10. Circular Anti-Robinson Matrices for Symmetric Proximity Data
Society for Industrial and Applied Mathematics eBooks · 2006
In the approximation of a proximity matrix P by one that is row/column reorderable to an AR form, the interpretation of the fitted matrix in general had to be carried out by identifying a set of subsets through an increasing threshold variable; each of the subsets contained objects that were contiguous with respect to a given linear ordering along a continuum and had a diameter defined by the maximum fitted value within the subset. To provide a further representation depicting the fitted values as lengths of paths in a graph, an approximation was sought that satisfied the additional constraints of an SAR matrix; still, the subsets thus identified had to contain objects contiguous with respect to a linear ordering. As one possible generalization of both the AR and SAR constraints, we can define what will be called circular anti-Robinson (CAR) and circular strongly anti-Robinson (CSAR) forms that allow the subsets identified from increasing a threshold variable to be contiguous with respect to a circular ordering of the objects around a closed continuum. Approximation matrices that are row/column reorderable to display an AR or SAR form, respectively, will also be (trivially) row/column reorderable to display what is formally characterized below as a CAR or a CSAR form but not conversely. (Historically, there is a large literature on the possibility of circular structures emerging from and being identifiable in a given proximity matrix. For a variety of references, the reader is referred to the American Psychological Association sponsored volume edited by Plutchik and Conte (1997), the discussion of metric circular unidimensional scaling (CUS) in Part I, Chapter 3, and in Hubert, Arabie, and Meulman (1997). The extension of CAR forms to those that are also CSAR, however, has apparently not been a topic discussed in the literature before the appearance of Hubert, Arabie, and Meulman (1998); this latter source forms the basis for much of the present chapter.)