  Compositional data are commonly known as multivariate observations carrying
relative information. Even though the case of vector or even two-factorial
compositional data (compositional tables) is already well described in the
literature, there is still a need for a comprehensive approach to the analysis
of multi-factorial relative-valued data. Therefore, this contribution builds
around the current knowledge about compositional data a general theory of work
with k-factorial compositional data. As a main finding it turns out that
similar to the case of compositional tables also the multi-factorial structures
can be orthogonally decomposed into an independent and several interactive
parts and, moreover, a coordinate representation allowing for their separate
analysis by standard analytical methods can be constructed. For the sake of
simplicity, these features are explained in detail for the case of
three-factorial compositions (compositional cubes), followed by an outline
covering the general case. The three-dimensional structure is analysed in depth
in two practical examples, dealing with systems of spatial and time dependent
compositional cubes. The methodology is implemented in the R package
robCompositions.
