Multidimensional scaling with the nested hypersphere model for percentile dissimilarities

In symbolic data analysis, we deal with the higher level objects, called “concepts.” In the framework, a dissimilarity between two objects can be described as not only a single value, but also an interval, a fuzzy number, a histogram, and so on. Various clustering and multidimensional scaling methods for such complex dissimilarity data, called symbolic MDS and symbolic clustering, have been proposed. In most symbolic MDS methods, objects are represented by regions in real space (e.g. hypersphere, hyperbox and, so on). Denoeux and Masson (2000) proposed the hypersphere and hyperbox model of MDS for interval dissimilarities. In the hypersphere (hyperbox) model, objects are described by hyperspheres (hyperboxes) in real space. Moreover, Masson and Denoeux (2002) proposed the MDS method for fuzzy dissimilarities. In this method, an object is represented by nested hyperspheres which have a same center point. This constraint is very strict condition and not unsatisfied in most cases. In this paper, we derive a necessary and sufficient condition for that two hyperspheres are nested and propose a new MDS model for dissimilarities described by percentile intervals. In our method, more general nested hyperspheres, which are not necessary to have the same center point, are used for representing an object.