Mathematical Geology

, Volume 24, Issue 4, pp 365–379

On criteria for measures of compositional difference

  • John Aitchison

DOI: 10.1007/BF00891269

Cite this article as:
Aitchison, J. Math Geol (1992) 24: 365. doi:10.1007/BF00891269


Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It will be shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible logratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.

Key words

scale invariancepermutation invarianceperturbation invariancecovariance structuresubcompositionequivalence classesgroups

Copyright information

© International Association for Mathematical Geology 1992

Authors and Affiliations

  • John Aitchison
    • 1
  1. 1.Statistics Division, Department of Mathematics, Mathematics-Astronomy BuildingUniversity of VirginiaCharlottesville