1. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Data Mining: Cluster Analysis This lecture node is modified based on Lecture Notes for Chapter 8 of Introduction to Data Mining by Tan, Steinbach, Kumar. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1

2. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 2 What is Cluster Analysis? l Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Inter-cluster distances are maximized Intra-cluster distances are minimized

3. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 3 Notion of a Cluster can be Ambiguous How many clusters? Four ClustersTwo Clusters Six Clusters

4. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 4 Types of Clusters l Well-separated clusters l Center-based clusters l Contiguity-based clusters l Density-based clusters l Described by an Objective Function

5. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 5 Types of Clusters: Well-Separated l A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 3 well-separated clusters

6. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 6 Types of Clusters: Center-Based l A cluster is a set of objects such that an object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster l The center of a cluster is often a centroid, the average of all the points in the cluster (for continuous attributes), or a medoid, the most “representative” point of a cluster (categorical attributes) 4 center-based clusters

7. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 7 Types of Clusters: Contiguity-Based l Contiguity Cluster (Nearest neighbor or Transitive) l A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster. 8 contigiguity clusters

8. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 8 Types of Clusters: Density-Based l A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. l Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clusters

9. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 9 Types of Clusters: Objective Function l Clusters Defined by an Objective Function –Find clusters that minimize or maximize an objective function. –Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard) – Can have global or local objectives.  Hierarchical clustering algorithms typically have local objectives  Partitional algorithms typically have global objectives

10. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 10 K-means Clustering l Partitional clustering approach l Each cluster is associated with a centroid (center point) l Each point is assigned to the cluster with the closest centroid l Number of clusters, K, must be specified l The basic algorithm is very simple

11. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 11 K-Means l cluster k i = {t i1, t i2, …, t im } l Means of the cluster k i, m i = (t i1 + t i2 + … + t im )/m l data {2, 4, 10, 12, 3, 20, 30, 11, 25} l K = 2 l m 1 = 2, m 2 = 4, –K 1 = {2, 3}, and K 2 = {4, 10, 12, 20, 30, 11, 25} l m 1 = 2.5, m 2 = 16 –K 1 = {2, 3, 4}, and K 2 = {10, 12, 20, 30, 11, 25} l m 1 = 3, m 2 = 18 –K 1 = {2, 3, 4, 10}, and K 2 = {12, 20, 30, 11, 25} l m 1 = 4.75, m 2 = 19.6 –K 1 = {2, 3, 4, 10, 11, 12}, and K 2 = {20, 30, 25} l m 1 = 7, m 2 = 25 –K 1 = {2, 3, 4, 10, 11, 12}, and K 2 = {20, 30, 25}

12. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 12 Two different K-means Clusterings Sub-optimal ClusteringOptimal Clustering Original Points

13. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 13 Importance of Choosing Initial Centroids

14. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 14 Evaluating K-means Clusters l Most common measure is Sum of Squared Error (SSE) –For each point, the error is the distance to the nearest cluster –To get SSE, we square these errors and sum them. –x is a data point in cluster C i and m i is the representative point for cluster C i  can show that m i corresponds to the center (mean) of the cluster –Given two clusters, we can choose the one with the smallest error –One easy way to reduce SSE is to increase K, the number of clusters  A good clustering with smaller K can have a lower SSE than a poor clustering with higher K

15. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 15 Importance of Choosing Initial Centroids …

16. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 16 Solutions to Initial Centroids Problem l Multiple runs –Helps, but probability is not on your side l Sample and use hierarchical clustering to determine initial centroids l Select more than k initial centroids and then select among these initial centroids –Select most widely separated l Bisecting K-means –Not as susceptible to initialization issues

17. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 17 Bisecting K-means l Bisecting K-means algorithm –Variant of K-means that can produce a partitional or a hierarchical clustering

18. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 18 Bisecting K-means Example

19. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 19 Limitations of K-means: Differing Sizes Original Points K-means (3 Clusters)

20. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 20 Limitations of K-means: Differing Density Original Points K-means (3 Clusters)

21. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 21 Limitations of K-means: Non-globular Shapes Original Points K-means (2 Clusters)

22. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 22 Overcoming K-means Limitations Original PointsK-means Clusters One solution is to use many clusters. Find parts of clusters, but need to put together.

23. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 23 Overcoming K-means Limitations Original PointsK-means Clusters

24. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 24 Overcoming K-means Limitations Original PointsK-means Clusters

25. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 25 Hierarchical Clustering l Produces a set of nested clusters organized as a hierarchical tree l Can be visualized as a dendrogram –A tree like diagram that records the sequences of merges or splits

26. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 26 Strengths of Hierarchical Clustering l Do not have to assume any particular number of clusters –Any desired number of clusters can be obtained by ‘cutting’ the dendogram at the proper level l They may correspond to meaningful taxonomies –Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

27. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 27 Agglomerative Clustering Algorithm l More popular hierarchical clustering technique l Basic algorithm is straightforward 1.Compute the proximity matrix 2.Let each data point be a cluster 3.Repeat 4.Merge the two closest clusters 5.Update the proximity matrix 6.Until only a single cluster remains l Key operation is the computation of the proximity of two clusters –Different approaches to defining the distance between clusters distinguish the different algorithms

28. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 28 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Similarity? l MIN l MAX l Group Average l Distance Between Centroids Proximity Matrix

29. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 29 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

30. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 30 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

31. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 31 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

32. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 32 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids 

33. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 33 Cluster Similarity: MIN or Single Link l Similarity of two clusters is based on the two most similar (closest) points in the different clusters –Determined by one pair of points, i.e., by one link in the proximity graph. I1I2I3I4I5 I10.000.240.220.370.34 I20.240.000.150.200.14 I30.220.150.000.150.28 I40.370.200.150.000.29 I50.340.140.280.290.00 I6 0.23 0.250.110.220.39 0.23 0.25 0.11 0.22 0.39 0.00

34. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 34 Strength of MIN Original Points Two Clusters Can handle non-elliptical shapes

35. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 35 Limitations of MIN Original Points Two Clusters Sensitive to noise and outliers

36. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 36 Cluster Similarity: MAX or Complete Linkage l Similarity of two clusters is based on the two least similar (most distant) points in the different clusters –Determined by all pairs of points in the two clusters I1I2I3I4I5 I10.000.240.220.370.34 I20.240.000.150.200.14 I30.220.150.000.150.28 I40.370.200.150.000.29 I50.340.140.280.290.00 I6 0.23 0.250.110.220.39 0.23 0.25 0.11 0.22 0.39 0.00

37. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 37 Strength of MAX Original Points Two Clusters Less susceptible to noise and outliers

38. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 38 Limitations of MAX Original Points Two Clusters Tends to break large clusters Biased towards globular clusters

39. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 39 Hierarchical Clustering: Problems and Limitations l Once a decision is made to combine two clusters, it cannot be undone l No objective function is directly minimized l Different schemes have problems with one or more of the following: –Sensitivity to noise and outliers –Difficulty handling different sized clusters –Breaking large clusters