Compositional Regression: An Overview

A general common statistical task in geosciences is to relate a compositional data set with a set of covariables, a task often dealt with a linear model. Within the Euclidean framework of the Aitchison geometry on the simplex, the sample space of compositional data, such models are easy to construct, fit, visualize, test and use for prediction. This contribution presents in a systematic way the three cases of compositional regression, namely: (1) regression and analysis of the variance with compositional response (composition as Y); (2) regression with compositional explanatory variables (composition as X); and (3) composition-to-composition regression (compositions as both X and Y). The construction of these models is based on common tools of linear algebra: (1) the linear vector operations of the simplex, perturbation and powering; (2) the Aitchison scalar product; and (3) the concept of a linear operator between vector spaces. Fitting and usage for prediction is straightforward to obtain in logratio coordinates of the compositional objects, using classical multivariate regression (cases 1,2 and 3). Visualization of the output is based on matrices of scatterplots or boxplots, the principle of parallel plotting and the usage of all possible pairwise logratios (1 and 2). Visualization of case 3 models is possible through biplots, based on ideas from correspondence analysis. Finally, testing should target hypotheses of subcompositional independence (i.e. that the linear dependence is limited to a given subcomposition), which requires joint testing techniques. Existing multivariate tests can be adapted to the cases 1 and 2, but case 3 is only possible with asymptotic likelihood ratio tests. These concepts are illustrated with data from GEMAS, a soil survey campaign covering whole Europe.