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Analytical representation of ellipses in the Aitchison geometry and its application. (English) Zbl 1294.62143

Summary: Compositional data, multivariate observations that hold only relative information, need a special treatment while performing statistical analysis, with respect to the simplex as their sample space. For the logratio approach to the statistical analysis of compositional data the so called Aitchison geometry was introduced and confirmed to be the meaningful one. It was shown in [J. J. Egozcue et al., Compositional data analysis in the geosciences: from theory to practice. Geological Society Special Publication 264, 145–159 (2006; Zbl 1156.86307)], [V. Pawlowsky-Glahn et al., Lecture notes on compositional data analysis. http://hdl.handle.net/10256/297, (2007)] that it is quite easy to express simple geometric objects like compositional lines, this is however not the case for ellipses, although they play a fundamental role within most statistical methods, for example in outlier detection [P. Filzmoser and the author, Math. Geosci. 40, No. 3, 233–248 (2008; Zbl 1135.62040)].
The aim of the paper is to introduce a way, based on coordinate representations of compositions, how to obtain an analytical representation of ellipses in the Aitchison geometry.

MSC:

62H99 Multivariate analysis
14P99 Real algebraic and real-analytic geometry
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References:

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