Exploratory compositional data analysis using the R-package robCompositions

Compositional data are multivariate observations that carry only relative information. This means that not the absolute values but the ratios between the variables are of interest. This is important also for an exploratory analysis of such data. We present two basic methods for the exploratory compositional data analysis (ECDA), namely multivariate outlier detection and the compositional biplot. The methods are illustrated at a small data example using the R package robCompositions. 1 Compositional data In practice, data frequently consist of percentages or, more general, not the absolute values but the ratios between the variables are of interest. Usually, this kind of observations is characterized with a positive constant sum constraint of variables (usually 1 or 100 in the case of proportions or percentages, respectively), however, this condition is obviously not necessary. Nowadays, multivariate observations that represent quantitative descriptions of the parts of some whole, conveying exclusively relative information, are known under the term compositional data or compositions for short. Obviously, the D-part composition x = (x1, . . . , xD) ′ and its positive real multiple cx, c > 0, convey essentially the same information. The sample space of compositions is a D-part simplex, a (D − 1)-dimensional subset of RD−1 that contain all D-part compositions that sum up to a prescribed constant sum constraint. The nature of compositions claim for a special geometry, called nowadays the Aitchison geometry with special operations of perturbation, power transformations and the Aitchison inner product with the usual Hilbert space properties [5]. The name of the geometry comes according to John Aitchison, a British statistician that proposed the first comprehensive theory for statistical analysis of compositional data [1]. A special treatment for compositions is necessary because of the different sample space. Thus, the usual statistical methods cannot be applied directly to compositions as they are designed for the Euclidean sample space, where the information is absolute and not relative. As a way out, J. Aitchison proposed the family of log-ratio transformations from the