The Geometry of a Segment in a Family of Cluster Partitions of a Finite Set
The paper considers the metric space of a family of all partitions of a finite set into non-empty disjoint subsets in the cluster distance proposed by the author in one of the previous papers. The relation between this space structure and the partial order generated by the inclusion on a family of partitions is investigated. It is found out that the segment it is coordinated with the partial order when the segment is determined in such space with the boundaries of A and B as the set of those C that the sum of the distances from it to A and B is equal to the distance from A to B. This is expressed by the fact that the distance between partitions corresponds to the smallest path length between them along the chains in the lattice of the corresponding partial order. Nevertheless, the segment defined this way has significant differences from ordinary segments in vector spaces. Therefore, it is not possible to completely carry out the analogy with theorems of usual geometry. The obtained results can be used in the construction of new algorithms for cluster analysis, as well as for finding the exact probability distributions of the distance between, in some sense, a correct partition and a partition constructed from the data of real observations.
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