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4 Parametric Classifiers 129

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Fig. 4-4 A classifier by Euclidean distances.

4-6(a) are very much separable by the Bayes classifier of (4.2), the bisector
classifier or simple correlation gives a poor classification. The narrow distribu-
tions of Fig. 4-6(a) occur when x1 and x2 are highly correlated. Particularly,
when xI,x2,. . . are the time-sampled values of waveforms, adjacent xi's are
usually highly correlated and show this type of distribution. The whitening
transformation changes these two distributions to the circular ones of Fig. 4-
6(b) such that the Bayes classifier becomes the bisector.

Other Bayes Linear Classifiers

The Bayes classifier becomes linear for some other distributions such as
independent exponential distributions and the distributions of independent
binary variables. We will discuss these cases in this section.

Independent exponential distributions: When the xi's are mutually
independent and exponentially distributed for both a, and a2, the Bayes
classifier becomes linear as shown in (3.18).

Independent binary variables: When the x,'s are binary, either +1 or
-1, the density function of xi for a, is expressed by