Principal component analysis biplots - Understanding Biplots - page 78

12

23

21 13

22 11

18 19

1

4

17

8

7

3

6

9

14

15

20

25 16

24

5

10

2

− 15 − 10 − 5 0 5

V 1

Figure 3.7 PCA display of the sample points of Table 3.3. The scaffolding axes are
the first two columns of the matrix V . The aspect ratio is unity.

In Figure 3.12 the biplot space L is the same subspace of the three-dimensional space
R as above. Any point z ∗ : r × 1 in terms of the scaffolding (the first two columns of V
in (3.5)) or basis for the biplot space, with r = 2 in Figure 3.12, is also a point x ∗ : p × 1
in terms of the basis for R ,where p = 3 in the figure. Such a point will project onto
itself, giving

x ∗ = x ∗ V r ( V r V r ) − 1 V r . (3.7)

In Section 3.2.2 it was shown that the interpolation of a point x ∗ ,isgivenby z ∗ = x ∗ V r .

Substituting into (3.7) above and since the columns of V r are orthonormal, we have

z ∗ V r x ∗ = (3.8)

as the prediction of z ∗ . For constructing a prediction biplot axis, a ( p - 1)-dimensional
hyperplane N perpendicular to the Cartesian axis is needed. In Figure 3.13 a two-
dimensional plane is constructed perpendicular to the first Cartesian axis through the
marker '20'. The intersection of L and N is an ( r − 1)-dimensional intersection
space, indicated by a line in Figure 3.13. All points on this line will predict the value
'20' for the X -axis. In Figure 3.14 the plane N has been shifted orthogonal to the
Cartesian axis X to go through marker '15'. All the points on this intersection space will
predict the value '15' for the X -axis.