Principal balances for three-way compositions

Orthonormal balances resulting from a sequential binary partition (SBP) are one of the preferred tools for transforming compositional data in real space coordinates. The interpretability of this approach, however, greatly depends on the relevance of the SBP. SBPs can be chosen with the help of expert knowledge or with data-oriented methods, such as Principal Balances analysis. This results in an SBP whose balances maximize the explained variance in a subsequent manner. Principal balances can be calculated in an exact way or in an approximate fashion by using methods based on PCA for compositional data. In this work a method for the approximation of principal balances in the more complex case of three-way compositions is proposed. Here the additional difficulty given by the introduction of third mode variability is dealt with. In particular an algorithm based on the Tucker3 model is used which allows to keep the variability of the third dimension separate in the definition of principal balances.