Canonical variate analysis biplots - Understanding Biplots - page 188

Ta b l e 4 . 1 9 Axis predictivities obtained with CVA biplots (weighted and unweighted)
of the copper froth data.

Weighted CVA

X1 X2 X3 X4 X5 X6 X7 X8

Dim_1 0.9478 0.3121 0.8361 0.2283 0.4331 0.4193 0.5522 0.3884

Dim_2 0.9478 0.9158 0.9992 0.9943 0.9929 0.9923 0.9942 0.4899

Dim_3 1.0000 0.9948 0.9999 0.9972 0.9971 0.9970 0.9968 0.6286

Dim_4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Unweighted I CVA

X1 X2 X3 X4 X5 X6 X7 X8

Dim_1 0.9163 0.3877 0.8003 0.1607 0.3434 0.3284 0.5025 0.2645

Dim_2 0.9224 0.8455 0.9990 0.9773 0.9878 0.9871 0.9802 0.5856

Dim_3 0.9998 0.9837 0.9993 0.9889 0.9902 0.9899 0.9889 0.6538

Dim_4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Unweighted Cent CVA

X1 X2 X3 X4 X5 X6 X7 X8

Dim_1 0.9185 0.3939 0.8067 0.1655 0.3553 0.3400 0.5137 0.2699

Dim_2 0.9232 0.8461 0.9990 0.9780 0.9890 0.9884 0.9811 0.6353

Dim_3 1.0000 0.9889 0.9993 0.9907 0.9910 0.9907 0.9910 0.7405

Dim_4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Note that the left-hand section of Table 4.17 combines estimates for the original
variables and therefore is subject to commensurability problems, so it is not considered
further. The right-hand section of Table 4.17 shows that a one-dimensional CVA fits quite
well and two dimensions very well. Nevertheless, the two-dimensional CVA shown in
Figure 4.18 shows that there is considerable overlap for two pairs of groups. The four
different views of the three-dimensional CVA shown in Figure 4.17 reveal that although
one group is well separated, the remaining four groups are not. The CVA biplot shown
in Figure 4.18, its zoomed version in Figure 4.20 and the density plot of Figure 4.19,
with 95% bags surrounding each group emphasize the problem, as do the numerical
values of the canonical means given in Table 4.16. The different methods for centring
make little difference so we shall focus our remarks on the weighted CVA analysis.
Table 4.18 on axis adequacies need not detain us (see Sections 3.3 and 4.6). Table 4.19
on axis predictivities, the most important of these efficiency measures, shows that it
requires two dimensions for good predictivity of the variables from the canonical means.
Exceptionally, X8 is poor even when we move to three dimensions; this draws attention
to a feature that needs further examination even though it shows no abnormality on
the biplot maps. As expected, the within-group predictivities of Table 4.20 are poor,
except for X4 . Figure 4.21 shows variants of uncertainty regions. Figure 4.22 shows
classification regions. In the top panel these represent the nearest canonical means in the
full four-dimensional space; in the bottom panel they show regions nearest to the two-
dimensional canonical means. Theoretically, the top panel would represent the optimal
discriminatory rule for assignment to a class, but the bottom panel seems to reflect better
the properties of the actual samples in the data. Now there is at least some separation of
the samples between the overlapping groups.