞ; ZðiÞÞ
¼ ð4:64Þ
PðZðIÞÞ
Pðzði þ 1Þ; ... ; zðIÞjxðiÞÞPðxðiÞ; ZðiÞÞ
¼
PðZðIÞÞ