SOME TECHNICAL CONSIDERATIONS IN MULTIDIMENSIONAL SCALING
Abstract
In an examination of problems arising in multidimensional scaling, the multivariate analysis of interpoint distances into coordinates in m- dimensional Euclidean space is first described. Problems encountered in application of the method are then presented, and solutions developed or suggested by the writer are discussed. The first difficulty was the occurrence of a complex additive constant during the course of Messick-Abelson interations for solutions of relatively low dimensionality. A regression approach is suggested as a reasonable alternative. This approach cannot yield a complex additive constant and was successfully used where the more common method failed. A second problem arose because the common Hotelling method of extracting characteristic roots and vectors does not find the vectors in the algebraic order of the roots. This difficulty, combined with the relative slowness of the Hotelling method, recommends the use of Jacobi's method. A third problem was the occurrence of negative 'distances' obtained when the final additive constant is added to the scale values. It was conjectured that this result may arise in samples even when a proper dimensionality has been defined and the solution sought. A model sampling study of a one-dimensional system supported the conjecture. Hence, the occurrence of such negative 'distances' is not necessarily indicative of an underlying structure of higher dimensionality. Finally, it was pointed out that the occurrence of negative 'distances' is actually built into the successive intervals and comparisons approaches. Alternative assumptions were discussed.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1965
- Accession Number
- AD0623047
Entities
People
- Robert F. Boldt