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Characterizations of pyramids and their generalizations
Abstract
Cluster Analysis is a collection of techniques whose goals are to try and suggest possible internal structures of a data set. It is a subfield of exploratory data analysis in which the goal is to find a starting point to investigate some collection of objects. A clustering technique takes a finite data set E with finitely many attributes or a collection of measurements called a dissimilarity coefficient and produces a single classification or a nested sequence of classifications of E. When one forms a nested sequence of partitions on the given set it is easily visualized as a hierarchy. Pyramids, developed by Diday (12), allow visual representation of output that has some overlap. It is a well known fact that weakly indexed pyramids are in one-to-one correspondence with definite Robinsonian dissimilarity coefficients. Pyramids allow some overlap between clusters. One drawback to pyramidal representations is the requirement that one must impose a linear order on the underlying set to be clustered. It will be shown that by examining a dissimilarity coefficient one is able to determine its compatible linear orders, if any, using the consecutive ones property. A generalization of pyramids, pseudo-pyramids, will be introduced. The concepts of weakly indexed and indexed pseudo-pyramids are constructed. Pyramids and their generalizations will be placed in the ordinal model developed by Janowitz (25). Characterizations of pyramids and their generalizations are given from set-theoretical, graph-theoretical, and lattice-theoretical viewpoints. In particular, a characterization of indexed pseudo-pyramids with respect to a collection of planar lattices will be introduced. Generalizations of dissimilarity coefficients called pseudo-dissimilarity coefficients will be given. A bijection between indexed (weakly indexed) pseudo-pyramids and strongly Robinsonian (Robinsonian) pseudo-dissimilarities is possible. This generalization removes the necessity of the minimal value on a dissimilarity being 0. Also, the output of a clustering technique using a pseudo-dissimilarity need not be reflexive at each level. In other words, it is not necessary to have all singleton subsets in the classifications.
Subject Area
Mathematics
Recommended Citation
Boucher, Catherine Dornback, "Characterizations of pyramids and their generalizations" (1998). Doctoral Dissertations Available from Proquest. AAI9841841.
https://scholarworks.umass.edu/dissertations/AAI9841841