Theory of Multidimensional Scaling

It is difficult to give a precise definition of MDS, because some people use the term for a very specific class of techniques while others use it in a much more general sense. Consequently it makes sense to distinguish between MDS in the broad sense and MDS in the narrow sense. MDS in the broad sense inc ludes various forms of cluster analysis and of linear multivariate analysis, MDS in the narrow sense represents dissimilarity data in a low-dimensional space. People who prefer the broad-sense definition want to emphasize the close relationships of clustering and scaling techniques. Of course this does not imply that they are not aware of the important differences. Clustering techniques fit a non-dimensional discrete structure to dissimilarity data, narrow-sense MDS fits a continuous dimensional structure. But both types of technique can be formalized as representing distance-like data in a particular metric space by minimizing some kind of loss function. The difference, then, is in the choice of the target metric space, the structure of the two problems is very much alike. The paper pioneering this point of view is Hartigan's (1967). In fact Hartigan proceeds the other way around, he takes clustering as the starting point and lets clustering in the broad sense include narrow-sense MDS. The same starting point and the same order are more or less apparent in the important review papers of Cormack (1971) and Sibson (1972). An influential broad-sense paper that uses narrow-sense MDS as the starting point is that of Carroll (1976). In fact Carroll discusses techniques that explicitly combine aspects of clustering and narrow-sense MDS, and find mixed discrete/continuous representations. In a very recent paper Carroll and Arabie (1980) propose a useful taxonomy of MDS data and methods which is very broad indeed. Investigators who are less interested in formal similarities of methods and more interested in substantial differences in models have naturally emphasized the choice of the space, and consequently the differences between clustering and narrow-sense MDS. Of course detailed comparison of the two classes of techniques already presupposes a common framework. This is obvious