Joint Biplots for CoDa

Compositional data (CoDa) consist of vectors of positive values summing to a unit, or in general, to some fixed constant for all vectors. They appear as proportions, percentages, concentrations, absolute and relative frequencies. Sometimes, compositions arise from non-negative data (such as counts, area, weights, volume) that have been scaled by the total of the components because the analyst is not interested in the total sum of the vector. The multidimensional analysis of this kind of data requires a careful consideration because the sample space for CoDa is the simplex. The first consistent methodological proposal to deal with CoDa was proposed by Aitchison (1986) when he introduced the log-ratio approach. Basically, the idea that this approach conveys is to move from the simplex space to the real space by using log.ratio transformations, applying standard statistical methods, and finally, by means of an inverse log-ratio transformation, to interpret the results in the simplex space. Starting from this paper, pairwise, centered, additive and isometric log-ratio transformations, in short plr, clr, alr (Aitchison, 1986) and ilr respectively, are proposed in literature (Egozucue et al., 2003). In the context of dimension-reducing techniques, Aitchison (1983) proposed applying principal component analysis (PCA) after having applied a centered log-ratio (clr) transformation to CoDa. Aitchison and Greenacre (2002) suggested an adaptation of the biplot to CoDa. The biplot is a well established graphical aid in other branches of statistical analysis and can prove to be a useful exploratory and expository tool for compositions. In literature many papers on dimensional-reduction techniques for CoDa are proposed. Based on log-ratio strategy, Gallo (2012a, 2012b, 2013) recently proposed to use three-mode analysis of compositional data. Starting from Gallo (2012b), we propose using of plr and clr joint biplots. Where in some cases the plr joint biplot is the only ones that show clearly the correlations.