Methodology and Statistics
International Conference, 16 - 18 September 2002
FDV, Ljubljana, Slovenia


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.