N-way partial least squares 
for compositional data

Partial least squares (PLS) is a method for building regression models between independent and dependent variables. When a set of independent variables is measured on several occasions, the samples can subsequently be arranged in three-way arrays. In this case N-way partial least squares (N-PLS) can be used. N-PLS decomposes three-way array of independent variables and establishing a relation between the three-way array of independent variables and the array of dependent variables. Sometimes, the set of independent variables are parts of the same whole, thus each observation consists of vectors of positive values summing to a unit, or in general, to some fixed constant. When these data, known as compositional data (CoDa), are analyzed by N-PLS, it is necessary to take into account the specific relationships between the parts that compositions are made of. The problems that potentially occur when one performs a N-way partial least squares analysis on compositional data are examined. A strategy based on the log-ratio transformations is suggested.