Page 216 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

LINEAR FEATURE EXTRACTION 205

T
matrices of the feature vector, i.e. Wm and WC k W , respectively. For
k
the sake of brevity, let m be the difference between expectations of z:
m ¼ m m 2 ð6:34Þ
1

T
Then, substitution of m, Wm and WC k W in (6.19) gives the Bhatta-
k
charyya distance of the feature vector:
1 T 1
T
J BHAT ðWzÞ¼ ðWmÞ WC 1 W þ WC 2 W T Wm
4
2 3
1 T T ð6:35Þ
6 jWC 1 W þ WC 2 W j 7
2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5
þ ln4
T
T
2 D jWC 1 W jjWC 2 W j
The first term corresponds to the discriminatory information of the
expectation vectors; the second term to the discriminatory information
of covariance matrices.
Equation (6.35) is in such a form that an analytic solution of (6.32) is
not feasible. However, if one of the two terms in (6.35) is dominant, a
solution close to the optimal one is attainable. We consider the two
extreme situations first.

Equal covariance matrices

In the case where the conditional covariance matrices are equal, i.e.
C ¼ C 1 ¼ C 2 , we have already seen that classification based on the
Mahalanobis distance (see (2.41)) is optimal. In fact, this classification
uses a 1 N dimensional feature extraction matrix given by:

T
W ¼ m C 1 ð6:36Þ
To prove that this equation also maximizes the Bhattacharyya distance is
left as an exercise for the reader.

Equal expectation vectors
If the expectation vectors are equal, m ¼ 0, the first term in (6.35)
vanishes and the second term simplifies to:
2 3
1 T T
J BHAT ðWzÞ¼ ln4 jWC 1 W þ WC 2 W j 7 ð6:37Þ
6
2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5
T
T
2 D jWC 1 W jjWC 2 W j