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84 4 Statistical Classification

Figure 4.6 shows the general structure of a maximum decision function
classifier.
Notice that the gk(x) are linear decision functions (compare with 2-2) with the
weight vector equal to the mean vector, and the bias term, wk,~, dependent on the
mean vector length.
We can get further insight into this linear discriminant system by referring to the
previous c=2 situation. Consider that the coordinate axes underwent a translation
so that we are now dealing with the new feature vectors y = x- 0.5(ml + m2). The
linear discriminant functions are now expressed as:

with m, and m2 evaluated in the new system of coordinates, we obviously have
llrn,ll = Ilm211, therefore the discriminant functions can be expressed simply as:

Since mily is simply the vector correlation (also known as dot product) between
m,' and y, the Euclidian linear discriminant is also known as maximum correlation
classijier. Notice that the vector correlation yields a value dependent on the angle
between the vectors. It increases with decreasing angle, reaching a maximum at a
zero angle. This allows an alternative interpretation (vectorial projection) of the
similarity measure. The technique we have just described for assessing class
membership of an unknown pattern x is one of the earliest known in pattern
recognition, called template matching. Each new pattern was matched against a
stored template (prototype), using a correlation measure.

Figure 4.7. Classification of a feature vector x by the maximum correlation
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approach: OA > OB - x E w, .