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An equivalence relation between correspondence analysis and classical metric multidimensional scaling for the recovery of Euclidean distances
Author(s) -
Carroll J. Douglas,
Kumbasar Ece,
Romney A. Kimball
Publication year - 1997
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/j.2044-8317.1997.tb01104.x
Subject(s) - multidimensional scaling , mathematics , equivalence (formal languages) , euclidean geometry , scaling , euclidean distance , metric (unit) , constant (computer programming) , matrix similarity , similarity (geometry) , transformation (genetics) , scale (ratio) , equivalence relation , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , statistics , computer science , geometry , operations management , economics , biochemistry , chemistry , physics , quantum mechanics , artificial intelligence , partial differential equation , image (mathematics) , gene , programming language
A theorem is proved showing that a special variant of correspondence analysis (CA), like classical two‐way metric multidimensional scaling (MMDS), recovers Euclidean distances (asymptotically, as a certain constant grows large) exactly, and in fact yields solutions equivalent up to a similarity transformation to MMDS, even in the case of ‘noisy’ data. Specifically, a slight modification of a use of CA for analysis of proximity data proposed independently by Gifi and by Weller & Romney, which depends on a certain additive constant, k , which should be ‘large’, is shown, as → ∞, to result in an R ‐dimensional solution equivalent, up to a scale factor, to that obtained by a certain form of MMDS. It is conjectured that this asymptotic result may account for the apparent success of the closely related ‘Gifi/Weller/Romney’ CA procedure in recovering multidimensional structure underlying proximity data.