A novel downscaling procedure for compositional data in the Aitchison geometry with application to soil texture data

In this work, we present a novel downscaling procedure for compositional quantities based on the Aitchison geometry. The method is able to naturally consider compositional constraints, i.e. unit-sum and positivity, accounting for the scale invariance and relative scale of these data. We show that the method can be used in a block sequential Gaussian simulation framework in order to assess the variability of downscaled quantities. Finally, to validate the method, we test it first in an idealized scenario and then apply it for the downscaling of digital soil maps on a more realistic case study. The digital soil maps for the realistic case study are obtained from SoilGrids, a system for automated soil mapping based on state-of-the-art spatial predictions methods.

Linear Association in Compositional Data Analysis

With compositional data ordinary covariation indexes, designed for real random variables, fail to describe dependence. There is a need for compositional alternatives to covariance and correlation. Based on the Euclidean structure of the simplex, called Aitchison geometry, compositional association is identied to a linear restriction of the sample space when a log-contrast is constant. In order to simplify interpretation, a sparse and simple version of compositional association is dened in terms of balances which are constant across the sample. It is called b-association. This kind of association of compositional variables is extended to association between groups of compositional variables. In practice, exact b-association seldom occurs, and measures of degree of b-association are reviewed based on those previously proposed. Also, some techniques for testing b-association are studied. These techniques are applied to available oral microbiome data to illustrate both their advantages and diculties. Both testing and measurements of b-association appear to be quite sensible to heterogeneities in the studied populations and to outliers.

Changing the Reference Measure in the Simplex and its Weighting Effects

Standard analysis of compositional data under the assumption that the Aitchison geometry holds assumes a uniform distribution as reference measure of the space. Weighting of parts can be done changing the reference measure. The changes that appear in the algebraic-geometric structure of the simplex are analysed, as a step towards understanding the implications for elementary statistics of random compositions. Some of the standard tools in exploratory analysis of compositional data analysis, such as center, variation matrix and biplots are studied in some detail, although further research is still needed. The main conclusion is that down-weighting some parts is approaching the geometry of the corresponding subcomposition, thus preserving a kind of coherence between standard and down-weighted analyses.