DUGi

Interpretation of Orthonormal Coordinates in Case of Three-part Compositions Applied to Orthogonal Regression for Compositional Data

http://dugi.udg.edu/item/http:@@@@hdl.handle.net@@2072@@273621

Orthonormal coordinates are very important tool for compositional data processing using standardstatistical methods. Namely, in order to express a D-part composition in the Euclidean real space weuse isometric log-ratio (ilr) transformation, which is an isometric mapping from the sample space ofcompositions, the simplex SD with the Aitchison geometry, to the (D −1)-dimensional Euclidean realspace RD−1. The ilr transformation results in coordinates of an orthonormal basis on the simplex.Advantages coming from this transformation, like the mentioned isometry between SD and RD−1, areclosely related with the problem of interpreting orthonormal coordinates, constructed by sequentialbinary partition. Their interpretation can be approached as balances between groups of parts of acomposition as well as by expressing their covariance structure by log-ratios of parts of the analyzedcomposition, i.e. in terms of ratios. Note that if we want to achieve interpretation of results ofstatistical analysis directly on the simplex (in terms of the original compositional parts), the backtransformationis required.The aim of the contribution is to analyze the interpretation of two coordinates (balances) obtainedby the ilr transformation of three-part compositions. Attention is focused on interpreting coordinatescoming from the description of their covariance structure. General conclusions will be usedfor analysing results from orthogonal regression for compositions. Its main idea is to fit a line explainingthe set of n compositional data points in coordinates in such a way that the sum of squareddistances from data points to the estimated line is minimal. By using the theory of linear regressionmodels with type II constraints, it is possible to construct confidence bounds or testing hypotheseson regression parameters. However, especially the mentioned parameters cannot be easily interpretedback on the simplex, the interpretation is only possible in sense of the orthonormal coordinates. Thetheoretical considerations will be illustrated on a real-world example

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada