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Rank Properties for Centred Three-Way Arrays

  • Casper J. AlbersEmail author
  • John C. Gower
  • Henk A. L. Kiers
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

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.

Keywords

Three-way analysis Multiway analysis Maximum rank Typical rank CANDECOMP/PARAFAC 

References

  1. 1.
    Albers, C.J., Gower, J.C.: A contribution to the visualisation of three-way arrays. J. Multivar. Anal. 132, 1–8 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Nelder, J.A.: A Reformulation of linear models. J. Roy. Stat. Soc. Ser. A (General) 140(1), 48–77(1977)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman & Hall/CRC, Boca Raton, Florida (1989)CrossRefGoogle Scholar
  7. 7.
    Kiers, H.A.L.: Towards a standardized notation and terminology in multiway analysis. J. Chemometr. 14, 105–122 (2000)CrossRefGoogle Scholar
  8. 8.
    TenBerge, J.M.F.: Simplicity and typical rank results for three-way arrays. Psychometrika 76, 3–12 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Smilde, A.K., Bro, R., Geladi, P.: Multi-way analysis with applications in the chemical sciences. Wiley, Hoboken, New Jersey (2004)CrossRefGoogle Scholar
  10. 10.
    Kroonenberg, P.M.: Applied Multiway Data Analysis. Wiley, Hoboken, New Jersey (2008)CrossRefGoogle Scholar
  11. 11.
    Lickteig, T.: Typical tensorial rank. Linear Algebra Appl. 69, 95–120 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    ten Berge, J.M.F.: The typical rank of tall three-way arrays. Psychometrika 65, 525–532 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Casper J. Albers
    • 1
    Email author
  • John C. Gower
    • 2
  • Henk A. L. Kiers
    • 1
  1. 1.Department of Psychometrics & StatisticsUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Mathematics & StatisticsThe Open UniversityMilton KeynesUK

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