Page 135 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

124 STATE ESTIMATION

The computation of this most likely state sequence is done efficiently by
means of a recursion that proceeds forwards in time. The goal of this
recursion is to keep track of the following subsequences:

x
^ x xð0Þ; .. . ; ^ xði 1Þ¼ arg max fPðxð0Þ; ... xði 1Þ; xðiÞjZðiÞÞg ð4:66Þ
xð0Þ;...; xði 1Þ

For each value of x(i), this formulation defines a particular partial
sequence. Such a sequence is the most likely partial sequence from time
zero and ending at a particular value x(i) at time i given the measure-
ments z(0), .. . , z(i). Since x(i) can have K different values, there are
K partial sequences for each value of i. Instead of using (4.66), we can
equivalently use

x
^ x xð0Þ; .. . ; ^ xði 1Þ¼ arg max fPðxð0Þ; .. . xði 1Þ; xðiÞ; ZðiÞÞg ð4:67Þ
xð0Þ;...; xði 1Þ
because P(X(i)jZ(i)) ¼ P(X(i), Z(i))P(Z(i)) and Z(i) is fixed.
In each recursion step the maximal probability of the path ending in
x(i) given Z(i) is transformed into the maximal probability of the path
ending in x(i þ 1) given Z(i þ 1). For that purpose, we use the following
equality:

Pðxð0Þ; .. . ; xðiÞ; xði þ 1Þ; Zði þ 1ÞÞ
¼ Pðxði þ 1Þ; zði þ 1Þjxð0Þ; ... ; xðiÞ; ZðiÞÞPðxð0Þ; ... ; xðiÞ; ZðiÞÞ
¼ P z ðzði þ 1Þjxði þ 1ÞÞP t ðxði þ 1ÞjxðiÞÞPðxð0Þ; .. . ; xðiÞ; ZðiÞÞ
ð4:68Þ
Here, the Markov condition has been used together with the assumption
that the measurements are memoryless.
The maximization of the probability proceeds as follows:

max fPðxð0Þ; ;xðiÞ;xði þ 1Þ;Zði þ 1ÞÞg
xð0Þ; ;xðiÞ
¼ max fP z ðzði þ 1Þjxði þ 1ÞÞP t ðxði þ 1ÞjxðiÞÞPðxð0Þ; ;xðiÞ;ZðiÞÞg
xð0Þ; ;xðiÞ

¼ P z ðzði þ 1Þjxði þ 1ÞÞmax P t ðxði þ 1ÞjxðiÞÞ:
xðiÞ

max fPðxð0Þ; ;xði 1Þ;xðiÞ;ZðiÞÞg
xð0Þ;...;xði 1Þ
ð4:69Þ