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2.1 Decision Regions and Functions 23

where llwll represents the vector w length.
Notice also that Id(z)JIJlwJJ is precisely the distance of any point z to the
hyperplane.

2.1 .I Generalized Decision Functions

In pattern classification we are not confined to using linear decision functions. As
long as the classes do not overlap one can always find a generalized decision
fhnction defincd in 3" that separate5 a class w, from a total of c classes, so that the
following decision rule applies:

For some generalized decision functions we will establish a certain threshold A
for class discrimination:

For instance, in a two-class one-dimensional classification problem with a
quadratic decision function d(x) = x2, one would design the classifier by selecting
an adequate threshold A so that the following decision rule would apply:

If d(x)=x22~ thenx~u~ else XEW~. (2-3b)

In this decision rule we chose to assign the equality case to class w,. Figure 2.3a
shows a two-class discrimination using a quadratic decision function with a
threshold A=49, which will discriminate between the class (0, = {x; XE [-7.71)
and 01, = {x; XE 1-7,7[}.
It is important to note that as far as class discrimination is concerned, any
functional composition of d(x) by a monotonic function will obviously separate the
classes in exactly the same way. For the quadratic classifier (2-3b) we may, for
instancc, usc a monotonic logarithmic composition:
If In(d(x)) 2 In(A) then XE a, else XE a,. (2-3c)