Page 231 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 213

values of n and MTM, we may generally conclude that vg is larger than v, for n
>> 1. This implies that many more samples are needed to properly design a
quadratic classifier than a linear classifier. It is believed in general that the
linear classifier is more robust (less sensitive to parameter estimation errors)
than the quadratic classifier, particularly in high-dimensional spaces. The
above results support this belief both theoretically and experimentally.
Also note that for large n, v lY! is proportional to Ilk. This indicates
that, as far as the design of a linear classifier is concerned, a fixed multiple
could be used to determine the sample size from the dimensionality. However,
(5.92) indicates that the value of the multiple depends on MTM, which meas-
ures the separability between two distributions with a common covariance
matrix 1. In particular, the less the separability between the two distributions,
the greater k must be for a fixed bias.

A
Variance: The variance of E may be computed from (5.59) and (5.64) as

1
- (E - -)2 . (5.93)
2

Applying the same approximation as (5.60) and keeping up to the second order
terms of Ah,

where

jm,
2
AC,(X) = Ah(X) + -Ah2(X) . (5.95)
Thus, (5.93) can be expanded to