Biplots of Compositional Data

Abstract

1 SummaryThe singular value decomposition and its interpretation as a linear biplot has proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the speci¯c case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is demonstrated on a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.2

Relationship between glyco-oxidation, antioxidant status and microalbuminuria in type 2 diabetic patients

“…A biplot graphic display was used to present the behaviour of variables in order to examine their correlation on the same chart. The length and the directions of vectors (rays) may be important in interpreting the data [35]. In this case, the most useful variable is the cosine of the eigenvectors that suggested correlations between different variables.…”

Compositional Data Analysis: Where Are We and Where Should We Be Heading?

“…For a compositional data set the biplot is based on a singular value decomposition of the doubly centered logratio matrix. For details of biplot construction see Aitchison (1990bAitchison ( , 1997Aitchison ( , 2001) and Aitchison and Greenacre (2002). Such biplots, consisting of vertices, rays, links and case markers, allow an overall view of compositional covariance structure, subcompositional analysis, the relationship of individual compositions to parts, and provide useful interpretations of near-coincident vertices, collinear vertices and orthogonal links.…”

“…The tools for such discovery are again either principal logcontrast analysis or, equivalently, singular value decomposition. For details of such discoveries through principal logcontrast analysis see Aitchison (1999) and through biplot analysis see Aitchison and Greenacre (2002).…”

Toward an Efficient and Integrative Analysis of Limited-Choice Behavioral Experiments: Figure 1.

“…A subject avoiding one choice altogether is represented by a point on the line opposite to the vertex representing this choice. ILR-transformed data are efficiently displayed in biplots (Aitchison and Greenacre, 2002), which, in the case of a ternary composition, are simple x-y scatterplots in which the two contrasts [say, "familiar/ stranger" and "(familiar, stranger)/neutral" in our example above] are plotted against each other. Both ternary diagrams and biplots give an ad hoc impression of data variability (or spread), and also of how individual variates and contrasts formed by them may relate to each other (Fig.…”