11 - Compositional Data

Summary

Compositional data are multivariate data whose components sum to a constant. They are very common in the earth sciences, and are typified by the weight percent oxides in major rock analyses. Statistical tools for their analysis have only become available over the past thirty years. Compositional data are closed, and subcompositions may be obtained by applying the closure operation to a subset of a larger data set. Their sample space is the simplex, of which the simplest example is ternary. A vector space for compositional data is obtained by defining the perturbation and powering operators that are analogous to addition and multiplication in a real vector space, along with an inner product and distance measure. Transformation between the simplex and a real vector space is facilitated by centered and isometric log ratio transformations. Transformed data may be analyzed using standard tools in a real space and then transformed back to the simplex. All of these concepts are illustrated using rock composition data.