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A NOTE ON THE REPRESENTATION OF ORDERED METRIC SCALING †
Author(s) -
Phillips J. P. N.
Publication year - 1971
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.2044-8317.1971.tb00469.x
Subject(s) - mathematics , scaling , simple (philosophy) , metric (unit) , solution set , representation (politics) , set (abstract data type) , ultrametric space , pure mathematics , metric space , computer science , geometry , philosophy , operations management , epistemology , politics , political science , law , economics , programming language
A unidimensional ordered metric scaling (Coombs, 1964) of a set of objects is an ordinal scaling of them and also a (partial or complete) ordinal scaling of the distances between them. The problem of representing such a scaling reduces to that of finding the general solution of a system of strict homogeneous linear inequalities. There are two possible approaches to the latter problem. The purely algebraic approach of Fourier (1890), which is a modification of the usual method (systematic elimination) for the solution of systems of linear equations, is computationally simple, but does not give a satisfactory representation of the general solution, only an inconvenient and non‐unique set of back substitutions. The coordinate‐geometric approach of Minkowski (1896), upon which Stokes' (1931) method of solution is based, gives a matrix representation of the general solution, but is computationally very laborious. However, an unpublished algorithm due to Goode (Coombs, 1964), although incomplete in its original form, turns out, when completed by the application of a theorem of Farkas (1902), to be essentially equivalent to Stokes' method, but computationally very simple, the only (slight) disadvantage being that it may give superfluous solutions, which are easily excluded by a simple check.