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Invariant co-ordinate selection

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

    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.

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    Bibliographic Info

    Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society: Series B (Statistical Methodology).

    Volume (Year): 71 (2009)
    Issue (Month): 3 ()
    Pages: 549-592

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    Handle: RePEc:bla:jorssb:v:71:y:2009:i:3:p:549-592

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    Cited by:
    1. Adolfo Álvarez & Daniel Peña, 2013. "Recombining partitions via unimodality tests," Statistics and Econometrics Working Papers ws130706, Universidad Carlos III, Departamento de Estadística y Econometría.
    2. 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.
    3. 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.

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