Simplifying compositional multiple regression: Application to grain size controls on sediment geochemistry

Modern geochemical data sets have typically around 20–30 compositional variables measured on some tens or hundreds of samples. A statistical analysis of data sets with so many variables should take as a priority the reduction of dimensionality of the model, in order to increase its reliability and enhance its interpretation. In the framework of compositional data analysis with multiple regression, such simplification can be achieved taking some geometric concepts into account. First, the sample space of compositions, the simplex, is given an Euclidean space structure by the compositional operations of perturbation, powering and Aitchison inner product. Then, given some qualitative information on which subcompositions might depend on each explanatory variable, one can decompose the simplex in a set of orthogonal subspaces, in such a way that the composition projected onto each subspace is independent of a subset of the explanatory variables. This is achieved with a series of singular value decomposition computations. The method is applied to a data set of 88 observations of six major oxides in molar proportions, from modern glacial and fluvio-glacial sediments, with grain size ranging from coarse sand to clay. The goal is to assess the influence of chemical weathering processes (expected to impose a linear relation of composition and grain size) against purely physical processes (expected to show step-wise functions following the largest characteristic crystal sizes of specific minerals in the source rock). We exhaustively explore all patterns of uncorrelation of the composition with three explanatory variables: grain size in ϕ scale, and two step functions for the silt and clay domains. The best pattern, chosen with a likelihood ratio test, has only a smooth trend of (Mg,Fe) vs. (Al,K,Ca+Na) enrichment towards finer grain sizes—explained as differential mechanical behaviour of phyllosilicates vs. feldspar—and coefficients for the two step functions related to the sharp decrease of quartz in silt fractions, and the sudden enrichment of mafic accessory minerals, alteration products and mechanically unstable phyllosilicates in the clay fraction. We could thus be confident that weathering is almost absent in this data set.