15.1 The Problem

Multidimensional scaling (MDS) is a mathematical tool that uses proximities between objects, subjects or stimuli to produce a spatial representation of these items. The proximities are defined as any set of numbers that express the amount of similarity or dissimilarity between pairs of objects, subjects or stimuli. In contrast to the techniques considered so far, MDS does not start from the raw multivariate data matrix $\data{X}$, but from a $(n \times n)$ dissimilarity or distance matrix, $\data{D}$, with the elements $\delta_{ij}$ and $d_{ij}$ respectively. Hence, the underlying dimensionality of the data under investigation is in general not known.

MDS is a data reduction technique because it is concerned with the problem of finding a set of points in low dimension that represents the ``configuration'' of data in high dimension. The ``configuration'' in high dimension is represented by the distance or dissimilarity matrix $\data{D}$.

MDS-techniques are often used to understand how people perceive and evaluate certain signals and information. For instance, political scientists use MDS techniques to understand why political candidates are perceived by voters as being similar or dissimilar. Psychologists use MDS to understand the perceptions and evaluations of speech, colors and personality traits, among other things. Last but not least, in marketing researchers use MDS techniques to shed light on the way consumers evaluate brands and to assess the relationship between product attributes.

Figure 15.1: Metric MDS solution for the inter-city road distances. 46048 MVAMDScity1.xpl
\includegraphics[width=1\defpicwidth]{MDS-city1new.ps}

In short, the primary purpose of all MDS-techniques is to uncover structural relations or patterns in the data and to represent it in a simple geometrical model or picture. One of the aims is to determine the dimension of the model (the goal is a low-dimensional, easily interpretable model) by finding the $d$-dimensional space in which there is maximum correspondence between the observed proximities and the distances between points measured on a metric scale.

Figure 15.2: Metric MDS solution for the inter-city road distances after reflection and $90^{\circ}$ rotation. 46056 MVAMDScity2.xpl
\includegraphics[width=1\defpicwidth]{MDScity2.ps}

Multidimensional scaling based on proximities is usually referred to as metric MDS, whereas the more popular nonmetric MDS is used when the proximities are measured on an ordinal scale.

EXAMPLE 15.1   A good example of how MDS works is given by Dillon and Goldstein (1984) (Page 108). Suppose one is confronted with a map of Germany and asked to measure, with the use of a ruler and the scale of the map, some inter-city distances. Admittedly this is quite an easy exercise. However, let us now reverse the problem: One is given a set of distances, as in Table 15.1, and is asked to recreate the map itself. This is a far more difficult exercise, though it can be solved with a ruler and a compass in two dimensions. MDS is a method for solving this reverse problem in arbitrary dimensions. In Figure 15.2 you can see the graphical representation of the metric MDS solution to Table 15.1 after rotating and reflecting the points representing the cities. Note that the distances given in Table 15.1 are road distances that in general do not correspond to Euclidean distances. In real-life applications, the problems are exceedingly more complex: there are usually errors in the data and the dimensionality is rarely known in advance.

Table 15.1: Inter-city distances.
  Berlin Dresden Hamburg Koblenz Munich Rostock
Berlin 0 214 279 610 596 237
Dresden   0 492 533 496 444
Hamburg     0 520 772 140
Koblenz       0 521 687
Munich         0 771
Rostock           0


EXAMPLE 15.2   A further example is given in Table 15.2 where consumers noted their impressions of the dissimilarity of certain cars.

Table: Dissimilarities for cars.
  Audi 100 BMW 5 Citroen AX Ferrari ...
Audi 100 0 2.232 3.451 3.689 ...
BMW 5 2.232 0 5.513 3.167 ...
Citroen AX 3.451 5.513 0 6.202 ...
Ferrari 3.689 3.167 6.202 0 ...
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\ddots$


The dissimilarities in this table were in fact computed from Table B.7 as Euclidean distances

\begin{displaymath}d_{ij}=\sqrt{\sum_{l=1}^{8}(x_{il}-x_{jl})^2}.\end{displaymath}

MDS produces Figure 15.3 which shows a nonlinear relationship for all the cars in the projection. This enables us to build a nonlinear (quadratic) index with the Wartburg and the Trabant on the left and the Ferrari and the Jaguar on the right. We can construct an order or ranking of the cars based on the subjective impression of the consumers.

Figure 15.3: MDS solution on the car data. 46068 MVAmdscarm.xpl
\includegraphics[width=1\defpicwidth]{mdsexam.ps}

What does the ranking describe? The answer is given by Figure 15.4 which shows the correlation between the MDS projection and the variables. Apparently, the first MDS direction is highly correlated with service(-), value(-), design(-), sportiness(-), safety(-) and price(+). We can interpret the first direction as the price direction since a bad mark in price (``high price'') obviously corresponds with a good mark, say, in sportiness (``very sportive''). The second MDS direction is highly positively correlated with practicability. We observe from this data an almost orthogonal relationship between price and practicability.

Figure 15.4: Correlation between the MDS direction and the variables. 46075 MVAmdscarm.xpl
\includegraphics[width=1\defpicwidth]{mdscorr.ps}

In MDS a map is constructed in Euclidean space that corresponds to given distances. Which solution can we expect? The solution is determined only up to rotation, reflection and shifts. In general, if $P_{1},...,P_{n}$ with coordinates $x_{i} =
(x_{i1}, ... , x_{ip})^{\top}$ for $i = 1, ... , n$ represents a MDS solution in $p$ dimensions, then $y_{i} = \data{A}x_{i} +b$ with an orthogonal matrix $\data{A}$ and a shift vector $b$ also represents a MDS solution. A comparison of Figure 15.1 and Figure 15.2 illustrates this fact.

Solution methods that use only the rank order of the distances are termed nonmetric methods of MDS. Methods aimed at finding the points $P_{i}$ directly from a distance matrix like the one in the Table 15.2 are called metric methods.

Summary
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MDS is a set of techniques which use distances or dissimilarities to project high-dimensional data into a low-dimensional space essential in understanding respondents perceptions and evaluations for all sorts of items.
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MDS starts with a $(n \times n)$ proximity matrix $\data{D}$ consisting of dissimilarities $\delta_{i,j}$ or distances $d_{ij}$.
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MDS is an explorative technique and focuses on data reduction.
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The MDS-solution is indeterminate with respect to rotation, reflection and shifts.
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The MDS-techniques are divided into metric MDS and nonmetric MDS.