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. A logarithmic isomorphism between quotient spaces induces a metric space structure for compositions. The logratio compositional analysis is the statistical analysis of compositions based on this structure, consisting of analysing logratio coordinates

This work has been partially nanced by the Ministerio de Econom a y Competitividad (Ref: MTM2015-65016-C2-1-R) and the Ag encia de Gesti o d'Ajuts Universitaris i de Re- cerca (AGAUR), Generalitat de Catalunya (Ref: 2014 SGR 551)