Page 295 - Introduction to Statistical Pattern Recognition
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6 Nonparametric Density Estimation 277
Distance to Neighbors
Distance to WN: From (6.25), the distance to the kth nearest neighbor
may be expressed in terms of the corresponding volume as
The distance is a random variable due to v. Using the first order approxima-
tion of u =pv and knowing the density function of u as (6.76), the mth order
moments of dfNN(X) can be obtained as [ 181
(6.108)
where
(__) +2
n
p / N
2 T(k+m/n) T(N+l)
V= (6.109)
xn”2 , Z , m12n r(k) r(N+l+m/n) ’
Note that A = C is used as the optimal matrix. The overall average of this dis-
tance in the entire space is
.
EXE(d;bN(X)J Z vEX{p-”’”(X)~ (6.110)
Ex (p-”’’”(X) 1 for normal and uniform distributions can be expressed as
(a) Normal:
I I (6.111)
Ex {p-””” (X) = (2~)”’~ n112f1
(b) Uniform [see (6.22)]:
where both the normal and uniform distributions have zero expected vector and
covariance matrix Z. Substituting (6.109), (6.1 1 I), and (6.1 12) into (6.1 IO),
(a) Normal:
