CoDa in three-way arrays and relative samples spaces

The object of these short notes is to give a set of convenient symbols to define the sample space for the different compositional vectors that can be arranged into a three-way array. For the exploratory analysis of three-way data, Parafac/Candecomp and Tucker3 are some of the most applied models for low-rank decomposition of three-way arrays. Here, in addition to the relative geometry, is presented a concise overview as to how the elements of a three-way array can be transformed into compositional form and the relative geometry is given.