Faithful consensus methods for n-trees
In this article are introduced and compared some new consensus methods for n-trees. Let the system of classifications of a fixed finite set be given in the form of the system \({\mathcal T}^ k_ n\) (tree primary classifications are given by hierarchical clusterings). A consensus rule is a function from \({\mathcal T}^ k_ n\) to \({\mathcal T}_ n\) (tree representing resulting clustering). A consensus rule is said to be faithful if a condition of betweenness is fulfilled. That means for any choice of the sets \(X_ i\) from the tree \(T_ i\) there exists some cluster B satisfying: \(\cap^{k}_{i=1}X_ i\subseteq B\subseteq \cup^{k}_{i=1}X_ i.\) Besides the betweenness axiom there are other reasonable properties of consensus rules as neutrality, monotonicity, Pareto axiom etc. A number of examples illustrates various consensus rules satisfying these axioms. The main result of this paper is the following: Only trivial consensus rules are simultaneously faithful and neutral.
scientific article; zbMATH DE number 3858218