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Transforming three‐way arrays to multiple orthonormality
Author(s) -
ten Berge Jos M. F.,
Kiers Henk A. L.,
Murakami Takashi,
van der Heijden René
Publication year - 2000
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 - 10.1002/1099-128x(200005/06)14:3<275::aid-cem588>3.0.co;2-r
Subject(s) - orthonormality , orthogonality , variety (cybernetics) , computer science , simplicity , convergence (economics) , simple (philosophy) , algorithm , cordic , core (optical fiber) , orthonormal basis , mathematics , arithmetic , theoretical computer science , artificial intelligence , telecommunications , philosophy , physics , geometry , epistemology , quantum mechanics , economics , field programmable gate array , computer hardware , economic growth
This paper is concerned with the question to what extent the concept of rowwise or columnwise orthonormality can be generalized to three‐way arrays. Whereas transforming a three‐way array to multiple orthogonality is immediate, transforming it to multiple orthonormality is far from straightforward. The present paper offers an iterative algorithm for such transformations, and gives a proof of monotonical convergence when only two modes are orthonormalized. Also, it is shown that a variety of three‐way arrays do not permit double orthonormalization. This is due to the order of the arrays, and holds regardless of the particular elements of the array. Studying three‐way orthonormality has proven useful in exploring the possibilities for simplifying the core, to guide the search for equivalent direct transformations to simplicity; see Murakami et al . ( Psychometrika 1998; 63 : 255–261) as an example. Also, it appears in various contexts of the mathematical study of three‐way analysis. Copyright © 2000 John Wiley & Sons, Ltd.