Measuring Subcompositional Incoherence

Abstract

Subcompositional coherence is a fundamental property of Aitchison’s approach to compositional data analysis, and is the principal justification for using ratios of components. For dimension reduction of a matrix of compositional data, either an unweighted (Aitchison & Greenacre 2002) or weighted (Greenacre & Lewi 2009; Greenacre 2010a: chapter 7) form of log-ratio analysis can be used, and these are both subcompositionally coherent. Many alternative methods that might be applied to compositional data are subcompositionally incoherent, but some can be judged to be less incoherent than others. In other words, either for a particular data set, or in general, a method might actually be quite subcompositionally “robust” in that its results for a subcomposition are quite close to its results for the same components as part of a full composition. So we propose that lack of subcompositional coherence, that is subcompositional incoherence, can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods, which might be better suited to cope with problems such as data zeros and outliers, while being only slightly incoheren