The purpose of multidimensional scaling (MDS) is to provide a visual representation of the pattern of proximities (i.e., similarities or distances) among a set of objects. For example, given a matrix of perceived similarities between various brands of air fresheners, MDS plots the brands on a map such that those brands that are perceived to be very similar to each other are placed near each other on the map, and those brands that are perceived to be very different from each other are placed far away from each other on the map.

The definition given by dmd is right, but a more general definition of MDS is that it attempts to take data points that are defined in many dimensions and allow them to be plotted in a simpler space, without losing too much information.

There are various situations in which this can be useful. As an example, if an IQ test has 10 sections, we might expect that the results in each test are really being driven by a couple of underlying variables (such as vocabulary and numeracy). We can measure the scores of 100 volunteers, and then produce a correlation matrix for the ten sections - i.e. ten dimensions - which characterises the similarities between the dimensions. MDS might then show that scaling down to two dimensions still explains most of the variance, and that the tests tend to cluster depending on whether they are semantic or arithmetic. On the other hand, it might show that the data can't easily be summarised - but this writeup isn't a critique of IQ tests (I hope there's a good one somewhere though...).

MDS has a very wide range of applications, including analysis of morphometric results in zoology, fMRI data, and many kinds of sociological studies.

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