Editing Multidimensional scaling (section)

{{Short description|Set of related ordination techniques used in information visualization}}
[[File:RecentVotes.svg|thumb|400px|An example of classical multidimensional scaling applied to voting patterns in the [[United States House of Representatives]]. Each blue dot represents one Democratic member of the House, and each red dot one Republican.]]

{{Data Visualization}}

'''Multidimensional scaling''' ('''MDS''') is a means of visualizing the level of [[Similarity measure|similarity]] of individual cases of a data set.  MDS is used to translate distances between each pair of <math display="inline"> n </math> objects in a set into a configuration of <math display="inline"> n </math> points mapped into an abstract [[Cartesian coordinate system|Cartesian space]].<ref name="MS_history">{{cite journal |last= Mead|first=A  |date= 1992|title= Review of the Development of Multidimensional Scaling Methods |journal= Journal of the Royal Statistical Society. Series D (The Statistician)|volume= 41|issue=1 |pages=27–39 |doi=10.2307/2348634 |quote= 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. |jstor=2348634  }}</ref>

More technically, MDS refers to a set of related [[Ordination (statistics)|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''-[[dimension]]al 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 plots|scatter plot]].<ref name="borg">{{cite book |last=Borg |first=I. |author2=Groenen, P. |author2-link=Patrick Groenen |title=Modern Multidimensional Scaling: theory and applications |publisher=Springer-Verlag |location=New York |year=2005 |pages=207–212 |edition=2nd |isbn=978-0-387-94845-4 }}</ref>

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]].<ref name="jsto_ACon">{{Cite journal
| title = A Conversation with James O. Ramsay
| journal = International Statistical Review / Revue Internationale de Statistique
| jstor = 43299752
| access-date = 30 June 2021
| url = https://www.jstor.org/stable/43299752
| quote = 
| last1 = Genest
| first1 = Christian
| last2 = Nešlehová
| first2 = Johanna G.
| last3 = Ramsay
| first3 = James O.
| year = 2014
| volume = 82
| issue = 2
| pages = 161–183
}}</ref>