Numerical Stability Analysis of the Centered Log-Ratio Transformation

Data have a compositional nature when the information content to be extracted and analyzed is conveyed into the ratio of parts, instead of the absolute amount. When the data are compositional, they need to be scaled so that subsequent analysis are scale-invariant, and geometrically this means to force them into the open Simplex. A common practice to analyze compositional data is to map bijectively compositions into the ordinary euclidean space through a suitable transformation, so that standard multivariate analysis techniques can be used. In this paper, the stability analysis of the Centered Log-Ratio (clr) transformation is performed. The purpose is to isolate areas of the Simplex where the clr transformation is ill conditioned and to highlight values for which the clr transformation cannot be accurately computed. Results show that the mapping accuracy is strongly affected by the closeness of the values to their geometric mean, and that in the worst case the clr can amplify the errors by an unbounded factor.