John C. Gower
The Open University, London
Biplots are concerned with presenting information on the variables
and units, typically for data presented in a data-martrix X. The archetypal
biplot is represented by Cartesian coordinate axes in which the calibrated
axes (usually orthogonal) represent the variables and the units are
represented as points. There are two questions that relate to Cartesian axes
(i) given a case x, that is a row of X, where is the corresponding point P
and (ii) given a point P what are the associated values of x?
Statistical biplots may differ from this classical set-up in several ways.
Firstly we usually have only an approximation to the full-dimensional
Cartesian representation. This induces non-orthogonal axes and complicates
the answers to (i) and (ii). Secondly, we may use unusual metrics based on
dissimilarity coefficients or on variants of Mahalanobis distance. Thus,
extensions are needed to embrace methodologies where X is represented by
various forms of metric and nonmetric multidimensional scaling. Thirdly, we
may wish to include categorical variables in X, and these may be either of
purely nominal form or they may be constrained to be ordered categories. An
extension to the Cartesian system is needed to cope with categories.
It will be shown how all these needs may be handled in a unified way.
Orthogonal projection is a key concept in the use of Cartesian axes; the more
general concept of the nearest point to a set is invoked to handle the
generalisations. For quantitative varibles we end up with calibrated axes,
possibly nonlinear, and for categorical variables, sets of labelled points
representing the different category-levels. The cases continue to be
represented by points.