| Title:
|
Analytical representation of ellipses in the Aitchison geometry and its application (English) |
| Author:
|
Hron, Karel |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
53-60 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 ([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. (English) |
| Keyword:
|
Aitchison geometry on the simplex |
| Keyword:
|
oordinates |
| Keyword:
|
ellipse |
| MSC:
|
14P99 |
| MSC:
|
15A03 |
| MSC:
|
15A63 |
| MSC:
|
62H99 |
| MSC:
|
62J05 |
| idZBL:
|
Zbl 05734887 |
| idMR:
|
MR2641947 |
| . |
| Date available:
|
2010-02-11T13:55:30Z |
| Last updated:
|
2013-09-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137515 |
| . |
| Reference:
|
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| Reference:
|
[2] Aitchison, J., Greenacre, M.: Biplots of compositional data.Applied Statistics 51 (2002), 375–392. Zbl 1111.62300, MR 1977249 |
| Reference:
|
[3] Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V.: Compositional data analysis in the geosciences: From theory to practice.Geological Society, London, Special Publications 264, 2006. Zbl 1155.86002 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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