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50 2 Pattern Discrimination

visiually identifiable clusters, with slope minus one and at a minimum distance from
the origin.

A classifier uses the following linear discriminant in a 3-dimensional space:

a) Compute the distance of the discriminant from the origin.
b) Give an example of a pattern whose classification is borderline (d(x)=O).
c) Compute the distance of the feature vector [I -1 01' from the discriminant

A pattern classification problem has one feature x and two classes of points: wl=(-2, 1,
1.5); @=[-1. -0.5, 0).
a) Determine a linear discriminant of the two classes in a two-dimensional space,
using features y, = x and y2= x2.
b) Determine the quadratic classifier in the original feature space that corresponds to
the previous linear classifier.
Consider a two-class one-dimensional problem with one feature x and classes wl=(-1,
I] and @=lo, 2).
a) Show that the classes are linearly separable in the two-dimensional feature space
with feature vectors y=[~2 x3]' and write a linear decision rule.
b) Using the previous linear decision rule write the corresponding rule in the original
one-dimensional space.

Consider the equidistant surfaces relative to two prototype patterns, using the city-
block and Chebychev metrics in a two-dimensional space, as shown in Figure 2.10. In
which case do the decision functions correspond to simple straight lines?
Draw the scatter plot of the +Cross data (Cltrsfer.xls) and consider it composed of two
classes of points corresponding to the cross arms, with the same prototype, the cross
center. Which metric must be used in order for a decision function, based on the
distance to the prototypcs, to achieve class separation'?
Compute the linear transformation y = Ax in a two-dimensional space using the sets of
points P=((0,0), (05.01, (0,1), (-1.5.0), (0,-2)) and Q=((1,1), (O,I), (1,0), (2,1), (1,211,
observing:
- Simple scaling for a2,=a12=0 with translation for set Q
- Simple rotation Tor a21=-a12=1 and all=a22=0.
- Simple mirroring for a21=i~12=1 and al l=a22=0.
This analysis can bc performcd using Microsoff Excel. Combinations of scaling,
translation with rotation or mirroring can also be observed.

Consider the linear transformation (2-12a) applied to pattern clusters with circular
shape, as shown in Figure 2.11. Compute the correlation between the transformed
features and explain which types of transformation matrices do not change the
correlation values.