Page 252 - Introduction to Statistical Pattern Recognition

234 Introduction to Statistical Pattern Recognition

the L and R errors is twice the bias between the L and true errors.
In order to confirm (5.156), the following experiment was conducted.

Experiment 8: Bias between the L and R error
Data: I-I (Normal, MTM = 2.562, E = 10%)
Dimensionality: n = 4, 8, 16, 32, 64
Classifier: Quadratic classifier of (5.54)
Sample size: N = N2 = kn, k = 3, 5, 10, 20, 40
I
No. of trials: z = 10
Results: Table 5-10, Fig. 5-3 [6]
The first line of Table 5-10 indicates the theoretical biases from (5.152) and
(5.156), and the second and third lines are the average and standard deviation
of the 10 trial experiment. Despite a series of approximations, the first and
second lines are close except for small k's and large n's, confirming the vali-
dity of our discussion.
An important fact is that, from (5.152) and (5.156), E(&] is roughly
proportional to n2/N for large n. A simpler explanation for this fact can be
obtained by examining (5.153) more closely. Assuming (5.155) and carrying
through the integration of (5.153) with respect to O,

(5.157)

It is known that d:(X) is chi-square distributed with an expected value of n and
standard deviation of d%, if X is normally distributed [see (3.59)-(3.61)].
This means that, when n is large, d?(X) is compactly distributed around the
expected value n (Le. n >> 6.) Therefore, d;(X) on the classification boun-
dary should be close to n2. Thus, should be roughly proportional to n2.
The analysis of the variance (5.154) is more complex. Though the order
of magnitude may not be immediately clear from (5.154), the experimental
results, presented in Fig. 5-4 and the third line of Table 5-10, show that the
standard deviation is roughly proportional to 1/N. The intuitive explanation
should be the same as that presented in Section 5.2.