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Title: Compositional scalar-on-function regression with a geological application Authors:  Ivana Pavlu - Palacky University Olomouc (Czech Republic) [presenting]
Renata Talska - Palacky University Olomouc (Czech Republic)
Karel Hron - Palacky University (Czech Republic)
Daniel Simicek - Palacky University Olomouc (Czech Republic)
Ondrej Babek - Palacky University Olomouc (Czech Republic)
Abstract: Regression between a real response and a density function as covariate has many practical motivations, e.g., to find a relationship between the geochemical composition of sediments and the distribution of particle sizes in soil (particle size distribution, PSD). In this case, the explanatory variable can be described in the form of the probability density function. At the same time, the response is a real variable (a meaningful scale-free representation of the original concentrations using log-ratios). Due to the relative character of densities, the Bayes space methodology was employed. Specifically, the centred log-ratio (clr) transformation played the role to represent the PSDs (densities) in the standard $L^2$ space. 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 examined with both simulated observations and real-world geological data; the latter were collected at four sites in the Czech Republic (Brodek u Prerova, Dobsice, Ivan, Rozvadovice). The regression model has proven to be a good tool for linking the grain size effect with geochemical signals (provenance, weathering, diagenesis, etc.).