Page 44 -

30 2 Pattern Discrimination

(2-1 le)

Parameter p controls the weight placed on any dimension dissimilarity.
Parameter r controls the distance growth of patterns that are further apart. The
special case r=p is called the Minkowsky norm. For r=p=2, one obtains the
Euclidian norm; for r=p=l, the city-block norm; for r=p + m, the Chebychev
norm.

Figure 2.9. Equidistant "surfaces" for Euclidian metric. The straight line is the set
of equidistant points from the means.

Notice that strictly speaking a similarity measure s(x, y) should respect the
inequality s(x,y) I so ~nstead of (2-10a). A distance is, therefore, a dissimilarity
measure.
For the Euclidian metric the equidistant surfaces are hyperspheres. Figure 2.9
illustrates this situation for d=2 and c=2 classes represented by the class means
[l 21' and [1.5 11'. The circles in this case represent distances of 1, 1.5 and 2.
Note that by using the Euclidian (or squared Euclidian) metric, the set of
equidistant points from the class means, i.e., the decision surface, is just one
hyperplane perpendicular to the straight-line segment between the class means and
passing through the middle point of this line segment.
With the city-block or Chebychev metric this excellent property is not always
verified as illustrated in Figure 2.10.