From Joint Graphical Display to Bi-Modal Clustering: [2] Dual Space Versus Total Space

Abstract

There are two ways to use dual scaling results of the condensed response-pattern matrix, one in terms of dual space and the other in total space. The main task of this paper is to explore the differences in dealing with dual space and total space, and it is hoped that the study will offer some insights into the characteristics of these two types of space for cluster analysis. Bi-modal clustering is defined as a method of cluster analysis to identify only between-set clusters (i.e., clusters consisting of both rows and columns of the contingency table), and the surest way to find such clusters is to use the between-set distance matrix as an input for clustering. Since this input matrix is typically rectangular, most traditional and currently popular methods of cluster analysis cannot handle rectangular distance matrices. Thus, the current study has chosen an intuitive method, called exploratory clustering, as a start. Since there are two ways to compute the between-set distance matrix, namely in dual space and total space, the current study aims at demonstrating the differences in clusters formed by two approaches. After examining the results, the study concludes that bi-modal clustering should be carried out in dual space. This conclusion is discussed in the current paper, and it is hoped that our conclusion can be supported by researchers. This conclusion also means that the initial object of total information analysis or comprehensive dual scaling by Nishisato and Clavel can be justified. This connection is also discussed in the current paper.