Covariance structure of principal components for three-part compositional data. (English) Zbl 1292.62092

Summary: Statistical analysis of compositional data, multivariate observations carrying only relative information (proportions, percentages), should be performed only in orthonormal coordinates with respect to the Aitchison geometry on the simplex. In case of three-part compositions it is possible to decompose the covariance structure of the well-known principal components using variances of log-ratios of the original parts. They seem to be helpful for the interpretation of these special orthonormal coordinates. Theoretical results are applied to real-world data containing relative structure of landscape use in German regions.


62H25 Factor analysis and principal components; correspondence analysis
15A18 Eigenvalues, singular values, and eigenvectors
62H99 Multivariate analysis
62J10 Analysis of variance and covariance (ANOVA)
62P12 Applications of statistics to environmental and related topics


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[1] Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London, 1986. · Zbl 0688.62004
[2] Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., Barceló-Vidal, C.: Isometric logratio transformations for compositional data analysis. Math. Geol. 35 (2003), 279-300. · Zbl 1302.86024 · doi:10.1023/A:1023818214614
[3] Egozcue, J. J., Pawlowsky-Glahn, V.: Groups of parts and their balances in compositional data analysis. Math. Geol. 37 (2005), 795-828. · Zbl 1177.86018 · doi:10.1007/s11004-005-7381-9
[4] Filzmoser, P., Hron, K.: Robustness for compositional data. Becker, C., Fried, R., Kuhnt, S. (eds.): Robustness and complex data structures, Springer, Heidelberg, 2013, 117-131.
[5] Fišerová, E., Hron, K.: On the interpretation of orthonormal coordinates for compositional data. Math. Geosci. 43 (2011), 455-468. · doi:10.1007/s11004-011-9333-x
[6] Fišerová, E., Hron, K.: Statistical inference in orthogonal regression for three-part compositional data using a linear model with type-II constraints. Communications in Statistics - Theory and Methods 41 (2012), 2367-2385. · Zbl 1270.62097 · doi:10.1080/03610926.2011.604145
[7] Golub, G. H.: Modified matrix eigenvalue problems. SIAM Review 15 (1973), 318-334. · Zbl 0254.65027 · doi:10.1137/1015032
[8] Pawlowsky-Glahn, V., Buccianti, A.: Compositional data analysis: Theory and applications. Wiley, Chichester, 2011. · Zbl 1103.62111
[9] R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, 2013.
[10] Härdle, W. K., Simar, L.: Applied Multivariate Statistical Analysis. Springer-Verlag, Berlin, Heidelberg, 2012. · Zbl 1266.62032 · doi:10.1007/978-3-642-17229-8
[11] Jackson, J. D., Dunlevy, J. A.: Orthogonal least squares and the interchangeability of alternative proxy variables in the social sciences. J. Roy. Statist. Soc. Ser. D (The Statistician) 37, 1 (1988), 7-14.
[12] Landesbetrieb für Statistik und Kommunikationstechnologie Niedersachsen: Kreiszahlen: Ausgewählte Regional Daten für Deutschland. Landesbetrieb für Statistik und Kommunikationstechnologie Niedersachsen, Hannover, 2012.
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