Page 241 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 223

ical form of (5.117) for the classifier, designing means to estimate the neces-
sary parameters - in this case two expected vectors by using the sample means,
Mi, of (5.8). Using the error counting procedure, each sample is tested by

n 01
(XfG J(Xf’-fi I) - (Xf)-k2)7jXf’-M2) >< I (5.1 18)
02

(i = 1,2 : k = 1,. . .,N;) .
If Xi!’ does not satisfy <, the sample is labeled as an error. Likewise, Xp),
which does not satisfy >, is labeled as an error. The R error is the number of
errors divided by the total number of samples.
On the other hand, in the L method, Xf) must be excluded from the
design set when Xf’ is tested. The mean estimate without Xf’, Mik, may be
computed as

(5.120)

Therefore, testing an oI -sample, Xi’), can be carried out as follows.
2
(XZ“-h Ik)T(Xp-h 1,) - (XiI’-M2) * 7- (X, (1) - fi )

(5.121)
,. ,.
Note that, when Xi’) is tested, only MI is changed and A42 is not changed.
Likewise, when an a2-sample, Xi2’, is tested,
(Xph ,7-(xp’-i I) - (XP’-~2,)T(Xi2’-fi2k)

><
12 * 01
= (Xp-M i)T(Xp-M )-(-)2(x~2)-h~)7-(x~2’-M*) t . (5.122)
N2-1 02
Equations (5.121) and (5.122) reveal that the modification from the R method
to the L method is simply to multiply a scalar [N,l(Ni-1)]2 with a distance.
Since the distance computation in a high-dimensional space needs much more