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4.1 Linear Discriminants 85

4.1.3 Mahalanobis Linear Discriminants

We know already from chapter 2 that the Mahalanobis metric is a generalization of
the Euclidian metric suitable for dealing with unequal variances and correlated
features. Let us assume that all classes have an identical covariance matrix C,
reflecting a similar hyperellipsoidal shape of the corresponding feature vector
distributions. The generalization of (4-3) is then written as:

or, d;(x) = X'C-~X - mkfc-'X - X'C-'mk + mkfc-lmk

Grouping, as we have done before, the terms dependent on mk, we obtain:

The decision functions are:

with wk = C1mk ; wkIO = -0.5mkfC-'mk . (4-5c)

We again obtain linear discriminant functions in the form of hyperplanes
passing through the middle point of the line segment linking the means. The only
difference from the results of the previous section is that the hyperplanes
separating class u, from class q are now orthogonal to the vector Ci(m,-mj).
In the particular case of C = s21 (uncorrelated features with the same variance
for all classes), the Mahalanobis classifier is identical to the Euclidian classifier (as
it should be).
In practice, it is impossible to guarantee that all class covariance matrices are
equal. Fortunately, the decision surfaces are usually not very sensitive to mild
deviations from this condition; therefore, in normal practice, one uses a pooled
covariance matrix computed as an average of the individual covariance matrices.
This is also done in all statistical software applications. We now exemplify the
previous results for the cork stoppers problem with two classes.

One feature, N

Given the similarity of both distributions, already pointed out in 4.1 .l, the
Mahalanobis classifier produces the same classification results as the Euclidian
classifier.
The decision function coefficients, as computed by Statistics, are shown in
Figure 4.8.