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80 4 Statislical Classification

As seen in previous chapters, it seems reasonable to take those sample means as
class prototypes and assign each cork stopper to its nearest prototype. This is the
essence of what is called the minimum distance or template matching
classzfication. The classification rule is then:

If 11 x - [55.28]11< 11 x - [79.74]11 then x E w, else x E w2 '

We assume, for the moment, that a Euclidian metric is used. Using the value at
half distance from the means, we can rewrite (4-1) as:

If x < 67.51 then XE al else XE a*. (4- 1 a)

The separating "hyperplane" is simply the point 67.51. Note that in the equality
case (x=67.51) the class assignment is arbitrary (a is a possibility, as in 4-la).
Let us now evaluate the performance of this simple classifier by computing the
error rate in the training set of n=50 cases per class. Figure 4.2 shows the
classification matrix (obtained with Statistics) with the predicted classifications
along the columns and the true (observed) classifications along the rows.

Figure 4.2. Classification matrix of two classes of cork stoppers using only one
feature, N. Both classes have equal probability of occurrence ("p=.5" in the listed
classification matrix).

We see that for this simple classifier the overall percentage of correct
classification in the training set is 77%, or equivalently, the overall training set
error is 23% (18% for wl and 28% for Q). For the moment we will not assess how
the classifier performs with independent patterns, i.e., we will not assess its test set
error.
Let us now use one more feature: PRTlO = PRTIIO. The feature vector is:

I We assume an underlying real domain for the ordinal feature N. Conversion to an ordinal
will be performed when needed. For instance, the practical threshold of the classifier in
(4-la) would be 68.