Classification of normalized cluster methods in an order theoretic model. (English) Zbl 0742.92025
Cluster analysis is a general method for classifying a finite collection of objects with respect to their dissimilarity. The Janowitz model of cluster method is a formalization based on isotope mappings of posets.
In the present paper, normalized cluster methods are investigated as a special class of residuated mappings. It is shown that there are eleven essential classes of normalized clustering methods with respect to Galois connection applied to residuated maps. In the concluding part a practical interpretation of the theoretical results is shown.
In the present paper, normalized cluster methods are investigated as a special class of residuated mappings. It is shown that there are eleven essential classes of normalized clustering methods with respect to Galois connection applied to residuated maps. In the concluding part a practical interpretation of the theoretical results is shown.
Reviewer: L.Niepel (Bratislava)
MSC:
| 91C20 | Clustering in the social and behavioral sciences |
| 06A15 | Galois correspondences, closure operators (in relation to ordered sets) |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
Keywords:
order model; dissimilarity; Janowitz model; isotope mappings of posets; normalized cluster methods; Galois connection; residuated mapsReferences:
| [1] | Baulieu, F. B., Order theoretic classification of cluster methods, Doctoral Dissertation (1981), University of Massachusetts: University of Massachusetts Amherst, MA |
| [2] | Baulieu, F. B., Order theoretic classification of monotone equivariant cluster methods, Algebra Universalis, 20, 351-367 (1985) · Zbl 0576.06005 |
| [3] | Blyth, T. S.; Janowitz, M. F., Residuation Theory (1972), Pergamon: Pergamon London · Zbl 0301.06001 |
| [4] | Janowitz, M. F., An order theoretic model for cluster analysis, SIAM J. Appl. Math., 34, 55-72 (1978) · Zbl 0379.62050 |
| [5] | Janowitz, M. F., Semiflat L-cluster methods, Discrete Math., 21, 47-60 (1978) · Zbl 0389.92001 |
| [6] | Janowitz, M. F., Monotone equivariant cluster analysis, SIAM J. Appl. Math., 37, 148-165 (1979) · Zbl 0419.62056 |
| [7] | Jardine, N.; Sibson, R., Mathematical Taxonomy (1971), Wiley: Wiley New York · Zbl 0322.62065 |
| [8] | Matula, D. M., Graph theoretic techniques for cluster analysis algorithms, Tech. Rept. CS 76015 (1976), Southern Methodist Univ.,: Southern Methodist Univ., Dallas, TX |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
