Fuzzy multidimensional scaling

Multidimensional scaling (MDS) is a data analysis technique for representing measurements of (dis)similarity among pairs of objects as distances between points in a low-dimensional space. MDS methods differ mainly according to the distance model used to scale the proximities. The most usual model is the Euclidean one, although a spherical model is often preferred to represent correlation measurements. These two distance models are extended to the case where dissimilarities are expressed as intervals or fuzzy numbers. Each object is then no longer represented by a point but by a crisp or a fuzzy region in the chosen space. To determine these regions, two algorithms are proposed and illustrated using typical data sets. Experiments demonstrate the ability of the methods to represent both the structure and the vagueness of dissimilarity measurements.