Augmented Biplot
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{file biaug.tex, Machine Graphics &Vision, Vol. 4 (1995), pp. 161–185 } THE AUGMENTED BIPLOT AND SOME EXAMPLES OF ITS USE Anna Bartk owia k, Adam Szustalewicz Institute of Computer Science, University of Wroc law Przesmyckiego 20, Wroc law 51-151, Poland Abstract. Biplot is an explorative method of data analysis permitting to represent graphically, usually in a plane, the interrelations among p oints -variable s and points- individuals located in a multi v aria te space. Thi s is done by mak ing projections from the multi v ariate space onto two- or three-dime nsi onal subspaces. The crucial issue is: to what extend the pro- jections 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 representa tion 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. Key wor ds and phrases: Data matrix, exploratory data analysis, reduction of di- mensionality, graphical representation of multivariate points and interrelations from mul- tivariate space. 1
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