Interpretation of orthonormal coordinates in case of three-part compositions applied to orthogonal regression for compositional data.

Abstract

Orthonormal coordinates are very important tool for compositional data processing using standard statistical methods. Namely, in order to express a D-part composition in the Euclidean real space we use isometric log-ratio (ilr) transformation, which is an isometric mapping from the sample space of compositions, the simplex S D with the Aitchison geometry, to the (D −1)-dimensional Euclidean real space RD−1 . The ilr transformation results in coordinates of an orthonormal basis on the simplex. Advantages coming from this transformation, like the mentioned isometry between S D and RD−1 , are closely related with the problem of interpreting orthonormal coordinates, constructed by sequential binary partition. Their interpretation can be approached as balances between groups of parts of a composition as well as by expressing their covariance structure by log-ratios of parts of the analyzed composition, i.e. in terms of ratios. Note that if we want to achieve interpretation of results of statistical analysis directly on the simplex (in terms of the original compositional parts), the backtransformation is required.