Page 288 - Introduction to Statistical Pattern Recognition

270 Introduction to Statistical Pattern Recognition

function of u, where u is the coverage of L(X) whose boundary is determined
by the kth NN.

(6.76)

That is, p,(u) is a Beta distribution Be(k,N-k+l). Also, note that the distribu-
tion of u is independent of the underlying distribution, p (X).
More generally, the joint density function of uI , . . . ,uk may be obtained
1171

(6.77)

where ui is the coverage of Li(X), the region extended until the ith NN is
found. Note that the joint density depends on uk only. The marginal density
of uk can be obtained by integrating (6.77) with respect to u I, . . . ,uk-l as
N!
.
6"' . . . c2p (u 1, . . . ,uk)duI . . . duk-l = ut-' ( I-u~)~-~ (6.78)
!
(k - 1 ) ! (N 4)
Equation (6.78) is the same as (6.76).
The relationship between u and v may be obtained by integrating (6.10)
over L(X) with respect to Y. That is,
1
u(X) Zp(X)v(X) + 2fr(V2p(X)[ (Y-X)(Y-X)'dY}
(X )
1
=p(x)v(x)[1 + -~(x)~.~(x)J , (6.79)
2
where a is given in (6.13). Note that j(Y-X)(Y-X)TdY = vr2A from (6.27).
The term [l+ar2/2] of (6.79) appeared in (6.18) in the Parzen case. Again,
u =pv gives the first order approximation, and (6.79) is the second order
approximation of u in terms of v.

Moments of p(X): When the first order approximation of u = pv is used,
from (6.68) and (6.76)

(6.80)

where the following formula is used