TY - JOUR

T1 - Complex Principal Component Analysis

T2 - Theory and Geometrical Aspects

AU - Denimal, Jean Jacques

AU - Camiz, Sergio

N1 - Publisher Copyright:
© 2022, The Author(s) under exclusive licence to The Classification Society.

PY - 2022/7

Y1 - 2022/7

N2 - 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.

AB - 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.

KW - Circularity

KW - Complex correlation coefficient

KW - Complex data tables

KW - Eccentricity

KW - Exploratory analysis

KW - Hermitian spaces

KW - Holomorphic planes

KW - Isoclinic planes

KW - Principal component analysis

UR - http://www.scopus.com/inward/record.url?scp=85129534984&partnerID=8YFLogxK

U2 - 10.1007/s00357-022-09412-0

DO - 10.1007/s00357-022-09412-0

M3 - Article

AN - SCOPUS:85129534984

SN - 0176-4268

VL - 39

SP - 376

EP - 408

JO - Journal of Classification

JF - Journal of Classification

IS - 2

ER -