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In-Depth Information
Ta b l e 3 . 1 5
Predictions for samples S10 and S35 of
both the original and the two added variables.
S10 predictions
S35 predictions
Ve s D
8 3
.
27
136
.
50
VesL
309
.
74
456
.
43
FibL
1211
.
16
1009
.
14
RayH
329
.
05
420
.
48
RayW
22
.
55
48
.
33
NumVes
13
.
33
18
.
76
VLDratio
3
.
84
3
.
47
RHWratio
13
.
97
7
.
58
Ta b l e 3 . 1 6
Axis predictivities for the newly added variables, VLDratio and RHWratio.
Axis predictivities for new variables according to formula
(3.26)
(3.27)
VLDratio
RHWratio
VLDratio
RHWratio
Dim 1
0.05
0.00
0.05
0.00
Dim 2
0.05
0.40
0.05
0.42
Dim 3
0.90
0.42
0.92
0.45
Dim 4
0.91
0.69
0.93
0.74
Dim 5
0.98
0.78
1.00
0.83
Dim 6
0.98
0.94
1.00
1.00
3.6 Scaling the data in a PCA biplot
In Section 2.5 we saw that different biplot representations of the aircraft data are obtained,
depending on the initial scaling. With the data in Table 1.1 it was essential to scale the
data since variable
PLF
is measured on a scale that is totally incommensurable with
those of the other variables. Comparing this situation to Table 3.5, it is clear why the
incommensurable variation in
PLF
did not contribute much to the PCA biplot display
of the unscaled data, resulting in very poor adequacy of representation of
PLF
.That
PCA is not scale-invariant is something that has to be addressed, usually as in Chapter
2, by normalizing the columns of
X
; naturally, normalization affects measures of quality.
When the columns of
X
are normalized, all variables have unit weights and then the
overall quality is just the average of the
p
values of
i
. Thus, relative qualities of the
contributions given by the individual variables can be easily assessed.
Figure 2.16 gives the PCA biplot for the aircraft data normalized to unit sum of
squares. For the normalized data set we have, from the output of
PCAbipl
,
.
−
.
.
41
0083
0
0
0
0
5587
0
3789
.
.
−
.
−
.
0
24
0087
0
0
0
5543
0
2740
=
and
V
2
=
0
0
9
.
2851
0
−
0
.
0557
0
.
8657
0
0
0
5
.
6979
−
0
.
6142
−
0
.
1785
