Biplots for Compositional Regression

In the statistical modeling of compositions, the latter can appear as response or explanatory variables, or even in both roles simultaneously. Such modeling is of multivariate nature because compositions consists of vectors of D elements, with potentially large D. Prior to fitting a model, the compositions are typically first transformed by a log-ratio transformation. The transformed compositions remain, however, of multivariate nature. Several techniques from the field of multivariate analysis can be used to analyze the data of which we mention multivariate regression, canonical correlation analysis and redundancy analysis. In multivariate regression the interpretation of the results is complicated by the large number of regression coefficients obtained, and collected in the matrix of regression coefficients B[p,q] with p explanatory and q response variables. Van den Boogaart and Tolosana-Delgado (2013) applied the singular value decomposition of the matrix of regression coefficients in order to find rank two approximations to B so that biplots can be constructed that summarize the relationships between the variables. However, the regression coefficients depend on the scale of the explanatory variables and their variances can be of very different order of magnitude. It is thus more natural to \standardize" the regression coefficient with a Mahanalobis-like transformation prior to biplot construction. Such an approach amounts to a redundancy analysis (also known as reduced-rank regression) of the data, and biplots for this method are discussed in detail by Ter Braak & Looman (1994). In this contribution we develop and apply redundancy analysis in a compositional context, discussing its biplots with empirical geological data and its compositional geometric interpretation.