Compositional scalar-on-function regression between geochemical composition and particle size distribution

To describe the relationship between the distribution of particle sizes in soil (particle size distribution, PSD) and the geochemical composition of sediment samples, specific attributes of the variables need to be considered.  In this case, the explanatory variable can be described in form of the probability density function while the response is a real variable represented by log-ratios of the original chemical concentrations. Due to the relative character of density functions, an adequate methodology must be used to satisfy their specific properties. Here, the Bayes space methodology was employed, specifically the centred logratio (clr) transformation played the role to represent the PSDs (densities) in the standard $L^2$ space which is suitable for multivariate statistical methods, including regression. The idea of smoothing splines was used to represent the discretized input densities while fulfilling the zero-integral constraint imposed by the clr transformation. The resulting regression parameters (densities) can be interpreted in both the original and clr space, however, in the latter the interpretation is more straightforward. The newly developed regression model, called compositional scalar-on-function regression was then used for real-world geological data consisting of samples from four loess-paleosol sequences (LPS) in the Czech Republic (Brodek u Přerova, Dobšice, Ivaň, Rozvadovice). The regression modeling allows to distinguish local effects on the PSD and elemental composition of loess, which were not apparent by the standard approach where the PSD and compositions are usually plotted separately. The high mixing capacity of the aeolian transport caused a similar mineral and chemical composition, despite the different source areas of the studied LPSs. Local variability in the PSDs and distribution of selected elements in different grain fractions reflect some microclimatic features, especially the annual precipitation totals, which affected the particle size distribution of dust material blown by wind as well as the intensity of subsequent post-deposition and pedogenic processes.