Abstract
Given a profile (family) Π of partitions of a set of objects or items X, we try to establish a consensus partition containing a maximum number of joined or separated pairs in X that are also joined or separated in the profile. To do so, we define a score function, S Π associated to any partition on X. Consensus partitions for Π are those maximizing this function. Therefore, these consensus partitions have the median property for the profile and the symmetric difference distance. This optimization problem can be solved, in certain cases, by integer linear programming. We define a polynomial heuristic which can be applied to partitions on a large set of items. In cases where an optimal solution can be computed, we show that the partitions built by this algorithm are very close to the optimum which is reached in practically all the cases, except for some sets of bipartitions.
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References
Barthélemy JP, Monjardet B (1981) The median procedure in cluster analysis and social choice. Theory Math Soc Sci 1:235–268
Barthélemy JP, Leclerc B (1995) The median procedure for partitions. DIMACS Ser Discret Math Theor Compu Sci 19: 3–34
Brenac T (2002) Contribution des méthodes de partition centrale à la mise en evidence expérimentale de catégories cognitives. INRETS, Arcueil
Celeux G, Diday E, Govaert G, Lechevalier Y, Ralambondrainy H (1989) Classification automatique des Données. Dunod, Paris
Charon I, Denoeud L, Guénoche A, Hudry O (2006) Maximum transfer distance between partitions. J Classif 23(1): 103–121
Day W (1981) The complexity of computing metric distances between partitions. Math Soc Sci 1: 269–287
Denoeud L (2008) Transfer distance between partitions. Adv Data Anal Classif 2: 279–294
Dubois D (1991) Sémantique et cognition—Catégories, prototypes et typicalité. Edition du CNRS, Paris
Filkov V, Skiena S (2003) Integrating microarray data by consensus clustering. In: Proceedings of International Conference on Tools with Artificial Intelligence (ICTAI), pp 418–425
Grötschel M, Wakabayashi Y (1989) A cutting plan algorithm for a clustering problem. Math Program 45: 59–96
Guénoche A (2010) Bootstrap clustering for graph partitioning. MARAMI proceedings (submitted)
Gusfield D (2002) Partition-distance. Inf Process Lett 82(3): 159–164
Jain AK, Moreau JV (1987) Bootstrap technique in cluster analysis. Pattern Recognit 20(5): 547–568
Krivanek M, Moravek J (1986) NP-hard problems in hierarchical-tree clustering. Acta Informatica 23: 311–323
Leclerc B (1984) Efficient and binary consensus functions on transitively valued relations. Math Soc Sci 8: 45–61
Mirkin BG (1975) On the problem of reconciling partitions. In: Quantitative Sociology. Academic Press, New York, pp 441–449
Monjardet B (1990) Arrowian characterization of latticial federation consensus functions. Math Soc Sci 20: 51–71
Newman ME (2004) A fast algorithm for detecting community structure in networks. Phys Rev E Stat Nonlin Soft Matter Phys 69: 066133
Nijenhuis A, Wilf H (1978) Combinatorial algorithms. Academic Press, New York
Pinto da Costa JF, Rao PR (2004) Central partition for a partition-distance and strong pattern graph. REVSTAT 2(2): 128–143
Régnier S (1965) Sur quelques aspects mathématiques des problèmes de classification automatique. Mathématiques et Sciences humaines 82:1983, 13–29 (reprint of I.C.C. bulletin 4: 1965, 175–191)
Zahn CT (1964) Approximating symmetric relations by equivalence relations. SIAM J Appl Math 12: 840–847
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Guénoche, A. Consensus of partitions : a constructive approach. Adv Data Anal Classif 5, 215–229 (2011). https://doi.org/10.1007/s11634-011-0087-6
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DOI: https://doi.org/10.1007/s11634-011-0087-6