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Body diagonalization of core matrices in three‐way principal components analysis: Theoretical bounds and simulation
Author(s) -
Henrion René
Publication year - 1993
Publication title -
journal of chemometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.47
H-Index - 92
eISSN - 1099-128X
pISSN - 0886-9383
DOI - 10.1002/cem.1180070604
Subject(s) - principal component analysis , superposition principle , core (optical fiber) , matrix (chemical analysis) , variance (accounting) , rotation (mathematics) , computer science , interpretation (philosophy) , rotation matrix , algorithm , component (thermodynamics) , contrast (vision) , mathematics , biological system , artificial intelligence , mathematical analysis , chemistry , physics , telecommunications , accounting , chromatography , biology , business , thermodynamics , programming language
In contrast with conventional PCA, a direct superposition and joint interpretation of loading plots is not possible in three‐way PCA, since there may be data variance which is described by unequal components of different modes. The contributions to variance of all possible combinations of components are described in the core matrix. Body diagonalization, which is achieved by appropriate rotation of component matrices, is an essential tool for simplifying the core matrix structure. The maximum degree of body diagonality which may be obtained from such transformations is analysed from both the mathematical and simulation viewpoints. It is shown that, at least in the average case, high degrees can be expected, which makes the procedure reasonable for many practical applications. Furthermore, simulation as well as theoretical derivation show that the success of body diagonality depends on the so‐called polarity of the core array. The methodology is illustrated by a three‐way data example from environmental chemistry.