Analytical representation of ellipses in the Aitchison geometry and its application

Hron, Karel. "Analytical representation of ellipses in the Aitchison geometry and its application." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 48.1 (2009): 53-60. <http://eudml.org/doc/35188>.

@article{Hron2009,
abstract = {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.},
author = {Hron, Karel},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Aitchison geometry on the simplex; oordinates; ellipse},
language = {eng},
number = {1},
pages = {53-60},
publisher = {Palacký University Olomouc},
title = {Analytical representation of ellipses in the Aitchison geometry and its application},
url = {http://eudml.org/doc/35188},
volume = {48},
year = {2009},
}

TY - JOUR
AU - Hron, Karel
TI - Analytical representation of ellipses in the Aitchison geometry and its application
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2009
PB - Palacký University Olomouc
VL - 48
IS - 1
SP - 53
EP - 60
AB - 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.
LA - eng
KW - Aitchison geometry on the simplex; oordinates; ellipse
UR - http://eudml.org/doc/35188
ER -