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34 2 Pattern Discrimination

to appreciating visually. As can be seen in Figure 2.13 there is a certain amount of
correlation between the features. One might wonder what would happen if the
features were not measured in approximately the same ranges, if for instance we
used the original PRT feature. We can do this by increasing the PRTlO scale ten
times as shown in Figure 2.14, where we also changed the axes orientation in order
to reasonably fit the plot in the book sheet.

-
0 20 40 80 80 100 120 140
PRTlO

Figure 2.14. Scatter plot for two classes of cork stoppers with PRTlO scale
increased ten times.

It is now evident that the measurement unit has a profound influence on the
Euclidian distance measures, and in the visual clustering of the patterns as well.
Namely, in Figure 2.14, the contribution of N to class discrimination is, in the
Euclidian distance sense, negligible.
The usual form of equalizing the features contributions consists of performing
some scaling operation. A well-known scaling method consists of subtracting the
mean and dividing by the standard deviation:

where mi is the sample mean and s, is the sample standard deviation of feature x,.
Using this common scaling method (yielding features with zero mean and unit
variance), the squared Euclidian distance of the scaled feature vector y relative to
the origin is expressed as:

Thus, the original squared Euclidian distance (x, - m, )2 has been scaled by
11 s: , shrinking large variance features with s, > 1 and stretching low variance
features with s, < I, therefore balancing the feature contribution to the squared
Euclidian distance. We can obtain some insight into this scaling operation by
imagining what it would do to an elliptical cluster with axes aligned with the
coordinate axes. The simple scaling operation of (2-14a) would transform
equidistant ellipses into equidistant circles. As a matter of fact, any linear