Multidimensional scaling

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of ${\textstyle n}$ objects in a set into a configuration of ${\textstyle n}$ points mapped into an abstract Cartesian space.^[1]

More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction.

Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot.^[2]

Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis.^[3]

^ Mead, A (1992). "Review of the Development of Multidimensional Scaling Methods". Journal of the Royal Statistical Society. Series D (The Statistician). 41 (1): 27–39. JSTOR 2348634. Abstract. Multidimensional scaling methods are now a common statistical tool in psychophysics and sensory analysis. The development of these methods is charted, from the original research of Torgerson (metric scaling), Shepard and Kruskal (non-metric scaling) through individual differences scaling and the maximum likelihood methods proposed by Ramsay.
^ Borg, I.; Groenen, P. (2005). Modern Multidimensional Scaling: theory and applications (2nd ed.). New York: Springer-Verlag. pp. 207–212. ISBN 978-0-387-94845-4.
^ Genest, Christian; Nešlehová, Johanna G.; Ramsay, James O. (2014). "A Conversation with James O. Ramsay". International Statistical Review / Revue Internationale de Statistique. 82 (2): 161–183. JSTOR 43299752. Retrieved 30 June 2021.

[MS_history-1] Mead, A (1992). "Review of the Development of Multidimensional Scaling Methods". Journal of the Royal Statistical Society. Series D (The Statistician). 41 (1): 27–39. JSTOR 2348634. Abstract. Multidimensional scaling methods are now a common statistical tool in psychophysics and sensory analysis. The development of these methods is charted, from the original research of Torgerson (metric scaling), Shepard and Kruskal (non-metric scaling) through individual differences scaling and the maximum likelihood methods proposed by Ramsay.

[borg-2] Borg, I.; Groenen, P. (2005). Modern Multidimensional Scaling: theory and applications (2nd ed.). New York: Springer-Verlag. pp. 207–212. ISBN 978-0-387-94845-4.

[jsto_ACon-3] Genest, Christian; Nešlehová, Johanna G.; Ramsay, James O. (2014). "A Conversation with James O. Ramsay". International Statistical Review / Revue Internationale de Statistique. 82 (2): 161–183. JSTOR 43299752. Retrieved 30 June 2021.

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