Note that r is invariant to translation and invariant to scale. This
suggests an important generalization of a minimum-Euclidean-distance classifier.
Let x(i) be the value for Feature i, let m(i,j) be the mean value of Feature
i for Class j, and let s(i,j) be the standard deviation of Feature i for
Class j. In measuring the distance between the feature vector x
and the mean vector mj for Class j, suppose
that we use the standardized distance
This distance has the important property that it is scale invariant.
That is, if we measure distance in this way, the units we use for the various
features will have no effect on the resulting distances, and thus no effect
on the final classification. The Mahalanobis distance
is a generalization of this standardized distance.
Back to Mean On to Transformations Up to Mahalanobis