Constructing Hierarchical Set Systems

Abstract.

In this note, it is shown that by applying ranking procedures to data that allow, for any three objects a₁, a₂, b in a collection X of objects of interest, to make consistent decisions about which of the two objects a₁ or a₂ is more similar to b, a family of cluster systems $\mathcal{A}^{(k)} (k = 0,1, \ldots )$ can be constructed that start with the associated Apresjan Hierarchy and keep growing until, for k = #X−1, the full set $\mathcal{P}(X)$ of all subsets of X is reached. Various ideas regarding canonical modifications of the similarity values so that these cluster systems contain as many clusters as possible for small values of k (and in particular for k := 0) and/or are rooted at a specific element in X, possible applications, e.g. concerning (i) the comparison of distinct dissimilarity data defined on the same set X or (ii) diversity optimization, and new tasks arising in ranking statistics are also discussed.