The Augmented Biplot and Some Examples of Its Use

Biplot is an explorative method of data analysis permitting to represent graphically, usually in a plane, the interrelations among points-variables and points-individuals located in a multivariate space. This is done by making projections from the multivariate space onto twoor three-dimensional subspaces. The crucial issue is: to what extend the projections in the lower dimension subspaces reflect the true relations of points-variables and points-individuals in the full data space? It happens that sometimes the representation given by the biplot is a good one, however sometimes it is a bad one and certainly not sufficient. We show exactly wherefrom (i.e. from which theorems) some inferential properties of a biplot can be deduced and under which circumstances the relations visualized in the biplot are trustworthy. We propose to construct the biplot in an extended mode which permits to judge the adequacy of the two-dimensional approximation visualized by the classical biplot. We call the biplot drawn in the extended mode the augmented biplot. Several real data examples illustrate the use of the augmented biplot and the broadness and diversity of problems which can be elucidated relatively simply by use of the elaborated technique.