Data Error Reduction in Nonmetric Multidimensional Scaling

The aim of Multidimensional Scaling (MDS) is to search for a geometrical pattern of n points, on the basis of experimental dissimilarities data between these points. For nonmetric MDS, one may use ordinal data as dissimilarities. In general, as these dissimilarities are empirical, they may be errorful. Thus, in order to obtain better scaling solutions, it is of great interest to reduce error in experimental data. In this paper, we suggest an approach to reduce the amount of error contained in interpoints dissimilarities evaluated in an ordinal scale. This approach is illustrated both on simulated and real data coming from 12 stimuli and 30 judges. 1.1 Introduction and Deenitions Multidimensional Scaling (MDS) is a generic name for techniques and algorithms from the area of multivariate data analysis, that start with matrixes of dissimilarities between objects in order to generate their connguration in a low dimensional geometrical space. Suppose a set of n objects is under consideration and between each pair of objects i and j, there is a measurement ij of dissimilarity. Multidimensional scaling is the search for a low dimensional space, usually Euclidean, in which points in the space represent the objects, and such that the distances between the points in the space, fd ij g, match as well as possible the original dissimilarities f ij g CC94]. There are two main categories of MDS methods : the metric and the non-metric ones. The diierence between these two families of MDS methods lies in the kind of input dissimilarities they used. For metric MDS methods Tor52] the dissimilarities f ij g are used in a quantitative metric sense, that means the dissimilarities are such that the value of each fd ij g approximates