Computational Analysis of the Random Components Induced by a Binary Equivalence Relation

Citation

De Balbine, Guy (1968) Computational Analysis of the Random Components Induced by a Binary Equivalence Relation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2A9K-P464. https://resolver.caltech.edu/CaltechETD:etd-11102005-145557

Abstract

The problem of partitioning into classes by means of a binary equivalence relation is investigated. Several algorithms for determining the number of components in the graph associated with a particular set of elements are constructed and compared. When the classification process operates on independently drawn samples of n distinct elements from a population, the expected number of components is shown to be obtainable recursively for a class of problems called separable; in all cases, estimates are available to reach any desired level of accuracy. Clustering models in Euclidean space are analyzed in detail and asymptotic formulas obtained to complement experiments. Conjectures concerning the general behavior of the expected number of components are presented also. Finally, several computational tools of general interest are improved significantly.

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