Biplot basics - Understanding Biplots - page 14

so that k is nearer to n , as is correctly displayed in the bottom panel. This example
clearly demonstrates how one can go seriously wrong by constructing biplots that do
not respect the aspect ratio. An aspect ratio of unity is not necessary for the validity
of reading the scales by projection but, in much of what follows, we shall see that the
relative scaling (or aspect ratio) of axes is crucial. The scatterplot in the bottom panel
of Figure 2.1 has an aspect ratio of one. The call to the plot function to reproduce
this scatterplot requires asp = 1 instead of the asp default. The window for plotting is
then set up so that one data unit in the x direction is equal in length to one data unit in
the y direction. If this precaution is not taken when constructing biplots the inter-point
distances in the biplot are distorted.
Figure 2.1 happens to be in two dimensions, but this is not necessary for a biplot.
Indeed, if we make a three-dimensional Cartesian plot of the first three variables, this
too would be a biplot (see Figure 2.2). The three-dimensional biplot in Figure 2.2
can be obtained by first using the following code and then interactively rotating and
zooming the biplot to the desired view by using the left and right mouse buttons,
respectively.
> library(rgl)
> open3d()
> view3d(theta = 180, phi = 45, fov = 40, zoom = 0.8)
> points3d(aircraft.mat, size = 10, col = "green", box = FALSE,
xlim = c(3,6), ylim = c(1,9), zlim = c(0,0.5))
> text3d(aircraft.mat, texts = dimnames(aircraft.data)[[1]],
adj = c(0.25, 1.2), cex = 0.75)
> axes3d(c("y","x","z-+"), cex = 0.75)
> aspect3d(1, 1, 0.5)
> title3d("","","SPR","RGF","PLF")
It is also possible to construct one-dimensional biplots, and although we consider such
biplots as well as three-dimensional biplots in later chapters; for the remainder of this
chapter we restrict ourselves to two-dimensional biplots.
2.2 The biplot as a multidimensional scatterplot
Although the plots in Figures 2.1 and 2.2 are commonly known as scatterplots, they
are simple examples of biplots. Suppose now that we wish to show all four variables of
Table 1.1. A perfect Cartesian representation would require four dimensions, so we would
find it convenient if we could approximate the information in a two-dimensional (say)
display. There are many ways of representing the aircraft by points in two dimensions
so that their actual inter-point distances in the four dimensions are approximated. This is
the concern of multidimensional scaling (MDS). We shall meet several methods of MDS
in later chapters, but here we use one of the simplest methods by expressing the data
matrix in terms of its singular value decomposition (SVD). We shall see that many of the
ideas introduced in this chapter carry over easily into various forms of biplot discussed
in later chapters.