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Complex Principal Component Analysis: Theory and Geometrical Aspects

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Abstract

The extension of exploratory use of real principal component analysis (RPCA) to complex data tables clears the deficiency existing in complex principal component analysis (CPCA), a method mainly developed in the statistical framework, but lacking effective interpretation tools. While often used in climatology, oceanography, and signal analysis among others, its use raises understanding difficulties, due to an intrinsic indeterminacy, which gets more tortuous its use. In this paper, a real framework in which CPCA may be embedded is proposed in order to solve its intricacies. This is obtained through the RPCA of a particular real table, derived from the complex at hand, whose double eigenvalues correspond to eigenplanes, that are proved to be both holomorphic and isoclinic. Relations existing between the two analyses lead to fix the intrinsic CPCA indetermination through a second RPCA, that optimizes the complex principal components inner structure. As a spin-off, appropriate interpretation aids derive, associated with statistics describing the structure of the clouds of units associated with the complex variables, which may get meaningful the issued graphical results. Eventually, CPCA is applied to a small wind speeds data table, to show both its use and the effectiveness of its interpretation aids, allowing an easier understanding of the CPCA’s abilities in the exploratory framework.

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Data Availability Statements

The original data had been downloaded from Internet as quoted; thus, they are public: their transformations are fully described. An R (R Core Team, 2013) package to run CPCA including these data is currently under development.

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Funding

For this work, the second author was partially supported by the Erasmus Plus agreement between Sapienza Università di Roma and the Université des Sciences et Technologies de Lille and by his Sapienza research grant 2016.

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Authors and Affiliations

  1. Université des Sciences et Technologies de Lille, Villeneuve-d’Ascq, France

    Jean-Jacques Denimal

  2. National Researches Council-ISPC, Roma, Italy

    Sergio Camiz

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  1. Jean-Jacques Denimal
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  2. Sergio Camiz
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Correspondence to Jean-Jacques Denimal.

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Denimal, JJ., Camiz, S. Complex Principal Component Analysis: Theory and Geometrical Aspects. J Classif 39, 376–408 (2022). https://doi.org/10.1007/s00357-022-09412-0

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