Including supplementary elements in a compositional biplot

The biplot is a widely and powerful methodology used with multidimensional data sets to describe and display the relationships between observations and variables in an easy way. Compositional data are vectors with positive components, whose sum is constant because they represent a relative contribution of different parts to a whole; due to this property standard biplots cannot be performed with compositional data, instead of a previous transformation of the data is performed. In this paper, we extend the compositional biplot defined by Aitchison and Greenacre (2002), in order to include in the display supplementary elements which are not used in the definition of the compositional biplot. Different types of supplementary elements are considered: supplementary parts of the composition, supplementary continuous variables external to the composition, supplementary categorical variables and supplementary observations. The projection of supplementary parts of the composition is done by means of the equivalence of clr and lr biplots. The other supplementary projections are done by classical methodology. An application example with a real geological data is included.