# 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.

## Keywords

Distance Matrix Multidimensional Scaling Similarity Matrice Euclidean Distance Matrix Virtual Point## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Young, F.W.: Encyclopedia of Statistical Sciences, vol. 5, chap. Multidimensional scaling. John Wiley & Sons, Inc (1985).Google Scholar
- [2]Torgerson, W.S.: Multidimensional scaling I. Theory and method. Psychometrika 17, 401–419 (1952).MathSciNetzbMATHCrossRefGoogle Scholar
- [3]Young, F.W., Hamer, R.M.: Theory and Applications of Multidimensional Scaling. Eribaum Associates, Hillsdale, NJ. (1994).Google Scholar
- [4]Shepard, R.: The Analysis of proximities: Multidimensional scaling with an unknown distance function I. Psychometrika 27, 125–140 (1962).MathSciNetzbMATHCrossRefGoogle Scholar
- [5]Shepard, R.: The Analysis of proximities: Multidimensional scaling with an unknown distance function II. Psychometrika 27, 219–246 (1962).MathSciNetCrossRefGoogle Scholar
- [6]Young, G., Householder, A.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938).zbMATHCrossRefGoogle Scholar
- [7]Borg, I., Groenen, P.: Modern multidimensional scaling: Theory and applications. Springer-Verlag, New York (1997).zbMATHGoogle Scholar
- [8]Cox, T.F., Cox, M.A.A.: Multidimensional Scaling. Chapman & Hall, London, New York (1994).zbMATHGoogle Scholar