Page 41 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

30 DETECTION AND CLASSIFICATION

Minimum distance classification
A further simplification is possible when the measurement vector equals
the class-dependent vector m corrupted by class-independent white
k
2
noise with covariance matrix C ¼ I.
2
( )
kz m k
k
^ ! !ðzÞ¼ ! i with i ¼ argmin 2ln Pð! k Þþ 2 ð2:27Þ
k¼1;...;K
The quantity k(z m )k is the normal (Euclidean) distance between z
k
and m . The classifier corresponding to (2.27) decides for the class whose
k
expectation vector is nearest to the observed measurement vector (with a
2
correction factor 2 log P(! k ) to account for the prior knowledge).
Hence, the name minimum distance classifier. As with the minimum
Mahalanobis distance classifier, the decision boundaries between com-
partments are linear (hyper)planes. The plane separating the compart-
ments of two classes ! i and ! j is given by:

1
2
T
2
2
log Pð! i Þ þ ðkm k km k Þþ z ðm m Þ¼ 0 ð2:28Þ
j
i
j
i
Pð! j Þ 2
The solution of this equation is a plane perpendicular to the line segment
connecting m and m . The location of the hyperplane depends on the
j
i
2
factor log (P(! i )/P(! j )). If P(! i ) ¼ P(! j ), the hyperplane is the perpen-
dicular bisector of the line segment (see Figure 2.8).
Figure 2.9 gives an example of the decision function of the minimum
distance classification. PRTools code to generate these figures is given in
Listing 2.4.
z 1
µ decision
j
boundary

µ i

z 0

Figure 2.8 Decision boundary of a minimum distance classifier