Page 218 - Introduction to Statistical Pattern Recognition

200 Introduction to Statistical Pattern Recognition

From (5.40) and (5.45)-(5.47),

a 1
E,{&} = - + PIEl - p2a2
2
1
1
1
= - + P*(E, - 2) - P2(, - E21 = E, (5.48)
2

p: 1 1 p: 1 1
-
= -[- - (E] - -)2] + -[- - (2 E2YI
NI 4 2 N2 4
(5.49)

A
That is, E is an unbiased and consistent estimate, no matter what h (X) is used.

Error counting approach: When the error counting procedure is used,
the effect of test samples can be analyzed in a more direct way. In order to
estimate E~, Nj samples are drawn from oj and tested by a given classifier. Let
ij be the number of misclassified samples. Then, the random variables i, and
A
q are independent, and each is binomially distributed as
A A 2 A
Pr{r, = 21,q = 22} = rIPr(2; =Ti)
i=l

(5.50)

I
The 0;-error, E~, is estimated by qlNj and, subsequently, the total probability
of error is estimated by

(5.51)

The expected value and variance of the binomial distribution are known, and
thus