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2.2 I'cature Space Metrics 31

Figure 2.10. Equidistant "surfaces" for city-block (a) and Chebychev (b) metrics.

In both cases the decision surfaces are stepwise linear. In the city-block case the
surfaces are parallel to the coordinate axis with smaller distance of the class means
(horizontal axis in Figure 2.10a); in the Chebychev case the surfaces bisect the
axes. Therefore, as we will see next chapter, the city-block metric is well suited for
separating flattened clusters aligned along the axes; the Chebychev metric is
adequate when the clusters are aligned along the quadrant bisectors.
The choice of one type of metric must take into account the particular shape of
the pattern clusters around the class means. This common sense rule will be further
explored in the following chapters. Let us present now a more sophisticated type of
metric, which can be finely adjusted to many cluster shapes. For that purpose
consider the following linear transformation in the feature space:

y = Ax, with symmetric matrix A. (2-12)

Let us assume a cluster of 2-dimensional patterns around the class mean [I I]'
with a circular structure, i.e., the patterns whose corresponding vectors fall on the
same circle are equally similar. Figure 2.1 1 shows one of these circles and
illustrates the influence of a linear transformation on the circular equidistant
surface, using the following matrix A: