The Aitchison geometry of the simplex and the statistical analysis of compositional data: from data exploration to spatial features

Vera Pawlowsky-Glahn & Juan-José Egozcue

Prof. Dr. Vera Pawlowsky-Glahn
IAMG 2007 Distinguished Lecturer
Department of Computer Science and Applied Mathematics
University of Girona, Spain
e-mail: vera.pawlowsky /at\ udg.es

Prof. Dr. Juan-José Egozcue
Department of Applied Mathematics III
Technical University of Catalonia, Spain
e-mail: juan.jose.ezgozcue /at\ upc.edu

Talk by Vera Pawlowsky-Glahn in MN 16, on Tuesday, 10th July 2007, 16:15

Compositional data are by definition parts of some whole which only carry relative in-formation. Typical units are parts per unit, percentages, ppm, ppb, and alike. Since John Aitchison introduced in 1982 the log-ratio approach for compositional data analysis, much work has been done to understand the algebraic-geometric structure of their sam-ple space, the D-part simplex. In this talk, the real complete inner product space struc-ture of the simplex is presented, and the implications for the (geo)statistical analysis of compositional data are illustrated with four geochemical data sets. The first one deals with samples of granitoid rocks used by Nesbitt and Markovics in 1977 to characterise a progressive chemical weathering profile developed on the Toorongo granodiorite in South Australia, the second with sediments of the world's major rivers and erosional products of some of the world's major denudation areas published by McLennan in 1993, and the third one with olivine analysis from Kimberlites, kindly provided by B. Kjarsgaard and E. Grunsky, of the Geological Survey of Canada, an ongoing study which aim is to relate compositional data analysis and stoichiometric equations. The last example, copper measured from a (non-productive) prospected zone in southern Spain, shows how to take the structure of the simplex into account when dealing with spatially dependent compositional data.