Analytical representation of ellipses in the Aitchison geometry and its application

Hron, Karel

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 48 (2009) no. 1, pp. 53-60.

Résumé

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 ([Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London, 1986.], [Aitchison, J., Greenacre, M.: Biplots of compositional data. Applied Statistics 51 (2002), 375–392.], [Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V. (eds): Compositional data analysis in the geosciences: From theory to practice. Geological Society, London, Special Publications 264, 2006.], [Filzmoser, P., Hron, K.: Outlier detection for compositional data using robust methods. Math. Geosci. 40 (2008), 233–248.], [Filzmoser, P., Hron, K.: Correlation analysis for compositional data. Math. Geosci., to appear.], [Filzmoser, P., Hron, K., Reimann, C.: Principal component analysis for compositional data with outliers. Environmetrics, to appear.], [Filzmoser, P., Hron, K., Reimann, C., Garrett, R.: Robust factor analysis for compositional data. Computers Geosciences, to appear.], [Pearson, K.: Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proceedings of the Royal Society of London 60 (1897), 489–502.]). 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 [Egozcue, J. J., Pawlowsky-Glahn, V.: Simplicial geometry for compositional data. In: Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V., (eds): Compositional data analysis in the geosciences: From theory to practice. Geological Society, London, Special Publications 264 (2006), 145–160.], [Pawlowsky-Glahn, V., Egozcue, J. J., Tolosana-Delgado, J.: 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 ([Filzmoser, P., Hron, K.: Outlier detection for compositional data using robust methods. Math. Geosci. 40 (2008), 233–248.]). 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.

Détail

Classification : 14P99, 15A03, 15A63, 62H99, 62J05
Keywords: Aitchison geometry on the simplex; oordinates; ellipse