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Invariant co-ordinate selection


  • David E. Tyler
  • Frank Critchley
  • Lutz Dümbgen
  • Hannu Oja


A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue-eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for "invariant co-ordinate selection". By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant co- ordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given. Copyright (c) 2009 Royal Statistical Society.

Suggested Citation

  • David E. Tyler & Frank Critchley & Lutz Dümbgen & Hannu Oja, 2009. "Invariant co-ordinate selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 549-592.
  • Handle: RePEc:bla:jorssb:v:71:y:2009:i:3:p:549-592

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    References listed on IDEAS

    1. Maronna, Ricardo A. & Stahel, Werner A. & Yohai, Victor J., 1992. "Bias-robust estimators of multivariate scatter based on projections," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 141-161, July.
    2. Caussinus, H. & Fekri, M. & Hakam, S. & Ruiz-Gazen, A., 2003. "A monitoring display of multivariate outliers," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 237-252, October.
    3. Nordhausen, Klaus & Oja, Hannu & Paindaveine, Davy, 2009. "Signed-rank tests for location in the symmetric independent component model," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 821-834, May.
    4. Annaliisa Kankainen & Sara Taskinen & Hannu Oja, 2007. "Tests of multinormality based on location vectors and scatter matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 16(3), pages 357-379, November.
    5. Filzmoser, Peter & Maronna, Ricardo & Werner, Mark, 2008. "Outlier identification in high dimensions," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1694-1711, January.
    6. Taskinen, S. & Sirkia, S. & Oja, H., 2007. "Independent component analysis based on symmetrised scatter matrices," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5103-5111, June.
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    Cited by:

    1. Ilmonen, Pauliina, 2013. "On asymptotic properties of the scatter matrix based estimates for complex valued independent component analysis," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1219-1226.
    2. Alashwali, Fatimah & Kent, John T., 2016. "The use of a common location measure in the invariant coordinate selection and projection pursuit," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 145-161.
    3. Álvarez, Adolfo & Peña, Daniel, 2013. "Recombining partitions via unimodality tests," DES - Working Papers. Statistics and Econometrics. WS ws130706, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Ilmonen, Pauliina & Nevalainen, Jaakko & Oja, Hannu, 2010. "Characteristics of multivariate distributions and the invariant coordinate system," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1844-1853, December.
    5. Dürre, Alexander & Vogel, Daniel & Tyler, David E., 2014. "The spatial sign covariance matrix with unknown location," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 107-117.
    6. Virta, J., 2016. "One-step M-estimates of scatter and the independence property," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 133-136.

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