Page 287 - Introduction to Statistical Pattern Recognition

6 Nonparametric Density Estimation 269

function whose size is adjusted automatically, depending on the location. That
is, with k fixed throughout the entire space, 1' becomes larger in low density
areas and smaller in high density areas. The kNN density estimate may be
rewritten from (6.1) as [12-14]

(6.68)

The reason why (k-I) is used instead of k will be discussed later.

Density of coverage: Although the density function of v is not available,
the density function of the coverage (the probability mass in the local region),
u, may be obtained as follows [ 171.

Let L (X) and AL (X) be defined by
L (X) = { Y :d(Y,X) I E} and AL (X) = { Y :!<d(Y,X) I <+Ai} (6.69)
and

where d'(Y.X) = (Y-X)'A-'(Y-X). Also, let two events G and H be defined
as

G = [ (k-I) samples in L(X)) , (6.71)

H = ( 1 sample in AL(X)} . (6.72)

Then, the probability of the h-th NN in AL (X) is
Pr(G andH} =Pr{G)Pr(HIG), (6.73)
where

(6.74)

N 4.
(6.75)

Note that the coverage of AL(X) in the complementary domain of L(X) is
Au/(l-u). Substituting (6.74) and (6.75) into (6.73) and using
{ I-Au/(I-u)} + 1 as Air + 0, the probability of (6.73) becomes the product
of Au and a function of u, pl,(u). Therefore, p,,(u) should be the density