Page 244 - Introduction to Statistical Pattern Recognition

226 Introduction to Statistical Pattern Recognition

Nj-1 * N, -2
I iik = (-y I E; [ 1 - -d; (Xi!’)] (5.127)
I
I
Nj-2 (N, - 1 )2
Or, taking the logarithm,
(5.128)

Let &(Xt)) and iL(Xf)) be the R and L discriminant functions with fi, and i,
replacing M, and C, for &(Xf)), and klL and ilk replacing MI and C, for
&(Xt’), respectively. Then, substituting (5.126) and (5.128) into (5.54),

+g(N ,$(Xi’’)) for ol
hL(Xf’) - iR(Xf)) = (5.129)
for
-g(~~,iS(xi~))) o2 ,

where

When the R method is used to count the number of misclassified samples,
-2 A
d, (Xf’) and hR(Xf)) must be computed for k = 1,. . . ,N, and i = 1,2. There-
fore, the additional computation of the scalar function of (5.130) for each k is a
negligible load for a computer in comparison with the computation of &(Xf))
for each k, which includes vector-matrix operations in a high-dimensional
space. Thus, the computation time of both the R and L methods becomes
-
almost equivalent to the computation time of the R method alone. Remember
that, in the R method, we are required to design only one classifier. Since hL
can be obtained from hR by a trivial perturbation equation, we do not need to
design the classifier N times in the L method.