The Mathematics of Compositional Analysis

The term compositional data analysis is historically associated to the approach based on the logratio transformations introduced in the eighties. Two main principles of this methodology are scale invariance and subcompositional coherence. New developments and concepts emerged in the last decade revealed the need to clarify the concepts of compositions, compositional sample space and subcomposition. In this work the mathematics of compositional analysis based on equivalence relation is presented. The two principles are essential attributes of the corresponding quotient space. A logarithmic isomorphism between quotient spaces induces a metric space structure for compositions. Using this structure, the statistical analysis of compositions consists of analysing logratio coordinates.