Abstract/Details

Characterizations of pyramids and their generalizations


1998 1998

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Abstract (summary)

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.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences, Cluster analysis, Dissimilarity, Pyramids, Robinsonian
Title
Characterizations of pyramids and their generalizations
Author
Boucher, Catherine Dornback
Number of pages
116
Publication year
1998
Degree date
1998
School code
0118
Source
DAI-B 59/07, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9780591960235, 0591960230
Advisor
Janowitz, Melvin
University/institution
University of Massachusetts Amherst
University location
United States -- Massachusetts
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9841841
ProQuest document ID
304461274
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/304461274
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