Abstract
When analysing three-way arrays, it is a common practice to centre the arrays. Depending on the context, centring is performed over one, two or three modes. In this paper, we outline how centring affects the rank of the array; both in terms of maximum rank and typical rank.
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Notes
- 1.
Technically, this is a matter of assessing the class’ Lebesgue measure, to which we have no clue. To give an example that generally performed transformations may alter ‘randomness’ properties, consider for instance squaring all values, which clearly affects the Lebesgue measure of subclasses of the class of such arrays. However, because [12]’s transformations, as our own, are rank preserving, we expect that the results that are only proven for the maximal rank, also hold for the typical rank of classes of arrays.
References
Albers, C.J., Gower, J.C.: A contribution to the visualisation of three-way arrays. J. Multivar. Anal. 132, 1–8 (2014)
Albers, C.J., Gower, J.C.: Visualising interactions in bi- and triadditive models for three-way tables. Chemometr. Intell. Lab. Syst. 167, 238–247 (2017)
Nelder, J.A.: A Reformulation of linear models. J. Roy. Stat. Soc. Ser. A (General) 140(1), 48–77(1977)
Gower, J.C..: The analysis of three-way grids. In: Slater, P. (ed.) Dimensions of Intra Personal Space. The Measurement of Intra Personal Space by Grid Technique, vol. 2, pp. 163–173. Wiley, Chichester (1977)
Carroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via an \(n\)-way generalization of ‘Eckart-Young’ decomposition. Psychometrika 35, 283–319 (1970)
McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman & Hall/CRC, Boca Raton, Florida (1989)
Kiers, H.A.L.: Towards a standardized notation and terminology in multiway analysis. J. Chemometr. 14, 105–122 (2000)
TenBerge, J.M.F.: Simplicity and typical rank results for three-way arrays. Psychometrika 76, 3–12 (2011)
Smilde, A.K., Bro, R., Geladi, P.: Multi-way analysis with applications in the chemical sciences. Wiley, Hoboken, New Jersey (2004)
Kroonenberg, P.M.: Applied Multiway Data Analysis. Wiley, Hoboken, New Jersey (2008)
Lickteig, T.: Typical tensorial rank. Linear Algebra Appl. 69, 95–120 (1985)
ten Berge, J.M.F., Sidiropoulos, N.D., Rocci, R.: Typical rank and Indscal dimensionality for symmetric threeway arrays of order \({I} \times 2 \times 2\) or \({I} \times 3 \times 3\). Linear Algebra Appl. 388, 363–377 (2004)
ten Berge, J.M.F.: The typical rank of tall three-way arrays. Psychometrika 65, 525–532 (2000)
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Albers, C.J., Gower, J.C., Kiers, H.A.L. (2018). Rank Properties for Centred Three-Way Arrays. In: Mola, F., Conversano, C., Vichi, M. (eds) Classification, (Big) Data Analysis and Statistical Learning. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Cham. https://doi.org/10.1007/978-3-319-55708-3_8
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DOI: https://doi.org/10.1007/978-3-319-55708-3_8
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