Page 149 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 131

n
.
Pr{X=Xloi} =~Pr{xj=xjIoi) (4.16)
/=I
Thus, the minus-log likelihood ratio of (4.16) becomes

I
P, { x = x 0, ]
h (X) = -In
P,{ x = x Io2 ]

This is a linear function of xi.

4.2 Linear Classifier Design

Linear classifiers are the simplest ones as far as implementation is con-
cerned, and are directly related to many known techniques such as correlations
and Euclidean distances. However, in the Bayes sense, linear classifiers are
optimum only for normal distributions with equal covariance matrices. In
some applications such as signal detection in communication systems, the
assumption of equal covariance is reasonable because the properties of the
noise do not change very much from one signal to another. However, in many
other applications of pattern recognition, the assumption of equal covariance is
not appropriate.
Many attempts have been made to design the best linear classifiers for
normal distributions with unequal covariance matrices and non-noma1 distribu-
tions. Of course, these are not optimum, but in many cases the simplicity and
robustness of the linear classifier more than compensate for the loss in perfor-
mance. In this section, we will discuss how linear classifiers are designed for
these more complicated cases.
Since it is predetermined that we use a linear classifier regardless of the
given distributions, our decision rule should be