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86 4 Statistical Classification

G-2: 2
p= 50000
,2772
4
Constant -6 00532 -11.7464
Figure 4.8. Decision functions coefficients for two classes of cork stoppers and
one feature, N.

Let us check these results. The class means are ml=[55.28] and m2=[79.74]. The
average variance is s2=287.6296. Applying formula (4-52) we obtain:

These results confirm the ones shown in Figure 4.8. Let us assume that a new
cork stopper has arrived and we measure 65 defects. To which class is it assigned?
As g1([65])=6.49 is greater than g2([65])=6.27 it is assigned to class 0,.

Two features, N and PRTlO

The training set classification matrix is shown in Figure 4.9. A significant
improvement was obtained in comparison with the Euclidian classifier results
mentioned in section 4.1.1 (namely an overall training set error of 10% instead of
18%). The Mahalanobis metric, taking into account the shape of the pattern
clusters, not surprisingly, performed better. The decision function coefficients are
shown in Figure 4.10. Using these coefficients we write the decision functions as:

g, (x) = w, 'x + wlv0 = [0.2616 -0.097831~ - 6.1382. (4-7a)
g2 (x) = w z'x + w~,~ = [0.0803 0.277601~ -12.8 166. (4-73)

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DISCRIM. Rows: Observed classifications
ANALYSIS Columns: Predicted classifications
G-1: 1 G-2: 2
Group p=. 50000 p=. 50000
98.00000 4 9 1
G-2 : 2 82.00000 9 4 1
Total 90.00000 5 8 4 2
Figure 4.9. Classification matrix for two classes of cork stoppers with two
features, N and PRT 10.