Automatic Cluster Number Selection using a Split and Merge K-Means Approach
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1 Automatic Cluster Number Selection using a Split and Merge K-Means Approach Markus Muhr and Michael Granitzer 31st August 2009 The Know-Center is partner of Austria's Competence Center Program COMET.
2 Agenda Introduction Contribution Split & Merge Spherical K-Means Internal Validity Indices Experimental Setup Results Discussion Conclusion 2
3 Introduction Text is high-dimensional, very sparse, large-scale Partitional methods especially Spherical K-Means Bag-of-word based Vector Space Model Cosine similarity Unit sphere Cluster number must be known a-priori! Dynamic K-Means X-Means Bisecting K-Means Bayesian Information Criterion Assess quality of bisecting splits 3
4 Contribution Extending X-Means Merge steps (reduce error propagation) Online updates Spherical optimization criteria Evaluation Standard text data sets (Cluto) Several different internal validity indices Batch and Online Updates Adapt internal validity indices 4
5 Spherical K-Means Basics Maximize cosine similarity as cost function Unit-length normalized centroids and data samples Calculate inner product between Data samples and their nearest cluster centroid 5
6 Batch Spherical K-Means Separate update and assignment step Samples are assigned to nearest centroid Normalized sum of assigned samples 6
7 Online Spherical K-Means Update step of winning cluster after assignment Competitive learning technique Winner-Take-All approach 7
8 Split & Merge Spherical K-Means Initial K-Means partitioning with given k (e.g. 2) Consecutive split and merge steps Criteria to select cluster candidates largest cluster (radius or size) for split step nearest centroid pair for merge step Assess quality of cluster result by internal validity index Bayesian Information Criterion, etc. No quality improvements Terminate Optional K-Means step for refinement 8
9 Split & Merge Spherical K-Means 9
10 Internal Validity Indices Bayesian Information Criterion (X-Means) Bayesian Information Criterion (heuristic) Calinski-Harabasz Index Hartigan Index Krzanowski-Lai Index 10
11 Experimental Setup Text data sets from CLUTO Toolkit 12 different data sets documents with 6 to 25 classes Stop words removed, stemmed, tf-idf Cluster performance using f-score 11
12 Experimental Setup 12
13 Experimental Setup Batch and Online Split & Merge K-Means 12 data sets and 5 validity indices Different initial partitioning 5, 15 or 35 clusters 2, 8 or 15 clusters Seeding mechanism using K-Means++ Square root annealing factor with initial at runs for each setting Mean, std. dev., maximum of f-score, cluster number 13
14 Results Batch Updates 14
15 Results Online Updates 15
16 Discussion F-Score Dynamic vs. Static cluster number selection* Similar f-scores in most cases Better results for few class problems (tr31, k1b) Worse results for many class problems (re0, re1) Online vs. Batch Updates Mean higher with online updates Standard deviation mostly lower with online updates Maximum value can be reached with both Online updates are better and more robust *Y. Zhao and G. Karypis, Evaluation of hierarchical clustering algorithms for document datasets 16
17 Discussion Cluster Number Cluster number close to the given class number Few class problem deviation is around 1 e.g. for data set k1b mean of cluster number is 5.4 Many class problems deviation increases Better results for good separable data sets (k1b, tr31) Standard deviation of cluster number is low robust F-Scores are always comparable with best static results Selected cluster number optimizes cost function 17
18 Discussion Validity Indices No clear winner quite equally good results Bayesian Information Criterion seems to be best X-Means and heuristic version provide similar results Slight favor for heuristic version Calinski-Harabasz index performs good as well Hartigan, Krzanowski-Lai index less suitable All indices approximate the cluster number quite well! 18
19 Discussion Dynamic Updates Initial partitioning does not alter results considerably Fewer initial clusters (2, 5) slight underestimation Many initial clusters (15, 35) slight overestimation Almost accurate initial clusters (5, 15) correct estimation Suitable for incremental work flow Good separable problems always correct number Bad separable problems Under- / Overestimations Many classes problem Underestimations 19
20 Discussion Runtime Split & Merge K-Means Faster than X-Means less splits and cheap merges Validity indices can be calculated fast Runtimes comparable with bisecting K-Means Online vs. batch updates Online updates converge faster, but take longer Batch sum of very sparse documents Online updates all non-zero dimensions of the centroid (rather dense) in each update step Online updates 10 times slower than batch updates 20
21 Discussion Runtime Outlook Heuristic approach c i=c i x n Weighted sum of documents in update step Instead of geometrical correct update c i=c i x n c i l2-norm exceeds threshold normalize centroids Increases convergence time, decreases update time 21
22 Conclusion Split & Merge K-Means guesses clusters quite good Appropriate for incremental work flow Online updates better than batch updates In real-world applications use heuristic online update Type of validity index of lesser concern Bayesian Information Criterion provides best results 22
23 Thank you for your attention! Questions? 23
Automatic Cluster Number Selection using a Split and Merge K-Means Approach
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