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0.1
6
RGF
r
5.5
0
5
g
SLF
q
4.5
4
u
v
3.5
0.1
h
4
3
j
c
m
p
2
3.5
i
f
1.5
b
1
n
0
0.2
0.5
a
0.5
0
3
d
k
w
s
2
e
0.3
t
4
6
0.4
SPR
PLF
Figure 2.18 Two-dimensional biplot of the normalized aircraft data as given in Table
2.2 with the origin moved to SPR =
0and SLF =
0. Calibrations on biplot axes in
terms of the original (unnormalized) data.
not possible to use the above biplots by erecting normals at the values x 1 , x 2 , x 3 ,
, x p
on the corresponding biplot axes because only in exceptional circumstances will these
normals be concurrent at a unique point, as they would be with Cartesian axes. Gower and
Hand (1996) show that in the biplots described earlier in this chapter, biplot axes may be
used for interpolation, provided they are calibrated with a scale that is inversely related to
the one used for prediction. The coordinates for one unit for interpolation on the k th axis
are given by e k V r ,the k th row of V r . These scales (converted back to their original units
of measurement) are shown in Figure 2.22 for the normalized data of Table 2.2. Notice
that the directions of the biplot axes are identical to those in Figure 2.17; it is only the
calibrations that differ (inversely related). Our biplot functions included in UBbipl have
an argument ax.type with default setting ax.type = "predictive" . Changing the
setting to ax.type = "interpolative" results in a biplot with interpolative biplot
axes like the one in Figure 2.22.
Not only do the scales for prediction and interpolation differ but also they are used
differently. We have that
...
1
p
p
p
x V r =
x k ( e k V r ) = p ×
x k ( e k V r )
= p × centroid .
(2.15)
k = 1
k = 1
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