Mapping the reliability of the additive log-ratio transformation

Data for which ratios of parts are more important than absolute values have a compositional nature and should be analysed in the (D - 1)-dimensional simplex, but in order to use standard multivariate analysis techniques they are often mapped bijectively from the simplex into the ordinary Euclidean space. The additive log-ratio (alr) is one popular ad hoc transformation, that has already been shown to amplify the relative errors in data, in some cases by an unbounded factor. In this paper an explicit mapping of the reliability of the alr transformed data is derived, given the desired accuracy and the relative error of the sensors acquiring them. To this purpose, first the relative condition number for the alr transformation has been defined and plotted in the 3-simplex, then a generalised characterisation of well conditioned compositions (with a condition number less than or equal to an arbitrary threshold) has been derived and finally isoconditioning surfaces have been defined, analytically derived and plotted in the 3-simplex.