Complex Principal Component Analysis: Theory and Geometrical Aspects

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.