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8 8 4 Statistical Classification

"hyperplane" leans along the regression direction of the features (see Figure 4.3 for
comparison).
Let us suppose that the previously mentioned cork stopper with 65 defects, and
assigned to class wl based on this feature only, had a total perimeter of the defects
of 520 pixels. To which class will it be assigned now? As g1([65 5211)=5.78 is
smaller than g2([60 48Iv)=6.84, it is assigned to class q. This cork stopper has a
total perimeter of the defects that is too big to be assigned to class 8,.
Notice that if the distributions of the feature vectors in the classes correspond to
different hyperellipsoidal shapes, they will be characterized by unequal covariance
matrices. The distance formula (4-5) will then be influenced by these different
shapes in such a way that we will obtain quadratic decision boundaries as shown in
Figure 2.12b. Table 4.1 summarizes the different types of minimum distance
classifiers, depending on the covariance matrix.

Table 4.1. Summary of minimum distance classifier types.
Equiprobability
Covariance Classifier Discriminants
surfaces
Hyperplanes orthogonal to the segment
ci=s21 Linear, Euclidian Hypersphcres
linking the means
Hyperplanes leaning along the
ci=C Linear. Mahalanohis Hyperellipsoids
regression
Ci Quadratic, Mahalanobis Hyperellipsoids Quadratic surfaces

4.1.4 Fisher's Linear Discriminant

In the previous chapter the problem of dimensionality reduction was addressed in
an unsupervised context. In a supervised context, we are able to use the
classification information of the training set in order to produce an optimised
mapping into a lower dimensional space, easing the classification task and
obtaining further insight into the class separability. The Fisher linear discriminant
provides the necessary tool for this mapping.
Consider two classes with sample means ml and m2 and an overall sample mean
m. We can measure the class separability in a way similar to the well-known
Anova statistical test by evaluating the volume of the pooled covariance matrix of
the classes relative to the separation of their means. To get a more concrete idea of
this, let us consider:

2
S, = x x (x - m, )(x - mk 1'. the within-class scatter matrix, and (4-9a)
k=lx€ck