Title: Weighting of parts in compositional data using Bayes Hilbert spaces
Authors: Karel Hron - Palacky University (Czech Republic) [presenting]
Alessandra Menafoglio - Politecnico di Milano (Italy)
Javier Palarea-Albaladejo - University of Girona (Spain)
Peter Filzmoser - Vienna University of Technology (Austria)
Juan Jose Egozcue - Universitat Politecnica de Catalunya (Barcelona, Spain) (Spain)
Abstract: It often occurs in practice that it is sensible to give different weights to the variables involved in multivariate data analysis. The same holds for compositional data as multivariate observations carrying relative information, such as proportions or percentages. It can be convenient to apply weights to, for example, better accommodate differences in the quality of the measurements, the occurrence of zeros and missing values, or generally to highlight some specific features of compositional variables (i.e. parts of a whole). The characterisation of compositional data as elements of a Bayes space with the Hilbert space structure enables the definition of a formal framework to implement weighting schemes for the parts of a composition. This is formally achieved by considering a reference measure in the Bayes space alternative to the common uniform measure via the well-known chain rule. Unweighted centred logratio (clr) coefficients and isometric logratio (ilr) coordinates then allow us to represent compositions in the real space equipped with the (unweighted) Euclidean geometry, where ordinary multivariate statistical methods can be used and interpreted. We present these formal developments and use them to introduce a general approach to weighting parts in compositional data analysis. We demonstrate its practical usefulness on simulated and real-world data sets in the context of the earth sciences.