Geodesic
Covariance Structure of Principal Components for Three-Part Compositional Data
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 2, pp. 61-69.

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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.
Classification : 15A18, 62H25, 62H99, 62J10
Keywords: compositional data; covariance structure; principal components; log-contrasts 
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Hrůzová, Klára; Hron, Karel; Rypka, Miroslav; Fišerová, Eva. Covariance Structure of Principal Components for Three-Part Compositional Data. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 2, pp. 61-69. http://geodesic.mathdoc.fr/item/AUPO_2013__52_2_a5/