Page 163 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 145

SSI + (I-s)Sz = [SCI + (I-s)&] + (l-S)MMT . (4.61)

Using (2. I60),

--I --I ( 1-s)2-'MMTt-'
s =c - (4.62)
1 + (1-s)M'z-'M '
where = [sS ,+( 1-s)Sz] and = [sXI+( 1-s)C2]. Multiplying M from the
right side,

-
- 1 5'M. (4.63)
1 + ( l-s)MTz-lM
That is, s-'M and Z-IM are the same vector except for their lengths.

Minimum Mean-Square Error

The mean-square error is a popular criterion in optimization problems.
Therefore, in this section, we will study how the concept of the mean-square
error may be applied to linear classifier design.
Let y(X) be the desired ourpur of the classifier which we woufd like to
design. The possible functional forms for y(X) will be presented later. Then,
the mean-square error between the actual and desired outputs is

We minimize this criterion with respect to V and Y,,. Since the third term of
(4.64) is not a function of V and tio, the minimization is carried out for the
summation of the first and second terms only.
Two different functional forms of y(X) are presented here as follows:

(1) y(X) = -1 for- X E 0, and +I for- X E 02: Since h (X) is supposed
to be either negative or positive, depending on X E wI or X E w2, -1 and +1
for y(X) are a reasonable choice. Then