Generalization of Hierarchical Crisp Clustering Algorithms to Fuzzy Logic
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1 Generalization of Hierarchical Crisp Clustering Algorithms to Fuzzy Logic Mathias Bank Faculty for Mathematics and Economics University of Ulm Dr. Friedhelm Schwenker Institute of Neural Information Processing University of Ulm
2 Page 2 Fuzzy Logic Question: Answer: Is there a salami sandwich in the refrigator? 0.5 If probability: If measure: If fuzzy: then there is or isn t, with probability one half then there is half a salami sandwich there then there is something there, but it isn t really a salami sandwich. Perhaps it is some other kind of sandwich, or salami without the bread... Prof. Solomon W. Golomb University of Southern California
3 Page 3 Motivation for Hierarchical Fuzzy Clustering Fuzzy Documents deal with more than one topic overlapping topic categories Hierarchical Topic Analysis on different abstraction levels
4 Page 4 Precondition for different Clustering Approach Hierarchical agglomerative Clustering Hierarchical agglomerative Fuzzy- Clustering Deterministic Global Optimum (based on similarity) Different Linkage Metrics available Deterministic Global Optimum (based on similarity) Different Linkage Metrics reusable Overlapping topics per document Multi- Assignment
5 Page 5 Generalized agglomerative Clustering Algorithm D1 D2 D3 D4 D5 D6 Group Clusters and with highest similarity D1 1 S(2,1) S(3,1) S(4,1) S(5,1) S(6,1) D2 1 S(3,2) S(4,2) S(5,2) S(6,2) D3 1 S(4,3) S(5,3) S(6,3) D4 1 S(5,4) S(6,4) D5 1 S(6,5) D6 1
6 Page 6 Generalized agglomerative Clustering Algorithm D1 D2 D3 D4 D5 D6 D1 1 S(2,1) S(3,1) S(4,1) S(5,1) S(6,1) Group Clusters and with highest similarity Delete all similarity values of and D2 1 S(3,2) S(4,2) S(5,2) S(6,2) D3 1 S(4,3) S(5,3) S(6,3) D4 1 S(5,4) S(6,4) D5 1 S(6,5) D6 1
7 Page 7 Generalized agglomerative Clustering Algorithm D1 D2 D3 D4 D5 D6 D1 1 0 S(3,1) S(4,1) S(5,1) S(6,1) D2 1 S(3,2) S(4,2) S(5,2) S(6,2) Group Clusters and with highest similarity Delete all similarity values of and Delete similarity value of and D3 1 S(4,3) S(5,3) S(6,3) D4 1 S(5,4) S(6,4) D5 1 S(6,5) D6 1
8 Page 8 Generalized agglomerative Clustering Algorithm D1 D2 D3 D4 D5 D6 C1 D1 1 0 S(3,1) S(4,1) S(5,1) S(6,1) 0 D2 1 S(3,2) S(4,2) S(5,2) S(6,2) 0 D3 1 S(4,3) S(5,3) S(6,3) S(C1,3) Group Clusters and with highest similarity Delete all similarity values of and Delete similarity value of and Insert new cluster into similarity matrix D4 1 S(5,4) S(6,4) S(C1,4) D5 1 S(6,5) S(C1,5) D6 1 S(C1,6) C1 1
9 Page 9 Linkage metrics Cluster Document Similarity Cluster Cluster Similarity Well known Linkage metrics reusable Single Linkage, Complete Linkage, Well known Linkage metrics reusable Single Linkage, Complete Linkage, Except to documents :, 0 Except to clusters :, 0 How to calculate,, if? Use common documents for similarity calculation?
10 Page 10 Linkage metrics Cluster Document Similarity Cluster Cluster Similarity Well known Linkage metrics reusable Single Linkage, Complete Linkage, Well known Linkage metrics reusable Single Linkage, Complete Linkage, Except to documents :, 0 Except to clusters :, 0 How to calculate,, if? Use common documents for similarity calculation?
11 Page 11 Linkage metrics Cluster Document Similarity Cluster Cluster Similarity Well known Linkage metrics reusable Single Linkage, Complete Linkage, Well known Linkage metrics reusable Single Linkage, Complete Linkage, Except to documents :, 0 Except to clusters :, 0 How to calculate,, if? Generalized subgraph property: don t use common data points for similarity calculation.
12 Page 12 Fuzzyness specification D1 D2 D3 D4 D5 D6 D1 1 0 f * f * f * f * S(3,1) S(4,1) S(5,1) S(6,1) D2 1 f * f * f * f * S(3,2) S(4,2) S(5,2) S(6,2) D3 1 S(4,3) S(5,3) S(6,3) D4 1 S(5,4) S(6,4) D5 1 S(6,5) D6 1 Fuzzifier specify fuzziness 0; 1 Applied to selected rows / columns If,, =0 Definable for each level separately (level counted by document level) Crisp level definition possible
13 Page 13 Generalized agglomerative Clustering Algorithm While (!empty(similaritymatrix)) Group Clusters and with highest similarity Delete all similarity values of and Delete similarity value of and Apply fuzzifier to similarity values fo and Insert new cluster into similarity matrix End
14 Page 14 Fuzzy Membership Calculation AB BC µ, s 1 s 1 s 2 s 2 µ, A B C µ,,, p: Parents
15 Interpretable hierarchy size 5,10 possible
16 Page 16 Generalized agglomerative Clustering Algorithm While (!empty(similaritymatrix)) Group Clusters and with highest similarity Delete all similarity values of and Delete similarity value of and Apply fuzzifier to similarity values fo and Insert new cluster into similarity matrix End Calculate Fuzzy Membership, Shrink cluster level size to predefined level size
17 Page 17 Evaluation measure for external labeled data 2A2B 4A2B2C1D 4A2B2C 2A2C COS-Quality-Index Propagate document labels via membership Cluster quality defined as statistics of all cos similarities of document label vector and cluster label vector Fuzziness Statistics AB AB AC AC D Recall in fuzzy mode difficult to define Fuzziness analysis for recall information Data 7 randomly chosen document collections Based on.. RCV1 (26501, 23309, docs) RCV2-german (16604, 13146, 11600, 9350 docs) Fuzzyfier 0.0, 0.1,, 0.8 for document level, 0.0 else Hiearchy level 5 Quality analysis for document parents necessary
18 Page 18 Evaluation Results COS-Quality for document parents
19 Page 19 Evaluation Results Fuzziness per document
20 Page 20 Conclusion Agglomerative Fuzzy Cluster Algorithm Agglomerative Crisp Cluster algorithm generalizable to Fuzzy Logic Deterministic hierarchical fuzzy cluster algorithm Global optimal cluster based on similarity function Topic Groups per level shrinkage possible (crisp & fuzzy) Quality Results Fuzziness does not highly influence cluster quality 25% - 50% of random chosen textual data affected by multi assignment
21 Page 21 Discussion
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