Tri-PLS for compositional data

Compositional data (CoDa, [1] and [2]) consist of vectors of positive values summing to a unit, or in general to some fixed constant. They can often be found in many disciplines and appear as proportions, percentages, concentrations, absolute and relative frequencies. Unfortunatly, the constant-sum constraint that characterizes compositions is frequently disregarded or improperly incorporated into statistical modeling and a misleading interpretation of the results is given. Due to these specifications, several difficulties arise when dealing with CoDa. The first word of warning came already in 1897 from Karl Pearson who showed the dangers of underestimating spurious correlations. There are several approaches to incorporate CoDa into statistical modeling when it is not realistic to assume a multinomial distribution of the data. Based on the log-ratio transformations, Aitchison [1] proposed preprocessing the compositional data by means of log-ratio transformations, and successively analyzing them in a straightforward way by ’traditional’ methods. Following Aitchison’s approach, the high dimensionality of CoDa in many scientific fields has encouraged the use of bilinear and trilinear decomposition models. Thus, in attempts to find adequate low-dimensional descriptions of compositional variability, CoDa are collected into two or three-way arrays ([3], [4], [5], [6], [7]). On the other side, Hinkle and Rayens [8] examined the problems that potentially occur when one performs a partial least squares (PLS) on compositional data. The principal goal of this talk is to extend the PLS regression to three-way compositional data, following the approach proposed by Bro [9] and Bro and al. [10]. Both Candecomp/Parafac (CP - [11] [12]) and Tucker3 [13] models can be viewed as latent variables models extending principal component analysis to three-way data set. However, the most fundamental properties of PCA cannot be extended to these two models. PCA is an optimal representation of a two-way array with respect to the criteria of best low-rank approximation in least squares sense and the best approximation of the data within a joint low-dimensional subspace, while Tucker3 is only the best approximation of a three-way array within a joint low-dimensional subspace and CP is the best low-rank approximation in a least squares sense. The proposed extension of PLS to three-way compositional data is illustrated on real data sets and a software implementation will be available in the R package rrcovHD.