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{{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]].
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=== Super multidimensional scaling (SMDS) ===
An extension of MDS, known as Super MDS, incorporates both distance and angle information for improved source localization. Unlike traditional MDS, which uses only distance measurements, Super MDS processes both distance and angle-of-arrival (AOA) data algebraically (without iteration) to achieve better accuracy.<ref>{{cite conference |last1=de Abreu |first1=G. T. F. |last2=Destino |first2=G. |title=Super MDS: Source Location from Distance and Angle Information |conference=2007 IEEE Wireless Communications and Networking Conference |___location=Hong Kong, China |pages=
The method proceeds in the following steps:
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* [[ELKI]] includes two MDS implementations.
* [[MATLAB]] includes two MDS implementations (for classical (''cmdscale'') and non-classical (''mdscale'') MDS respectively).
* The [[R (programming language)|R programming language]] offers several MDS implementations, e.g. base ''cmdscale'' function, packages [https://CRAN.R-project.org/package=smacof smacof]<ref>{{Cite journal|
* [[scikit-learn]] contains function [http://scikit-learn.org/stable/modules/generated/sklearn.manifold.MDS.html sklearn.manifold.MDS].
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== Bibliography ==
{{refbegin}}
* {{cite book |last1=Cox |first1=T.F. |last2=Cox |first2=M.A.A. |editor1-last=Unwin |editor1-first=A |editor2-last=Chen |editor2-first=C |editor3-last=Hardle |editor3-first=W. K. |title=
* {{cite book |author=Coxon, Anthony P.M. |title=The User's Guide to Multidimensional Scaling. With special reference to the MDS(X) library of Computer Programs |publisher=Heinemann Educational Books |___location=London |year=1982 }}
* {{cite journal |author=Green, P. |title=Marketing applications of MDS: Assessment and outlook |journal=Journal of Marketing |volume=39 |pages=24–31 |date=January 1975 |doi=10.2307/1250799 |issue=1 |jstor=1250799 }}
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