Computational Intelligence Winter Term 2017/18
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1 Computational Intelligence Winter Term 2017/18 Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund
2 Plan for Today Fuzzy Clustering 2
3 Cluster Formation and Analysis Introductory Example: Textile Industry production of T-shirts (for men) best for producer : one size vs. best for consumer: made-to-measure compromize: S, M, L, XL, 2XL 5 sizes OK, but which lengths for which size? 3
4 Cluster Formation and Analysis idea: select, say, 2000 men at random and measure their body lengths arrange these 2000 men into five disjoint groups such that deviations from mean of group as small as possible arm s length, collar size, chest girth, differences between group means as large as possible in general: arrange objects into groups / clusters such that elements within a cluster are as homogeneous as possible elements across clusters are as heterogeneous as possible 4
5 Cluster Formation and Analysis numerical example: 1000 points uniformly sampled in [0,1] x [0,1] form 5 cluster 5
6 Hard / Crisp Clustering given data points x 1, x 2,, x N objective: group data points into cluster such that - points within cluster are as homogeneous as possible - points across clusters are as heterogeneous as possible crisp clustering is just a partitioning of data set { x 1, x 2,, x N }, i.e., { x 1, x 2,, x N } and where is Cluster and denotes the number of clusters. Constraint: hence 6
7 Hard / Crisp Clustering Complexity: How many choices to assign N objects into clusters? more precisely: objects are distinguishable / labelled clusters are nondistinguishable / unlabelled and nonempty Stirling number of 2nd kind enumeration hopeless! iterative improvement procedure required! 7
8 Hard / Crisp Clustering idea: define objective function that measures compactness of clusters and quality of partition elements in cluster C j should be as homogeneous as possible! sum of squared distances to unknown center y should be as small as possible (Euclidean norm) 8
9 Hard / Crisp Clustering elements in each cluster C j should be as homogeneous as possible! Definition Theorem 9
10 Crisp K-Means Clustering 10
11 From Crisp to Fuzzy Clustering objective for crisp clustering: rewrite objective: expresses membership objective for fuzzy clustering: 11
12 Fuzzy K-Means Clustering where subject to 12
13 Fuzzy K-Means Clustering two questions: ad a) weighted mean! 13
14 Fuzzy K-Means Clustering ad b) apply Lagrange multiplier method: 14
15 Fuzzy K-Means Clustering after insertion: problems: - choice of K calculate quality measure for each #cluster; then choose best - choice of m try some values; typical: m=2; use interval fuzzy type-2 15
16 Example: Special Case J i > 1 black dot is center of - red cluster - blue cluster - yellow cluster in case of equal weights u ij = 1 / J i for j J i appears plausible but: different values algorithmically better cluster centers more likely to separate again ( tiny randomization?) 16
17 Measures for Cluster Quality Partition Coefficient ( larger is better ) crisp partition entirely fuzzy Partition Entropy ( smaller is better ) entirely fuzzy crisp partition 17
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