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2.1 Decision Regions and Functions 25

circular limits as shown in Figure 2.4a. A quadratic decision function capable of
separating the classes is:

d(x)= (x, -1)' + (x, - 1)2 +0.25 . (2-5)

Instead of working with a quadratic decision function in the original two-
dimensional feature space, we may decide to work in a transformed one-
dimensional feature space:

y* = [I ,y]' with = f(x)= (x, -I)? + (x2 - 1)2 . (2-5b)

In this one-dimensional space we rewrite the decision function simply as a
linear decision function:

Figure 2.4b illustrates the class discrimination problem in this transformed one-
dimensional feature space. Note that if there are small scaling differences in the
original features xl and XI, as well as deviations from the class centres, it would be,
in principle, easier to perform the discrimination in the y space than in the x space.
A particular case of interest is the polynomial expression of a decision function
d(x). For instance, the decision function (2-5a) can be expressed as a degree 2
polynomial in xi and x2. Figure 2.5 illustrates an example of 2-dimensional classes
separated by a decision boundary obtained with a polynomial decision function of
degree four:

Figure 2.5. Decision regions and boundary for a degree 4 polynomial decision
function.