Introduction - Understanding Biplots - page 1

1 Introduction

Biplots have been with us at least since Descartes, if not from the time of Ptolemy who
had a method for fixing the map positions of cities in the ancient world. The essential
ingredients are coordinate axes that give the positions of points. From the very begin-
ning, the concept of distance was central to the Cartesian system, a point being fixed
according to its distance from two orthogonal axes; distance remains central to much of
what follows. Descartes was concerned with how the points moved in a smooth way as
parameters changed, so describing straight lines, conics and so on. In statistics, we are
interested also in isolated points presented in the form of a scatter diagram where, typi-
cally, the coordinate axes represent variables and the points represent samples or cases.
Cartesian geometry soon developed three-dimensional and then multidimensional forms
in which there are many coordinate axes. Although two-dimensional scatter diagrams
are invaluable for showing data, multidimensional scatter diagrams are not. Therefore,
statisticians have developed methods for approximating multidimensional scatter in two,
or perhaps three, dimensions. It turns out that the original coordinate axes can also be
displayed as part of the approximation, although inevitably they lose their orthogonality.
The essential property of all biplots is the two modes, such as variables and samples. For
obvious reasons, we shall be concerned mainly with two-dimensional approximations but
should stress at the outset that the bi - of biplots refers to the two modes and not the
usual two dimensions used for display.
Biplots, not necessarily referred to by name, have been used in one form or another
for many years, especially since computer graphics have become readily available. The
term 'biplot' is due to Gabriel (1971) who popularized versions in which the variables
are represented by directed vectors. Gower and Hand (1996) particularly stressed the
advantages of presenting biplots with calibrated axes, in much the same way as for
conventional coordinate representations. A feature of this topic is the wealth of examples
of different kinds of biplots. Although there are many novel ideas in this topic, we
acknowledge our debts to many others whose work is cited either in the current text or
in the bibliography of Gower and Hand (1996).