Page 246 - Introduction to Statistical Pattern Recognition

228 Introduction to Statistical Pattern Recognition

1
n-1
Ni-1
-_ 1 + - ln- Ni + - >O (5.134)
1
-
In-
2 Ni(N;-l) 2 Ni-1 2 Ni-2
for Ni > 2. The inequality of (5.134) holds since the numerators of the second
and third terms are larger than the corresponding denominators.
Comments: Since g of (5.130) is always positive, hL > hR for 01-
A , .
samples and hL < hR for y-samples. Since h > 0 is the condition for ol-
samples to be misclassified and h < 0 is the condition for 02-samples to be
misclassified, the L method always gives a larger error than the R method.
This is true for any test distributions, and is not necessarily limited to normal
distributions. Note that this conclusion is a stronger statement than the ine-
quality of (5.116), because the inequality of (5.1 16) holds only for the expecta-
tion of errors, while the above statement is for individual samples of individual
tests.
Since we have the exact perturbation equation of (5.130), the use of this
equation is recommended to conduct the R and L methods. However, for
further theoretical analysis, (5.130) is a little too complex. An approximation
-2
of (5.130) may be obtained by assuming N; >> dj and Ni >> 1. When X is
distributed normally, it is know that d2(X) has the chi-square distribution with
an expected value of n and standard deviation of 6, where
d2(X) = (X-M)TC-’(X-M) [see (3.59)-(3.61)]. Therefore, if N >> n, the
approximation based on N >> d2 is justified. Also, In( 1+6) Z6 for a small 6 is
used to approximate the second and third terms of (5.130). The resulting
approximation is
-2 1 -4
g (Ni,di (Xf))) E -[d; (Xf’)+n] . (5.135)
2Ni
In order to confirm that the L and R methods give the upper and lower
bounds of the Bayes error, the following experiment was conducted.

Experiment 7: Error of the quadratic classifier, L and R
Data: I-A (Normal, n = 8, E = 1.9%)
Classifier: Quadratic classifier of (5.54)
Sample size: N1 = N2 = 12, 50, 100, 200, 400
No. of trials: z = 40
Results: Table 5-9 [ 141
As expected, the L and R methods bound the Bayes error.