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Gabriel (1971). Figure 2.4 is obtained by adding the following R code to the code given
above for Figure 2.3:
> plot(x = x, y = y, xlim = c(-6,4), ylim = c(-2,2), pch = 15,
col = "green", cex = 1.2, xlab = "V1", ylab = "V2",
frame.plot = FALSE)
> text(x = x, y = y, label = dimnames(aircraft.mat)[[1]], pos = 1)
> text(x = svd.X.centered$v[,1], y = svd.X.centered$v[,2], label
= dimnames(aircraft.mat)[[2]], pos = 2, offset = 0.4, cex = 0.8)
> windows()
> PCAbipl(cbind(x,y), reflect = "y", colours = c("green",
rep("black",8)), pch.samples = 15, pch.samples.size = 1.2,
exp.factor = 1.4, n.int = c(5,3), offset = c(0, 0, 0.5, 0.5),
pos.m = c(1,4), offset.m = c(-0.25, -0.25), pos = "Hor")
> arrows(0, 0, svd.X.centered$v[-3,1], svd.X.centered$v[-3,2],
length = 0.15, angle = 15, lwd = 2, col = "red")
> text(x = -svd.X.centered$v[,1], y = svd.X.centered$v[,2],
label = dimnames(aircraft.mat)[[2]], pos = 2, offset = 0.075,
cex = 0.8)
X
[
r
]
=
UJ
JV
that
X
[
r
]
can be written as
We note from the approximation
X
[
r
]
=
(
)
UJ
)(
VJ
)
=
(
UJ
Q
)(
VJQ
(2.8)
=
A
[r]
B
[r]
.
Since (2.8) is valid for any
p
×
p
orthogonal matrix
Q
, it follows that the configurations in
Figures 2.3 and 2.4 may be subjected to orthogonal rotations and/or reflections about the
horizontal or vertical axes without violating the inner product representation above. The
same code on different computers can thus result in apparently different representations,
but one is just an orthogonal rotation and/or reflection of the other.
What are the practical implications of the biplot representation (2.8)? Instead of
answering this question immediately we turn to our standpoint of understanding a biplot
as an extension of an ordinary scatterplot. Although Figure 2.4 is a biplot, there are no
calibrated axes representing the variables as in Figure 2.1. Therefore, in the next section
we address the problem of converting the markers or arrows representing the variables
in Figure 2.4 into calibrated axes analogous to ordinary scatterplots.
2.3 Calibrated biplot axes
We have seen in Section 2.2 that the biplot of Figure 2.4 uses an inner product repre-
sentation. This inner product interpretation can be described as follows. The biplot axes
are shown as vectors
v
k
whose end-points V
k
have coordinates given by the first two
elements of the
k
th row of
V
. Then, the value
x
ik
associated with a point P
i
and a vector
v
k
is the product of the lengths OP
i
and OV
k
and the cosine of the angle
subtended at
the origin. The matrix
X
gives all
np
inner product values. Although a unit aspect ratio
θ
