multidimensional scaling

Multidimensional scaling (MDS) are a set of related statistical technique often used in data visualisation. 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. The technique is also used in marketing (see Multidimensional scaling (in marketing)) Different MDS algorithms that fall into a taxonomy, depending on the meaning of the input matrix:

Classical multidimensional scaling --- In classical MDS, the input matrix is assumed to be exactly an item-item distance matrix. Analogous to Principal Components Analysis, an eigenvector problem is solved to find the locations that minimize
Metric multidimensional scaling --- A superset of classical MDS, metric MDS assumes that there is a known parametric relationship between the elements of the item-item dissimilarity matrix and the Euclidean distance between the items.
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

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.

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