Robust Coordinates for Compositional Data Using Weighted Balances

Multivariate observations which carry exclusively relative information are known under the name compositional data, and they have very specific geometrical properties. In order to represent them in the usual Euclidean geometry, they need to be expressed in orthonormal coordinates prior to their possible further statistical processing. As it is not possible to construct Cartesian coordinates for the compositions, that would assign a coordinate for each of the parts separately, a choice of interpretable orthonormal coordinates is of particular interest. Although recent experiences show clear advantages of such coordinates, where the first coordinate aggregates information from logratios with a particular compositional part of interest, their usefulness is limited if there are distortions like rounding errors or other data problems in the involved parts. The aim of the paper is thus to introduce a “robust” version of these coordinates, where the role of the remaining parts (with respect to the part of interest) is weighted according to their relevance for the purpose of the statistical analysis. Theoretical considerations are accompanied by examples with data sets from chemistry and geochemistry, pointing out the role of robust estimation in the context of regression with compositional covariates.