Homogeneity Analysis Using Absolute Deviations

Homogeneity analysis is a technique for making graphical representations of categorical multivariate data sets. Such data sets can also be represented by the adjacency matrix of a bipartite graph. Homogeneity analysis optimizes a weighted least squares criterion and the optimal graph drawing is computed by an alternating least squares algorithm. Heiser (1987) looked at homogeneity analysis under a weighted least absolute deviations criterion. In this paper, we take a closer look at the mathematical structure of this problem and show that the graph drawings are created by reciprocal computation of multivariate medians. Several algorithms for computing the solution are investigated and applications to actual data suggest that the resulting p-dimensional drawings (p >= 2) are degenerate, in the sense that all object points are clustered in p + 1 locations. We also examine some variations of the criterion used and conclude that the generate solutions observed are a consequence of the normalization contraint employed in this class of problems