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Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Donevska, S.
FiÅ¡erovÃ¡, E. Hron, Karel 

Orthonormal coordinates are very important tool for compositional data processing using standardstatistical methods. Namely, in order to express a Dpart composition in the Euclidean real space weuse isometric logratio (ilr) transformation, which is an isometric mapping from the sample space ofcompositions, the simplex SD with the Aitchison geometry, to the (D âˆ’1)dimensional Euclidean realspace RDâˆ’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 RDâˆ’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 logratios 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 threepart 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 realworld example  
http://hdl.handle.net/2072/273621  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
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EstadÃstica matemÃ tica  Congressos
Mathematical statistics  Congresses AnÃ lisi multivariable  Congressos Multivariate analysis  Congresses 

Interpretation of Orthonormal Coordinates in Case of Threepart Compositions Applied to Orthogonal Regression for Compositional Data  
info:eurepo/semantics/conferenceObject  
Recercat 