Analyzing proximity matrices: The assessment of internal variation in combinatorial structure

A data analysis strategy is discussed for evaluating the degree to which a subset D of a larger object set S satisfies a particular algebraic property. Based on a set measure f(D) and a proximity function on S × S, two separate evaluation tasks, referred to as confirmatory and exploratory, are considered. In a confirmatory task the subset D is identified a priori and f(D) is compared against the distribution of f(·) over all subsets containing the same number of objects. The exploratory task, on the other hand, treats f(·) as an objective function to be optimized over all subsets of a given size. Examples of these two notions include the assessment of symmetry, cluster compactness, and the extent to which D satisfies the error-free conditions for a hierarchical model or a unidimensional scale.