4. Clustering Methods

1. 4. Clustering Methods Concepts Partitional (k-Means, k-Medoids) Hierarchical (Agglomerative & Divisive, COBWEB) Density-based (DBSCAN, CLIQUE) Large size data (STING, BIRCH, CURE) Data Mining – Clustering G Dong (WSU)

2. The Clustering Problem • The clustering problem is about grouping a set of data tuples into a number of clusters. Data in the same cluster are highly similar to each other and data in different clusters are highly different from each other. • About clusters • Inter-clusters distance  maximization • Intra-clusters distance  minimization • Clustering vs. classification • Which one is more difficult? Why? • Various possible ways of clustering, which way is the best? Data Mining – Clustering G Dong (WSU)

3. Different ways of representing clusters • Division with boundaries • Venn diagram or spheres • Probabilistic • Dendrograms • Trees • Rules 1 2 3 I1 I2 … In 0.5 0.2 0.3 Data Mining – Clustering G Dong (WSU)

4. Major Categories of Algorithms • Partitioning: Divide into k partitions (k fixed); regroup to get better clustering. • Hierarchical: Divide into different number of partitions in layers - merge (bottom-up) or divide (top-down). • Density-based: Continue to grow a cluster as long as the density of the cluster exceeds a threshold • Grid-based: First divide space into grids, then perform clustering on the grids. Data Mining – Clustering G Dong (WSU)

5. k-Means • Algorithm • Given k • Randomly pick k instancesas the initial centers • Assign the rest instances to closest one of k clusters • Recalculate the mean of each cluster • Repeat 3 & 4 until means don’t change • How good the clusters are • Initial and final clusters • Within-cluster variation diff(x,mean)^2 • Why don’t we consider inter-cluster distance? Data Mining – Clustering G Dong (WSU)

6. Example • For simplicity, 1 dimensional objects and k=2. • Objects: 1, 2, 5, 6,7 • K-means: • Randomly select 5 and 6 as initial centroids; • => Two clusters {1,2,5} and {6,7}; meanC1=8/3, meanC2=6.5 • => {1,2}, {5,6,7}; meanC1=1.5, meanC2=6 • => no change. • Aggregate dissimilarity = 0.5^2 + 0.5^2 + 1^2 + 1^2 = 2.5 Data Mining – Clustering G Dong (WSU)

7. Discussions • Limitations: • Means cannot be defined for categorical attributes; • Choice of k; • Sensitive to outliers; • Crisp clustering • Variants of k-means exist: • Using modes to deal with categorical attributes • How about distance measures • Is it similar to or different from k-NN? • With and without learning Data Mining – Clustering G Dong (WSU)

8. k-Medoids • k-Means algorithm is sensitive to outliers • Is this true? How to prove it? • Medoid – the most centrally located point in a cluster, as a representative point of the cluster. • In contrast, a centroid is not necessarily in a cluster. • An example Initial Medoids Data Mining – Clustering G Dong (WSU)

9. Partition Around Medoids • PAM: • Given k • Randomly pick k instances as initial medoids • Assign each instance to the nearest medoid • Calculate the objective function • the sum of dissimilarities of all instances to their nearest medoids • Randomly select an instance y • Swap some medoid x by y if the swap reduces the objective function • Repeat (3-6) until no change Data Mining – Clustering G Dong (WSU)

10. k-Means and k-Medoids • The key difference lies in how they update means or medoids • Both require distance calculation and reassignment of instances • Time complexity • Which one is more costly? • Dealing with outliers Outlier (100 unit away) Data Mining – Clustering G Dong (WSU)

11. EM (Expectation Maximization) • Moving away from crisp clusters as in k-Means by allowing an instance to belong to several clusters • Finite mixtures – a statistical clustering model • A mixture is a set of k probability distributions, representing k clusters • The simplest finite mixture: one feature with a Gaussian • When k=2, we need to estimate 5 parameters: 2 pairs of μ, 2 pairs of σ, and pA, where pB = 1- pA • EM • Estimate using instances • Maximize the overall likelihood that data came from this data set Data Mining – Clustering G Dong (WSU)

12. Agglomerative • Each object is viewed as a cluster (bottom up). • Repeat until the number of clusters is small enough • Choose a closest pair of clusters • Merge the two into one • Defining “closest”: Centroid (mean of cluster) distance, (average) sum of pairwise distance, … • Refer to the Evaluation part • A dendrogram is a tree that shows clustering process. Data Mining – Clustering G Dong (WSU)

13. Dendrogram • Cluster 1, 2, 4, 5, 6, 7 into two clusters (centriod distance) 1 2 4 5 6 7 Data Mining – Clustering G Dong (WSU)

14. An example to show different Links • Single link • Merge the nearest clusters measured by the shortest edge between the two • (((A B) (C D)) E) • Complete link • Merge the nearest clusters measured by the longest edge between the two • (((A B) E) (C D)) • Average link • Merge the nearest clusters measured by the average edge length between the two • (((A B) (C D)) E) B A E C D Data Mining – Clustering G Dong (WSU)

15. Divisive • All instances belong to one cluster (top-down) • To find an optimal division at each layer (especially the top one) is computationally prohibitive. • One heuristic method is based on the Minimum Spanning Tree (MST) algorithm • Connecting all instances with MST (O(N2)) • Repeatedly cut out the longest edges at each iteration until some stopping criterion is met or until one instance remains in each cluster. Data Mining – Clustering G Dong (WSU)

