Classical Multidimensional Scaling

Abstract

Classical multidimensional scaling (CMDS) is a technique that displays the structure of distance-like data as a geometrical picture. It is a member of the family of MDS methods. The input for an MDS algorithm usually is not an object data set, but the similarities of a set of objects that may not be digitalized. The input distance matrix of CMDS is of Euclidean-type. There exists an r-dimensional vector set H such that the Euclidean distance matrix of H is equal to the input one. The set H is called a configuration of the input matrix. In the case that the dimension of the set H is too high to be visualized, we then need to reduce the dimension of the configuration to 2 or 3 for visualization. Sometimes, a little higher dimension is acceptable. CMDS is equivalent to PCA when the input of CMDS is a data set. The data model in CMDS is linked to a complete weighed graph, in which the nodes are objects and the weights are the similarities between the objects. This chapter is organized as follows. In Section 6.1, we introduce multidimensional scaling. In Section 6.2, we discuss the relation between Euclidean distance matrices and Gram matrices. The description of CMDS is in Section 6.3. The algorithm of CMDS is presented in Section 6.4.