A procedure for the three-mode analysis of compositions


Abstract


The Tucker3 model is one of the most widely used tools for factorial analysis of three-way data arrays. When orthogonal factors are extracted this model can be seen as a three-way PCA (principal component analysis). The Tucker3 model is characterized by extreme flexibility as it allows for the use of a different number of factors in each mode and it yields non-unique results. This adaptability makes the Tucker3 model extremely effective for decomposition and compression of data in many applications and fields. When this model is applied to vectors of non-negative values with a sum constraint all problems connected with the statistical analysis of compositions must be taken into consideration. Like other standard statistical techniques, this model cannot be directly applied. The aim of this paper is to present the theory behind the correct application of the Tucker3 model on compositional data and to describe the TUCKALS3 algorithm.

DOI Code: 10.1285/i20705948v6n2p202

Keywords: Compositional data, simplex space, log-ratio transformation, Tucker models, TUCKALS3.

References


. Pawlowsky-Glahn, V., Egozcue, J.J. (2001). Geometric approach to statistical analysis on the simplex. Stochastic Environmental Research and Risk Assessment, 15 (5), 384-398.

. Billheimer, D., Guttorp, P., Fagan, W. (2001), Statistical interpretation of species composition, Journal of American Statistics Association, 96 (456), 1205–1214.

. Aitchison, J. (1982). The statistical analysis of compositional data (with discussion). Journal of the Royal Statistical Society, Series B (Methodological), 44(2), 139-177.

. Aitchison, J. (1986). The statistical analysis of compositional data, Monographs on Statistics and Applied Probability. Chapman & Hall, Ltd, London, UK.

. Egozcue, J.J., Pawlowsky-Glahn, V. (2005). Groups of parts and their balances in compositional data analysis, Mathematical Geology, 37 (7), 795–828.

. Egozcue, J.J., Pawlowsky-Glahn, V. (2006). Exploring compositional data with the coda-dendrogram, in IAMG 2006: The XIth annual conference of the International Association for Mathematical Geology, Liege - Belgium, 3-8 September 2006.

. Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G., Barcelo-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis, Mathematical Geology, 35 (3), 279-300.

. Egozcue, J.J., Barceló-Vidal, C., Martín-Fernández, J.A., Jarauta-Bragulat, E., Díaz-Barrero, J.L., Mateu-Figueras, G. (2011). Elements of Simplicial Linear Algebra and Geometry, in Compositional Data Analysis: Theory and Applications, eds. V. Pawlowsky-Glahn and A. Buccianti, John Wiley & Sons, Ltd, Chichester, UK.

. Gallo, M. (2012). Log-ratio and parallel factor analysis: an approach to analyze three-way compositional data, in Advanced Dynamic Modeling of Economic and Social Systems. Studies In Computational Intelligence, vol. 448, p. 209-221, Springer, ISSN: 1860-949X, doi: 10.1007/978-3-642-32903-6.

. Gallo, M. (2013). Tucker3 model for compositional data, Commun. Statist. Theor. Meth. In Press.

. Gallo, M., Buccianti, A. (2013). Weighted principal component analysis for compositional data: application example for the water chemistry of the Arno river (Tuscany, central Italy). Environmetrics, ISSN: 1180-4009, doi: 10.1002/env.2214

. Gallo, M. (2012). CoDa in three-way arrays and relative sample spaces. Electronic Journal Of Applied Statistical Analysis, 5, 401-406, ISSN: 2070-5948, doi: 10.1285/i20705948v5n3p400

. Smilde, K.A., Bro, R., Geladi, P. (2004). Multi-way analysis: applications in the chemical sciences, Wiley, Chichester, UK.

. Kiers, H. A. L. (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics, 14, 105–122.

. Barceló-Vidal, C., Martín-Fernández, J.A., Mateu-Figueras, G. (2011). Compositional differential calculus on the simplex, in Compositional Data Analysis: Theory and Applications, eds. V. Pawlowsky-Glahn and A. Buccianti, John Wiley & Sons, Ltd, Chichester, UK.

Kroonenberg, P.M. (2007). Applied Multiway Data Analysis. Hoboken: Wiley Series in Probability and Statistics.


Full Text: pdf
کاغذ a4
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.