16. COBWEB • Building a conceptual hierarchy incrementally • Each cluster has a probabilistic description • Category Utility: kijP(fi=vij)P(fi=vij|ck)P(ck|fi=vij) • All categories ck, all features fi, all feature values vij • It attempts to maximize both the probability that two objects in the same category have values in common and the probability that objects in different categories will have different property values Data Mining – Clustering G Dong (WSU)

17. A tree of clusters produced by COBWEB: Data Mining – Clustering G Dong (WSU)

18. Processing one instance at a time by choosing best among • Placing the instance in the best existing category • Adding a new category containing only the instance • Merging of two existing categories into a new one and adding the instance to that category • Splitting of an existing category into two and placing the instance in the best new resulting category Grandparent Grandparent Split Parent Merge Child 1 Child 2 Child 1 Child 2 Data Mining – Clustering G Dong (WSU)

19. Cobweb Demo http://kiew.cs.uni-dortmund.de:8001/mlnet/instances/81d91eaae317b2bebb Data Mining – Clustering G Dong (WSU)

20. Density-based • DBSCAN –Density-Based Clustering of Applications with Noise • It grows regions with sufficiently high density into clusters and can discover clusters of arbitraryshape in spatial databases with noise. • Many existing clustering algorithms find spherical shapes of clusters • DBSCAN defines a cluster as a maximal set of density-connected points. Data Mining – Clustering G Dong (WSU)

21. Q M R S P O • Defining density and connection • -neighborhood of an object x (core object) (M, P, Q) • MinPts of objects within -neighborhood (say, 3) • directly density-reachable (Q from M, M from P) • density-reachable (Q from P, P not from Q) [asymmetric] • density-connected (O, R, S) [symmetric] <for border points> • What is the relationship between DR and DC? Data Mining – Clustering G Dong (WSU)

22. Clustering with DBSCAN • Search for clusters by checking the -neighborhood of each instance x • If the -neighborhood of x contains more than MinPts, create a new cluster with x as a core object • Iteratively collect directly density-reachable objects from these core object and merge density-reachable clusters • Terminate when no new point can be add to any cluster • DBSCAN is sensitive to the thresholds of density, but it is many folds faster than CLARANS • Time complexity O(N log N) if a spatial index is used, O(N2) otherwise Data Mining – Clustering G Dong (WSU)

23. Dealing with Large Data • Key ideas • Reducing the number of instances to be maintained, and yet to maintain the distribution • Identifying relevant subspaces where clusters possibly exist • Using summarized information to avoid repeated data access • Sampling • CLARA (Clustering LARge Applications) working on samples instead of the whole data • CLARANS (Clustering Large Applications based on RANdomized Search) Data Mining – Clustering G Dong (WSU)

24. Grid: STING (STatistical INformation Grid) • Statistical parameters of higher-level cells can easily be computed from those of lower-level cells • Attribute-independent: count • Attribute-dependent: mean, standard deviation, min, max • Type of distribution: normal, uniform, exponential, or unknown • Irrelevant cells can be removed Data Mining – Clustering G Dong (WSU)

25. Representatives • BIRCH using Clustering Feature (CF) and CF tree • A cluster feature is a triplet about sub-clusters of instances (N, LS, SS) • N - the number of instances, LS – linear sum, SS – square sum • Two thresholds: branching factor (the max number of children per non-leaf node) and diameter threshold • Two phases • Build an initial in-memory CF tree • Apply a clustering algorithm to cluster the leaf nodes in CF tree • CURE (Clustering Using REpresentitives) is another example Data Mining – Clustering G Dong (WSU)

26. B: Branching factor L: Threshold: max diameter of subclusters at leaf nodes CF1 CF2 CF3 CF6 child1 child2 child3 child6 CF Tree Root B = 7 L = 6 Non-leaf node CF1 CF2 CF3 CF5 child1 child2 child3 child5 Leaf node Leaf node prev CF1 CF2 CF6 next prev CF1 CF2 CF4 next Data Mining – Clustering G Dong (WSU)

27. Taking advantage of the property of density • If it’s dense in higher dimensional subspaces, it should be dense in some lower dimensional subspaces • CLIQUE (CLustering In QUEst) • With high dimensional data, there are many void subspaces • Using the property identified, we can start with dense lower dimensional data • CLIQUE is a density-based method that can automatically find subspaces of the highest dimensionality such that high-density clusters exist in those subspaces Data Mining – Clustering G Dong (WSU)

28. Drawbacks of Distance-Based Method • Drawbacks of square-error based clustering method • Consider only one point as representative of a cluster • Good only for convex shaped, similar size and density, and if k can be reasonably estimated Data Mining – Clustering G Dong (WSU)

29. Chameleon • A hierarchical Clustering Algorithm Using Dynamic Modeling • Observations on the weakness of pure distance based methods • Basic steps: • Build K nearest neighbor graph • Partition the graph • Merge the “strongly connected partitions,” in terms of strength of connections between partitions Data Mining – Clustering G Dong (WSU)

30. Summary • There are many clustering algorithms • Good clustering algorithms maximize inter-cluster dissimilarity and intra-cluster similarity • Without prior knowledge, it is difficult to choose the best clustering algorithm. • Clustering is an important tool for outlier analysis. Data Mining – Clustering G Dong (WSU)

31. Bibliography • I.H. Witten and E. Frank. Data Mining – Practical Machine Learning Tools and Techniques with Java Implementations. 2000. Morgan Kaufmann. • M. Kantardzic. Data Mining – Concepts, Models, Methods, and Algorithms. 2003. IEEE. • J. Han and M. Kamber. Data Mining – Concepts and Techniques. 2001. Morgan Kaufmann. • M. H. Dunham. Data Mining – Introductory and Advanced Topics. Data Mining – Clustering G Dong (WSU)