Algorithms For Approximation Proc, Chester 2005 by Iske A , Levesley J (Eds)

By Iske A , Levesley J (Eds)

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Yet, in convex domains the constant C2 in (1) is geometry independent [4]. Defining the distance defect ratio of a pair of points x, y ∈ cl(Ω) = Ω ∪ ∂Ω (with ∂Ω the boundary of Ω) by µ(x, y)Ω = ρ(x, y)Ω |x − y| (2) where ρ(x, y)Ω is the length of the shortest path inside cl(Ω) connecting x and y, we observe that in the domains {Ωǫ } of the example, there exist pairs of points with distance defect ratio growing as ǫ → 0. Note that there is no upper bound for the distance defect ratio of arbitrary domains, while in convex domains the distance defect ratio is 1.

Both the similarity and the dissimilarity should be examinable in a clear and meaningful way. Here, we give the simple mathematical descriptions of partitional clustering and hierarchical clustering, based on [40]. Given a set of N input patterns X = {x1 , . . , xj , . . , xN }, where xj = (xj1 , xj2 , . . , xjd )T ∈ Rd and each xji measure is said to be a feature (attribute, dimension, or variable), 32 R. Xu, D. Wunsch II • (Hard) partitional clustering attempts to seek a K-partition of X, C = {C1 , .

Plenum Press, New York, 1981. 9. C. Bishop: Neural networks for pattern recognition. Oxford University Press, 1995. Computational Intelligence in Clustering Algorithms 47 10. J. Burke, D. Davison, and W. Hide: d2 Cluster: a validated method for clustering EST and full-length cDNA sequences. Genome Research, 1999, 1135–1142. 11. G. Carpenter and S. Grossberg: A massively parallel architecture for a selforganizing neural pattern recognition machine. Computer Vision, Graphics, and Image Processing, 1987, 54–115.

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