Page 297 - Introduction to Statistical Pattern Recognition

6 Nonparametric Density Estimation 279

TABLE 6-3

THE AVERAGE DISTANCE TO THE kth NEAREST NEIGHBOR

Normal Uniform
n=5 1.147 0.966
10 2.319 2.078
20 3.886 3.626
~
N = 50 2.484 2.226
100 2.319 2.078
200 2.164 1.940
400 2.020 1.810
k=l 2.319 2.078
2 2.55 1 2.286
3 2.679 2.400

effects of N and p (X) on v~NN. Since we estimate a density function by (6.68),
if k or N is changed, v must be changed accordingly. Because of the nth
power, a reasonable change of the volume is translated to a very small change
of the distance for a large n.
In order to show the effect of the location on E(dkNN(X)}, the following
experiment was conducted.

Experiment 1: NN distance
Data: N(O,I), n = 10
kNN: k = 1
Sample size: N = 100
No. of trials: 2 = 10
Results: Fig. 6-2 [ 181

Figure 6-2 shows the averaged NN distances and the standard deviations of 10
trails vs. the distance from the center, !. Also, theoretical curves computed
from (6.108) are plotted by dotted lines. The theoretical and experimental
curves match closely until L = 4, where most samples are located. Also, note
that the standard deviation is very small. This is predicted theoretically,