Computational Intelligence for Knowledge-Based Systems by Bernadette Bouchon-Meunier, Giulianella Coletti,

By Bernadette Bouchon-Meunier, Giulianella Coletti, Marie-Jeanne Lesot, Maria Rifqi (auth.), Eyke Hüllermeier, Rudolf Kruse, Frank Hoffmann (eds.)

The ebook constitutes the refereed complaints of the thirteenth overseas convention on info Processing and administration of Uncertainty in Knowledge-Based structures, IPMU 2010, held in Dortmund, Germany from June 28 - July 2, 2010. The seventy seven revised complete papers have been conscientiously reviewed and chosen from 320 submissions and replicate the richness of study within the box of Computational Intelligence and signify advancements on themes as: laptop studying, information mining, development acceptance, uncertainty dealing with, aggregation and fusion of knowledge in addition to common sense and information processing.

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Additional info for Computational Intelligence for Knowledge-Based Systems Design: 13th International Conference on Information Processing and Management of Uncertainty, IPMU 2010, Dortmund, Germany, June 28 - July 2, 2010. Proceedings

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Using the partition coefficient P C(U ) = 1 n n c u2ik (5) k=1 i=1 Comparing partitions produced by different methods, parameters, or initializations (obtained by classification, clustering, or in other ways) requires functions f (U1 , U2 , . . , Uj ) ∈ R, j > 1. Here, we restrict to functions comparing pairs of ˜ A popular measure for comparing crisp partipartitions f (U1 , U2 ) or f (U, U). tions is the Rand index [7]. Extensions of the Rand index to fuzzy partitions were proposed very recently by Campello [8] and by H¨ ullermeier and Rifqi [9].

References 1. : Monotone equivariant cluster analysis. SIAM J. Appl. Math. 37, 148–165 (1979) 2. : Indice de similarit´e et pr´eordonnance associ´ee. In: S´eminaire sur les ordres totaux finis, Aix-en-Provence, pp. 233–243 (1967) 3. : A classification of presence/absence based dissimilarity coefficients. Journal of Classification 6, 233–246 (1989) 4. : Comparing resemblance measures. Journal of Classification 12, 73–90 (1995) 5. : Ranking invariance based on similarity measures in document retrieval.

Cc ⊆ X, c > 0, Ci ∩ Cj = ∅, i, j = 1, . . , c, so that C1 ∪ C2 ∪ . . ∪ Cc = X. Obviously, for n = 1, there is only one partition C1 = X, and for n = 2, there are two partitions C1 = X and C1 = {x1 }, C2 = {x2 }. For arbitrary n > 0, the number of partitions is given by the Bell number Bn which can be recursively computed as n Bn+1 = k=0 n k Bk , B1 = 1 (1) For convenience, we represent a c–tuple of partition sets using a partition matrix ⎞ ⎛ u11 · · · u1n ⎟ ⎜ U = ⎝ ... . ... ⎠ (2) uc1 · · · ucn uik ∈ {0, 1}, i = 1, .

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