Changing the reference measure in the simplex and its weighting effects

Abstract

Under the assumption that the Aitchison geometry holds in the simplex, standard analysis of compositional data assumes a uniform distribution as reference measure of the space. Changing the reference measure induces a weighting of parts. The changes that appear in the algebraic-geometric structure of the simplex are analysed, as a step towards understanding the implications for elementary statistics of random compositions. Some of the standard tools in exploratory analysis of compositional data, such as center, variation matrix and biplots are studied in some detail, although further research is still needed. The main result is that through a progressive down-weighting of some parts, the geometry of the space approaches that of the corresponding subcomposition. In this way, the coherence between standard and down-weighted analyses is preserved.