Log-ratio and parallel factor analysis: an approach to analyze three-way compositional data

For the exploratory analysis of three-way data, Parafac/Candecomp model (CP) is one of the most applied models to study three-way arrays when the data are approximately trilinear. It is a three-way generalization of PCA (Principal Component Analysis). CP model is a common name for low-rank decomposition of three-way arrays. In this approach, the three-dimensional data are decomposed into a series of factors, each relating to one of the three physical ways. When the data are particular ratios, as in the case of compositional data, this model should consider the special problems that compositional data pose. The principal aim of this paper is to describe how an analysis of compositional data by CP is possible and how the results should be interpreted