Compositional Cubes: A New Concept for Multi-factorial Compositions

TL;DR

This paper introduces a comprehensive theory for analyzing multi-factorial compositional data, extending existing methods to higher dimensions and demonstrating practical applications with compositional cubes.

Contribution

It develops a general framework for multi-factorial compositional data, including orthogonal decomposition and coordinate representation, with implementation in R.

Findings

Multi-factorial compositional data can be orthogonally decomposed into independent and interactive parts.

A coordinate representation enables separate analysis of these parts using standard methods.

The methodology is demonstrated with practical examples involving spatial and temporal compositional cubes.

Abstract

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.