Page 194 - Introduction to Statistical Pattern Recognition

176 Introduction to Statistical Pattern Recognition

Y = [l XI . . . xn (XIX2) . . . (XIX2.. .xn)]'. (4.167)
2"
Then, the sample matrix for this extended vector becomes a square matrix as

Uy = [Yo Y1. . .Y~L~]}Y (4.168)
.
2"
The row vectors of Uy are also orthonormal, such that
uyu; = 2"1 . (4.169)

A linear discriminant function for Y is
2"- 1 n
+
wjyj = wo + x wjxj + C ~W!X;X~ . . . + ~2,#-1 I . . . X, . (4.170)
x
j=O j=l i i
In accordance with the reasoning applied to derive (4.165), we can determine
W of (4.170) by
1
w = -uyr (4.171)
2"
The following should be noted here:
(1) Any desired output is expressed by W'Y without error.
-2
(2) Since yt's are mutually orthonormal, E due to the elimination of
is
wlyl from W~Y wf.
(3) The E2 determined by the linear discriminant function of V'X + Vo
is

-2 2"- I
E = xw;. (4.172)
j=n+l

Computer Projects

1. Repeat Example 4, and obtain Fig. 4-8.

2. Repeat Experiment 1 for Ni = 50, 100, 200,400 and plot the error vs. s.