Page 247 - Introduction to Statistical Pattern Recognition
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5 Parameter Estimation 229
TABLE 5-9
THE BOUNDS OF THE BAYES ERROR BY THE L AND R METHODS
No. Resubstitution method Leave-one-out method Bias
of samples between
per class Standard Standard two
N I = N2 Mean (%) deviation (%) Mean (%) deviation (%) means (%)
12 0.21 1.3 18.54 7.6 18.33
so 1.22 0.9 2.97 1.7 I .75
100 I .44 0.8 2.1s 1 .o 0.71
200 1.56 0.7 2.00 0.7 0.44
400 1.83 0.5 1.97 0.5 0.14
Effect of removing one sample: Generalizing the above discussion, let
us study how the removal of one sample affects the estimate off which is a
function of M and Z. Let kR and iR the sample mean and sample covari-
be
A A
ance computed from all available samples, and let ML and ZL be the
corresponding estimates without a sample Y. From (5.119) and (5.124), we
..
..
IIn
may express ML and C, in terms of MR, CR and Y as
where N >> 1 is assumed. The terms associated with 1/N are small deviation
A
terms. Therefore, we may approximate MR and CR by the true parameters A4
and C whenever they appear in [.]lN, resulting in
