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28 2 Pattern Discrimination

In a multi-class PR problem complying, at least approximately, with an absolute
separation structure, design criteria derived for a two-class setting can be easily
generalized. We will have the opportunity to apply such generalization when
discussing the dimensionality ratio issue in section 4.2.4.
Painvise separation

Sometimes absolute separation is not achievable, however, the classes may be
separated into pairs as shown in Figure 2.8.
In this case one can establish c decision surfaces separating pairs of classes,
using decision functions defined as follows:

d,/ (x) > 0 Vx E wi and d,,(x) < 0 Vx E W, (dl (x) = -d,(x)) (2-9)

The decision regions are now defined in the following way:

Figure 2.8. Pairwise separation of three classes in a two-dimensional space.

Notice that pairwise separability may not lend itself to hierarchical
implementation, since decision surfaces at each level may cut through the classes.
Also, in this case, the hyperplanes separating the classes work in a coupled way,
which makes it risky to generalize for this situation design criteria established for a
two-class setting.