Page 38 -

24 2 Pattern Discrimination

Figure 2.3. (a) Quadratic decision function, d(x) = x2; (b) Logarithmic decision
function, g(x) = In(d(x)).

Figure 2.3b illustrates this logarithmic decision function for the quadratic
classifier example using the new threshold value ln(49)=3.89. The resulting class
discrimination is exactly the same as before the logarithmic transformation. The
benefit that can be derived from the use of a monotonic transformation will
become clear in later chapters.
It is sometimes convenient to express a generalized decision function as a
functional linear combination:

Figure 2.4. A two-class discrimination problem in the original feature space (a)
and in a transformed one-dimensional feature space (b).

In this way, an arbitrarily complex decision function is expressed linearly in the
space of the feature vectors y. Imagine, for instance, that we had two classes with