Page 234 - Introduction to Statistical Pattern Recognition

216 Introduction to Statistical Pattern Recognition

P:Var,(GI } + PiVar,{& 1 as seen in (5.49). On the other hand, when test dis-
* *
tributions are fixed and the classifier varies, and are strongly correlated
* *
with a correlation coefficient close to -1. That is, when increases,
decreases and vice versa. Thus, when P I = P2, Vard( 6 1 = Ed( AE~
}
+ A&; } + 2(0.5)2 Ed { AEI AE~ g(0.5)2 [Ed ( AEy I+ Ed { )' }+
}
2Ed{A~i (-AE~) 11 = 0. The covariance of 2, and G2 cancels the individual
* *
variances of and E~.
Effect of Independent Design and Test Samples

When both design and test sample sizes are finite, the error is expressed
as

(5.98)

where

(5.99)
I*
That is, the randomness comes from h due to the finite design samples as well
as from the test samples XI!').
*
The expected value and variance of E can be computed as follows:
- * * 1
&=E(&} =EtEd{&} =-+PIEl -P2Z2 (5.100)
2
where

(5.101)

Substituting (5.101) into (5.100),