1. 4/8/2002Copyright Daniel Barbara Clustering by impact Daniel Barbará George Mason University ISE Dept. http://www.ise.gmu.edu/~dbarbara (joint work with P. Chen, J. Couto, and Y. Li)

2. Problem Organizations are constantly acquiring and storing new data (data streams) The need to quickly extract knowledge from the newly arrived data (and compare it with the old) is pressing. Applications: Intrusion detection Tuning Intelligence analysis

3. Outline Clustering data streams Our method Continuous data: Fractal Clustering Categorical (nominal) data: Entropy-based Tracking clusters Future work

4. Clustering and data streams To cluster continuously arriving data streams a clustering algorithm should behave incrementally: make the decision based on the newly arrived point and a concise description of the clusters encountered so far. Concise  bounded amount of RAM to describe the clusters, independently of the number of data points processed so far…

5. Problem (cont.) Most algorithms in the literature do not have that property: They look at the entire set of points at once (e.g., K-means) They cannot make decisions point by point. The description of the clusters is usually the set of points in them. Some of the algorithms have high complexity

6. Some inroads Paper by U. Fayyad, D. Bradley and C. Reina: “Scaling Clustering algorithms to large databases” (KDD’98) Main idea: keep descriptions of centroids + set descriptions that are likely and unlikely to change given a new data point. Papers by Motwani, et al. Incrementally updating centroids while receiving a data stream. The goal is to have an approximation to “min squares” whose performance is bounded.

7. Our proposal Find functions that naturally define clusters and that can be easily computed given a new point and a concise representation of the current clusters. Place a new point in the cluster for which the evaluated function shows a minimum (or a maximum) – less impact---

8. “Impact” functions Numerical data points: fractal dimension Measures the self-similarity of points. The idea is that the lower the change in the fractal dimension (when the point is included), the more self-similar the point is w/respect to the cluster Categorical data points: entropy. Also measures similarity Lower entropy means similar points.

9. Fractal Clustering Fractal dimension, is a (not necessarily integer) number that characterizes the number of dimensions ``filled'' by the object represented by the dataset. The object on the upper right corner, called the Menger sponge (when complete) has a F.D. equal to 2.73 (less than the embedding space, whose dimension is 3) Conjecture: if part of a dataset brings about a change in the overall fractal dimension of the set, then this part is ``anomalous'' (exhibits different behavior) with respect to the rest of the dataset.

10. Fractal dimension r = grid size Probability distribution

11. Box Counting Cantor Dust Set

12. Box counting (cont.)  log 2 n D 1 = - lim n->  = 0.63  log (  ) n

13. Initialization Algorithm Take an unlabelled point in the sample and start a cluster. Find close neighbors and add them to the cluster. Find close neighbors to points in the cluster… If you can’t go to first step.

14. Space management

15. Space in RAM is not proportional to the size of the dataset, but rather to the size of the grid and number of grid levels kept. These vary with: Dimensionality Accuracy (odd-shaped clusters may require more levels).

16. Experiments Dataset1

17. Scalability results with Dataset1

18. Quality of clusters (Dataset1)

19. High dimensional set 10 dimensions, 2 clusters

20. Results with the noisy dataset 92 % of the noise gets filtered out.

21. Memory usage vs. dimensions

22. Memory reduction Space taken by the boxes is small, but it grows with the number of dimensions. Memory reduction techniques: Use boxes with # points > epsilon. Cache boxes Have only smallest granularity boxes and derive the rest. None of them causes a significant degradation of quality. (2 and 3 have an impact on running time.)

23. Memory reduction

24. Comparison with other algorithms

25. Entropy-based Clustering (COOLCAT) For Categorical data Place new point where it minimizes some function of the entropies of the individual clusters (e.g., min (max (entropy Ci))) Heuristic (problem is NP-Hard) Entropy of each cluster: Minimize expected entropy

26. Initialization Need to seed “k” clusters: Select a sample Find 2 points that are the most dissimilar (their joint entropy is the highest). Place them in 2 different clusters Find another point that is the most dissimilar (pairwise) to the ones selected, and start another cluster.

27. Incremental phase For a given point and k current clusters: Compute the expected entropy as the new point is placed in each cluster. Choose the one that minimizes the expected entropy After finishing with a batch of points, re- process m% of them (take the ``worse’’ fits out and re-cluster them): helps with the issue of order dependency

28. Conciseness Notice that the current cluster description is concise: Counts of Vij for every i= 1,.., d (number of attributes), and for every j (domain of each attribute)

29. COOLCAT and the MDL MDL = minimum description length. Widely used to argue about how good a classifier is: how many bits does it take to send to a receiver the description of your classifier + the exceptions (misclassifications)

30. MDL (cont.)

31. Experimental results Real and synthetic datasets Evaluate quality and performance Quality: Category utility function (how much “better” is the distribution probability in the individual clusters w/respect to the original distribution) External entropy: take an attribute not used in the clustering and compute the entropy of each cluster w/respect to it, then the expected external entropy

32. Experimental results Archaeological data set Alg.mCUExt. EExp E Coolcat00.762604.8599 Coolcat100.762604.8599 Coolcat200.762604.8599 Brute F.-0.762604.8599 ROCK-0.33120.96-

33. KDD99 Cup data (intrusion detection) k Exp E CU Ext E

34. Performance (synthetic data) N x 1000 T (sec.)

35. Tracking clusters Clustering data streams as they come: Consider r.v X = 0 if new point is outlier; 1 otherwise.Using Chernoff bounds: Must see s “successes” – not outliers– in a window w If you don’t, it is time for new clusters…

36. FC, COOLCAT and Tracking Find a good definition of outlier: FC: if the min change in FD exceeds a threshold. COOLCAT: mutual information of new point with respect to clusters

37. One tracking experiment with FC

38. One tracking experiment with COOLCAT (intrusion detection) Mutual Information density attacks No attacks

39. Hierarchical clustering More tracking experiments Hybrid data: numeric and categorical Indexing based on clustering …

40. Bibliography ``Using the Fractal Dimension to Cluster Datasets,'' Proceedings of the the ACM-SIGKDD International Conference on Knowledge and Data Mining, Boston, August 2000. D. Barbara, P.Chen. ``Tracking Clusters in Evolving Data Sets,'' Proceedings of FLAIRS'2001, Special Track on Knowledge Discovery and Data Mining, Key West, FL, May 2001. D. Barbara, P. Chen. ``Fractal Characterization of Web Workloads,'' Proceedings of the 11th International World Wide Web Conference, May 2002. D. Menasce, V. Almeida, D. Barbara, B. Abrahao, and F. Ribeiro. ``Using Self-Similarity to Cluster Large Data Sets,’’ to appear in Journal of Data Mining and Knowledge Discovery, Kluwer Academic pub. D. Barbara, P.Chen ``Requirements for Clustering Data Streams,'' SIGKDD Explorations (Special Issue on Online, Interactive, and Anytime Data Mining), Vol. 3, No. 2, Jan 2002. D. Barbara ``COOLCAT: An Entropy-Based Algorithm for Categorical Clustering,’’ Submitted for publication. D. Barbara, J. Couto, Y. Li.