Information Technology Reference
In-Depth Information
is essential (see Section 2.3.1), we have seen in (2.8) that it is legitimate to rotate and
reflect diagrams based on inner products. Thus, at first glance, biplot representations of
the same data matrix may seem to differ, but one is merely a rotation or reflection of the
other: essentially the inter-sample distances and the projections of the samples onto the
axes remain unchanged.
This inner product calculation is not easy to visualize except when comparing the
relative values of two points P
i
and P
j
onthesamevariableV
k
. Then, one only has
to compare the lengths of the projections of P
i
and P
j
onto OV
k
. This process does
not work when comparing across variables
h
and
k
, because then one has to take into
account the different lengths of OV
k
and OV
h
.
All points P that project onto the same point on OV
k
will have the same inner prod-
uct. It follows that we may label that point with an appropriate unique value. This is
the basis for the recommendation of Gower and Hand (1996) that the biplot axes be
calibrated like ordinary coordinate axes. Figure 2.5 shows Figure 2.4 (reflected about
the horizontal scaffolding axis) augmented in this manner. The four variables are now
represented by four nonorthogonal axes, known as biplot axes, which extend through-
out the diagram and are concurrent at, but not rooted in, the origin. The principal axes
are of no further interest so have been removed. The biplot axes are used in precisely
the same way as the Cartesian axes they approximate. That is, when a point represent-
ing an aircraft is projected orthogonally onto an axis, one may read off the value of
the corresponding variable. This process will give approximate values that do not in
general agree precisely with those given in Table 1.1 but reproduce the entries in the
matrix
X
[
r
]
.
Figure 2.5 can be reproduced using the following function call (see Chapter 3 for a
detailed discussion of the function
PCAbipl
):
PCAbipl(aircraft.data[,-1], colours = "green", pch.samples = 15,
pch.samples.size = 1.2, n.int = c(5,3,5,3), reflect = "x",
offset = c(1.2, 1.2, 0.3, 0), side.label = c(rep("right",3),
"left"), pos.m = c(1,4,4,1), offset.m = rep(-0.15, 4))
In Figure 2.5, the scale markers are in the units of the variables of Table 1.1. Thus
the biplot allows one to draw a scatter diagram and relate samples (here aircraft) to the
values of associated variables. It gives a visualization of Table 1.1 that can be inspected
for any interesting features. The salient feature of Figure 2.5 is the way that most of
the aircraft are regularly placed from
a
to
w
. Table 1.1 lists the aircraft in the temporal
order of their development, and the ordering reflects increasing flight range coupled with
increasing payloads. In this respect
r
, the F-8A, is in an anomalous position because its
specific power is very low, even lower than those of much earlier aircraft.
It should be apparent that this figure has all the characteristics of more familiar
scatterplots:
•
points, representing the 21 samples;
•
labelled axes;
•
calibrated axes.
