Multidimensional scaling

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The technique is also used in marketing, see Multidimensional scaling in marketing

Multidimensional scaling (MDS) is a set of related statistical techniques often used in data visualisation for exploring similarities or dissimilarities in data. An MDS algorithm starts with a matrix of item-item similarities, then assigns a location of each item in a low-dimensional space, suitable for graphing or 3D visualisation.

Applications include scientific visualisation and data mining in fields such as cognitive science, psychophysics, psychometrics and ecology.

MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:

• Classical multidimensional scaling -- assumes the input matrix is just an item-item distance matrix. Analogous to Principal components analysis, an eigenvector problem is solved to find the locations that minimize distortions to the distance matrix.
• Metric multidimensional scaling -- A superset of classical MDS that assumes a known parametric relationship between the elements of the item-item dissimilarity matrix and the Euclidean distance between the items.
• Generalized multidimensional scaling (GMDS) -- A superset of metric MDS that allows for the target distances to be non-Euclidean.
• Non-metric multidimensional scaling -- In contrast to metric MDS, non-metric MDS both finds a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distance between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression.

References

• Torgerson, W. S. (1958). Theory & Methods of Scaling. New York: Wiley.
• Kruskal, J. B., and Wish, M. (1978), Multidimensional Scaling, Sage University Paper series on Quantitative Application in the Social Sciences, 07-011. Beverly Hills and London: Sage Publications.
• Cox, M.F., Cox, M.A.A., (2001), Multidimensional Scaling, Chapman and Hall.
• Coxon, Anthony P.M. (1982): "The User's Guide to Multidimensional Scaling. With special reference to the MDS(X) library of Computer Programs." London: Heinemann Educational Books.
• Bronstein, A. M, Bronstein, M.M, and Kimmel, R. (2006), Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching, Proc. National Academy of Sciences (PNAS), Vol. 103/5, pp. 1168-1172.

External links

• An elementary introduction to multidimensional scaling
• Evaluation of multidimensional scaling algorithms
• NewMDSX: Multidimensional Scaling Software
• PERMAP, free software for making multidimensional analyses
• Relational Perspective Map: MDS on closed manifolds
• Multigrid MDS
• MDS pagees:Escalado multidimensional

it:Scaling multidimensionale de:Multidimensionale Skalierung zh:多维标度