×

Fuzzy partition models for fitting a set of partitions. (English) Zbl 1293.62243

Summary: Methodology is described for fitting a fuzzy consensus partition to a set of partitions of the same set of objects. Three models defining median partitions are described: two of them are obtained from a least-squares fit of a set of membership functions, and the third (proposed by Pittau and Vichi) is acquired from a least-squares fit of a set of joint membership functions. The models are illustrated by application to both a set of hard partitions and a set of fuzzy partitions and comparisons are made between them and an alternative approach to obtaining a consensus fuzzy partition proposed by Sato and Sato; a discussion is given of some interesting differences in the results.

MSC:

62P15 Applications of statistics to psychology
62H30 Classification and discrimination; cluster analysis (statistical aspects)
03E72 Theory of fuzzy sets, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahuja, R.K., Magnanti, T.L., & Orlin, J.B. (1993).Network flows: Theory, algorithms, and applications. Englewood Cliffs, NJ: Prentice-Hall. · Zbl 1201.90001
[2] Barthélemy, J.-P., & Leclerc, B. (1995). The median procedure for partitions. In I.J. Cox, P. Hansen & B. Julesz (Eds.),Partitioning data sets (DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 3–34). Providence, RI: American Mathematical Society. · Zbl 0814.62031
[3] Barthélemy, J.-P., Leclerc, B., & Monjardet, B. (1986). On the use of ordered sets in problems of comparison and consensus of classifications.Journal of Classification, 3, 187–224. · Zbl 0647.62056 · doi:10.1007/BF01894188
[4] Bezdek, J.C. (1974). Numerical taxonomy with fuzzy sets.Journal of Mathematical Biology, 1, 57–71. · Zbl 0403.62039 · doi:10.1007/BF02339490
[5] Bezdek, J.C. (1981).Pattern recognition with fuzzy objective functions. New York, NY: Plenum Press. · Zbl 0503.68069
[6] Bezdek, J.C. (1987). Some non-standard clustering algorithms. In P. Legendre & L. Legendre (Eds.),Developments in numerical ecology (pp. 225–287). Berlin, Germany: Springer.
[7] Bezdek, J.C., Windham, M.P., & Ehrlich, R. (1980). Statistical parameters of cluster validity functionals.International Journal of Computer and Information Sciences, 9, 323–336. · Zbl 0468.62051 · doi:10.1007/BF00978164
[8] Caliński, T., & Harabasz, J. (1974). A dendrite method for cluster analysis.Communications in Statistics, 3, 1–27. · Zbl 0273.62010 · doi:10.1080/03610928308827180
[9] Carpento, G., Martello, S., & Toth, P. (1988). Algorithms and codes for the assignment problem. In B. Simeone, P. Toth, G. Gallo, F. Maffioli, & S. Pallottino (Eds.). Fortan codes for network optimization.Annals of Operations Research, 13, 193–224.
[10] Daskin, M.S. (1995).Network and discrete location models, algorithms and applications. New York, NY: Wiley. · Zbl 0870.90076
[11] Day, W.H.E. (1986). (Ed.). Special issue on consensus classifications.Journal of Classification, 3(2).
[12] Dunn, J.C. (1974). A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters.Journal of Cybernetics, 3(3), 32–57. · Zbl 0291.68033 · doi:10.1080/01969727308546046
[13] Fisher, W.D. (1958). On grouping for maximum homogeneity.Journal of the American Statistical Association, 53, 789–798. · Zbl 0084.35904 · doi:10.1080/01621459.1958.10501479
[14] Gill, P.E., Murray, W., & Wright, M.H. (1981).Practical optimization. London, UK: Academic Press. · Zbl 0503.90062
[15] Gordon, A.D. (1980). On the assessment and comparison of classifications. In R. Tomassone (Ed.),Analyse de données et informatique (pp.149–160). Le Chesnay, France: INRIA. · Zbl 0446.62054
[16] Gordon, A.D. (1999).Classification (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC.
[17] Gordon, A.D., & Vichi, M. (1998). Partitions of partitions.Journal of Classification, 15, 265–285. · Zbl 0912.62072 · doi:10.1007/s003579900034
[18] Jain, A.K., & Dubes, R.C. (1988).Algorithms for clustering data. Englewood Cliffs, NJ: Prentice-Hall. · Zbl 0665.62061
[19] Kaufman, L., & Rousseeuw, P.J. (1990).Finding groups in data: An introduction to cluster analysis. New York, NY: Wiley. · Zbl 1345.62009
[20] Leclerc, B. (1998). Consensus of classifications: The case of trees. In A. Rizzi, M. Vichi & H.-H. Bock (Eds.),Advances in data science and classification (pp. 81–90). Berlin, Germany: Springer. · Zbl 1051.91530
[21] Marcotorchino, F., & Michaud, P. (1982). Agrégation de similarités en classification automatique.Revue de Statistique Appliquée, 30, 21–44. · Zbl 0537.62006
[22] Mirkin, B. (1996).Mathematical classification and clustering. Dordrecht, Netherlands: Kluwer. · Zbl 0874.90198
[23] Pittau, M.G., & Vichi, M. (1998).Fitting a fuzzy partition to a set of fuzzy partitions (Working paper 6, DMQTE). Manuscript submitted for publication.
[24] Powell, M.J.D. (1983). Variable metric methods for constrained optimization. In A. Bachen, M. Grötschel & B. Korte (Eds.),Mathematical programming: The state of the art (pp.288–311). Berlin, Germany: Springer. · Zbl 0536.90076
[25] Régnier, S. (1965). Sur quelques aspects mathématiques des problèmes de classification automatique.International Computation Centre Bulletin, 4, 175–191.
[26] Rosenberg, S. (1982). The method of sorting in multivariate research with applications selected from cognitive psychology and person perception. In N. Hirschberg & L. G. Humphreys (Eds.),Multivariate applications in the social sciences (pp. 117–142). Hillsdale, NJ: Erlbaum.
[27] Rosenberg, S., & Kim, M.P. (1975). The method of sorting as a data-gathering procedure in multivariate research.Multivariate Behavioral Research, 10, 489–502. · doi:10.1207/s15327906mbr1004_7
[28] Roubens, M. (1978). Pattern classification problems and fuzzy sets.Fuzzy Sets and Systems, 1, 239–253. · Zbl 0435.68064 · doi:10.1016/0165-0114(78)90016-7
[29] Sato, M., & Sato, Y. (1994). On a multicriteria fuzzy clustering method for 3-way data.International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 127–142. · Zbl 06006913 · doi:10.1142/S0218488594000122
[30] Trauwaert, E., Kaufman, L., & Rousseeuw, P. (1991). Fuzzy clustering algorithms based on the maximum likelihood principle.Fuzzy Sets and Systems, 42, 213–227. · Zbl 0741.62065 · doi:10.1016/0165-0114(91)90147-I
[31] Windham, M.P. (1981). Cluster validity for fuzzy clustering algorithms.Fuzzy Sets and Systems, 5, 177–185. · Zbl 0456.62053 · doi:10.1016/0165-0114(81)90015-4
[32] Windham, M.P. (1982). Cluster validity for the fuzzy c-means clustering algorithm.IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-4, 357–363. · doi:10.1109/TPAMI.1982.4767266
[33] Xie, X.L., & Beni, G. (1991). A validity measure for fuzzy clustering.IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 841–847. · Zbl 05112080 · doi:10.1109/34.85677
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.