Page 324 - Introduction to Statistical Pattern Recognition

306 Introduction to Statistical Pattern Recognition

where
qi(X) = Pr ( XEW, I X ) : a posteriori probability . (7.10)

The 2nd line of (7.9) is obtained because the two events in the first line are
mutually exclusive. The 3rd line is obtained because X and XNN are mutually
independent. When an infinite number of samples is available, XN~ is located
so close to X that qi(XNN) can be replaced by qi(X). Thus, the asymptotic con-
ditional risk of the NN method is
I.; (X) = 2q 1 (X)q2(X) = 2W) (7.1 1)

(7.12)

2NN: When k is even, kl =k2 may occur and a decision cannot be
made. In this case, we may set a rule that X be rejected and not counted as an
error. In the simplest case of k = 2, the rejection occurs when XNN&wl and
X2NN~~2, XNNeo2 and XZNN&al. On the other hand, X is misclassified,
or
when X&ol but XNN,XwN&o2, or XEW~ but XNN,XZNN&wI. Therefore, the con-
ditional risk is

For the asymptotic case with qi(X) = qi(XNN) qi(XZNN),
=

where q I (X) + q2(X) = 1 is used.

RNN: Extending the above discussion to larger values of k, the asymp-
totic conditional risks for odd k and even k are