Principal component analysis biplots - Understanding Biplots - page 100

Ta b l e 3 . 1 3 Sample predictivities for the first five and last five of the original samples
in the Figure 3.23 biplot.

S1 S2 S3 S4 S5 ... S33 S34 S35 S36 S37
Dim 1 0.8694 0.8258 0.8397 0.7154 0.5523 ... 0.4401 0.0903 0.0610 0.2064 0.0841
Dim 2 0.8754 0.8558 0.8588 0.7502 0.5579 ... 0.5070 0.7492 0.8784 0.3497 0.0887
Dim 3 0.9272 0.8902 0.8987 0.7545 0.7637 ... 0.5369 0.7940 0.9054 0.5535 0.9616
Dim 4 0.9482 0.9478 0.9899 0.9281 0.9331 ... 0.7910 0.7940 0.9124 0.7214 0.9621
Dim 5 0.9868 0.9845 0.9901 0.9673 0.9959 ... 0.9989 0.8038 0.9917 0.9182 0.9666
Dim 6 1.0000 1.0000 1.0000 1.0000 1.0000 ... 1.0000 1.0000 1.0000 1.0000 1.0000

Ta b l e 3 . 1 4 Sample predictivities for the three new
interpolated points in the Figure 3.23 biplot.

O.bul O.ken O.por

Dim 1 0.5411 0.7790 0.0032

Dim 2 0.5901 0.9074 0.5127

Dim 3 0.9506 0.9175 0.9754

Dim 4 0.9525 0.9412 0.9904

Dim 5 0.9920 0.9717 0.9939

Dim 6 1.0000 1.0000 1.0000

regression decomposition using all p variables (i.e. when J = I ) is (see Section 2.7)

x ∗ = XVb + ( x ∗ − XVb ). (3.24)

Now, XVb may be partitioned into orthogonal components (i) XVJb and (ii) XV ( I − J ) b .
In (i) XVJ represents the r -dimensional data, while Jb selects the r coefficients of b r .
The term (ii) XV ( I − J ) b represents an equivalent decomposition in the remaining
p − r dimensions. Thus we have a full orthogonal breakdown,

x ∗ = XVJb + XV ( I - J ) b + ( x ∗ − XVb ) ,

(3.25)

with an equivalent analysis of variance with three terms, the first of which represents the
sum of squares in the r -dimensional regression approximations, the second the remaining
sum of squares in the full p -dimensional space, while the final term is a residual sum of
squares not accounted for by regression.
Axis predictivity may be evaluated, as usual, as the ratio of the fitted sum of squares
to the original sum of squares:

x = ( b JV X )( XVJb )
x ∗ x ∗

. (3.26)

Alternatively, we may replace x ∗ x ∗ by its fitted regression sum of squares,

2 = ( b JV X )( XVJb ) . (3.27)

b V X XVb