Multidimensional scaling: Difference between revisions

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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 webjournal

| title = A Conversation with James O. Ramsay on JSTOR

| journal = International Statistical Review / Revue Internationale de Statistique

| author =

| workjstor = jstor.org43299752

| date =

| 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

| authorissue = 2

| pages = 161–183

}}</ref>

.

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===Generalized multidimensional scaling (GMD)===

{{main|Generalized multidimensional scaling}}

An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another.<ref name="bron">{{cite journal |vauthors=Bronstein AM, Bronstein MM, Kimmel R |title=Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=103 |issue=5 |pages=1168–72 |date=January 2006 |pmid=16432211 |pmc=1360551 |doi=10.1073/pnas.0508601103 |bibcode=2006PNAS..103.1168B |doi-access=free }}</ref>

==Details==

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# '''Formulating the problem''' – What variables do you want to compare? How many variables do you want to compare? What purpose is the study to be used for?

# '''Obtaining input data''' – For example, :- Respondents are asked a series of questions. For each product pair, they are asked to rate similarity (usually on a 7-point [[Likert scale]] from very similar to very dissimilar). The first question could be for Coke/Pepsi for example, the next for Coke/Hires rootbeer, the next for Pepsi/Dr Pepper, the next for Dr Pepper/Hires rootbeer, etc. The number of questions is a function of the number of brands and can be calculated as <math>Q = N (N - 1) / 2</math> where ''Q'' is the number of questions and ''N'' is the number of brands. This approach is referred to as the “Perception data : direct approach”. There are two other approaches. There is the “Perception data : derived approach” in which products are decomposed into attributes that are rated on a [[semantic differential]] scale. The other is the “Preference data approach” in which respondents are asked their preference rather than similarity.

# '''Running the MDS statistical program''' – Software for running the procedure is available in many statistical software packages. Often there is a choice between Metric MDS (which deals with interval or ratio level data), and Nonmetric MDS<ref>{{cite journal|first1=J. B.|last1=Kruskal| author-link=Joseph Kruskal| title=Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis|journal=Psychometrika|pages=1–27| volume=29| issue=1| year=1964| doi=10.1007/BF02289565|s2cid=48165675}}</ref> (which deals with ordinal data).

# '''Decide number of dimensions''' – The researcher must decide on the number of dimensions they want the computer to create. Interpretability of the MDS solution is often important, and lower dimensional solutions will typically be easier to interpret and visualize. However, dimension selection is also an issue of balancing underfitting and overfitting. Lower dimensional solutions may underfit by leaving out important dimensions of the dissimilarity data. Higher dimensional solutions may overfit to noise in the dissimilarity measurements. Model selection tools like AIC/BIC, Bayes factors, or cross-validation can thus be useful to select the dimensionality that balances underfitting and overfitting.

# '''Mapping the results and defining the dimensions''' – The statistical program (or a related module) will map the results. The map will plot each product (usually in two-dimensional space). The proximity of products to each other indicate either how similar they are or how preferred they are, depending on which approach was used. How the dimensions of the embedding actually correspond to dimensions of system behavior, however, are not necessarily obvious. Here, a subjective judgment about the correspondence can be made (see [[perceptual mapping]]).

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== Bibliography ==

{{refbegin}}

* {{cite book |lastlast1=Cox |firstfirst1=T.F. |last2=Cox |first2=M.A.A. |title=Multidimensional Scaling |publisher=Chapman and Hall |year=2001 }}

* {{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 }}