Page 193 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 175

TABLE 4-2
ALL POSSIBLE BINARY INPUTS

.UOI1 1 1 1 1 1 1 1
s -1 1 -1 1 -1 1 -1 1
I
x2 -1 -1 1 1 -1 -1 I 1
-1 -1 -1 -1 1 1 1 1
-r 3
1 -1 -1 1 1 -1 -1 1
1 -1 1 -1 -1 1 -1 1
1 1 -1 -1 -1 -1 1 1
-1 1 1 -1 1 -1 -1 1

aE2
2
1
-- - -u(uTw - r) = 2(w - -ur) = 0, (4.164)
aw 2" 2"
1
w=-ur. (4.165)
2"

Thus, the coefficients of the linear discriminant function are given by the corre-
lation between the desired output and the input X. The above discussion is
identical to that of the general linear discriminant function. However, it should
be noted that for binary inputs UUT = NI is automatically satisfied without
transformation.
As an example of y(X), let us use

The term y(X) would be positive for P ,p I (X) < P *p2(X) or q I (X) < q 2(X),
and be negative otherwise. Also, the absolute value of y(X) depends on p I (X)
and p2(X). When n is large but the number of observed samples N is far less
than 2", the correlation of (4.165) can be computed only by N multiplications
and additions, instead of 2" [7].
Table 4-2 suggests that we can extend our vector X = [xI . . . xnIT to