zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Clustering in Banach spaces. (English) Zbl 0786.62059
Novák, Vilém (ed.) et al., Fuzzy approach to reasoning and decision- making. Selected papers of the international symposium held at Bechyně, Czechoslovakia, 25-29 June 1990. Dordrecht, Prague: Kluwer Academic Publishers, Akademia. Theory Decis. Libr., Ser. D. 8, 173-184 (1992).

Summary: We extend the Hard and Fuzzy c-Means (HCM/FCM) clustering algorithms to the case where the (dis)similarity measure on pairs of numerical vectors includes two members of the Minkowski or p-norm family, viz., the p=1 and p= (or “sup”) norms. We note that a basic exchange algorithm can be used to find approximate critical points of the new objective functions. This method broadens the applications horizon of the FCM family by enabling users to match “discontinuous” multidimensional numerical data structures with similarity measures which have nonhyperelliptical topologies.

For example, data drawn from a mixture of uniform distributions have sharp or “boxy” edges; the (p=1 and p=) norms have open and closed sets that match these shapes. We illustrate the technique with a small artificial data set, and compare the results with the c-means clustering solution produced using the Euclidean (inner product) norm.

62H30Classification and discrimination; cluster analysis (statistics)
91C20Clustering (Social and behavioral sciences)
46N30Applications of functional analysis in probability theory and statistics