Asymmetic Multidimensional Scaling and Related Topics

Abstract

Asymmetric multidimensional scaling(MDS)is reviewed which embeds objects in a certain distance space, given a square asymmetric data matrix whose elements denote(dis-)similarities between them, along with techniques related to MDS. A body of extant asymmetric MDS's are contrasted; these have been proposed in order to overcome difficulties encountered in traditional (symmetric)MDS's. They are(1)augmented distance models, (2)non-distance models, and(3)extended distance models. The augmented distance models are those which append some quantity to some traditional distance measure or square one. The non-distance models are those which approximate some quantity such as inner product to(dis-)similarity data. The extended distance models are those which generalize traditional distance measures in a certain manner. Next, a technique closely related to asymmetric MDS, namely, asymmetric cluster analysis, is reviewed. Finally, problems which still remain to be solved and possible future developments in asymmetric MDS and related techniques are discussed.