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4.1 Linear Discriminants 83

This last equation, which is linear in x, represents a hyperplane perpendicular to
(m, -m2)' and passes through the point 0.5(ml +m2) halfway between the
means. It is a linear discriminant separating w, from ~ 2 .

Figure 4.5. Decision region for w , (hatched area) showing linear discriminants
relative to three other classes.

For c classes the minimum distance discriminant is piecewise linear, composed
of segments of hyperplanes with pairwise separation as discussed in 2.1.2, and
illustrated in Figure 4.5 with an example of a decision region for class 12, in a
situation of c=4.
Minimizing the distance dk2(x), as in (4-3a), is equivalent to maximizing the
following decision function:

with wk = m,; w ,.,, = -0.5 (1 mk (I2

%(XI t+ Select Class
m2
.
X- . the
maximum
m

Figure 4.6. Maximum decision function classifier for an input feature vector x.