Clustering

1. Clustering CIS 601 Fall 2004 Longin Jan Latecki Lecture slides taken/modified from: Jiawei Han (http://www-sal.cs.uiuc.edu/~hanj/DM_Book.html) Vipin Kumar (http://www-users.cs.umn.edu/~kumar/csci5980/index.html)

2. Clustering • Cluster: a collection of data objects • Similar to one another within the same cluster • Dissimilar to the objects in other clusters • Cluster analysis • Grouping a set of data objects into clusters • Clustering is unsupervised classification: no predefined classes • Typical applications • to get insight into data • as a preprocessing step • we will use it for image segmentation

3. Inter-cluster distances are maximized Intra-cluster distances are minimized What is Cluster Analysis? • 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

4. How many clusters? Six Clusters Two Clusters Four Clusters Notion of a Cluster can be Ambiguous

5. Types of Clusters: Contiguity-Based • Contiguous Cluster (Nearest neighbor or Transitive) • 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 contiguous clusters

6. Types of Clusters: Density-Based • Density-based • A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. • Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clusters

7. Euclidean Density – Cell-based • Simplest approach is to divide region into a number of rectangular cells of equal volume and define density as # of points the cell contains

8. Euclidean Density – Center-based • Euclidean density is the number of points within a specified radius of the point

9. Data Structures in Clustering • Data matrix • (two modes) • Dissimilarity matrix • (one mode)

10. Interval-valued variables • Standardize data • Calculate the mean squared deviation: where • Calculate the standardized measurement (z-score) • Using mean absolute deviation could be more robust than using standard deviation

11. Similarity and Dissimilarity Between Objects • Euclidean distance: • Properties • d(i,j) 0 • d(i,j)= 0 iff i=j • d(i,j)= d(j,i) • d(i,j) d(i,k)+ d(k,j) • Also one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures.

12. Covariance Matrix The set of 5 observations, measuring 3 variables, can be described by its mean vector and covariance matrix. The three variables, from left to right are length, width, and height of a certain object, for example. Each row vector Xrow is another observation of the three variables (or components) for row=1, …, 5.

13. The mean vector consists of the means of each variable. The covariance matrix consists of the variances of the variables along the main diagonal and the covariances between each pair of variables in the other matrix positions. where n = 5 for this example 0.025 is the variance of the length variable, 0.0075 is the covariance between the length and the width variables, 0.00175 is the covariance between the length and the height variables, 0.007 is the variance of the width variable.

14. Mahalanobis Distance  is the covariance matrix of the input data X For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.

15. Mahalanobis Distance Covariance Matrix: C A: (0.5, 0.5) B: (0, 1) C: (1.5, 1.5) Mahal(A,B) = 5 Mahal(A,C) = 4 B A

16. Cosine Similarity • If x1 and x2 are two document vectors, then cos( x1, x2 ) = (x1 x2) / ||x1|| ||x2|| , where  indicates vector dot product and || d || is the length of vector d. • Example: x1= 3 2 0 5 0 0 0 2 0 0 x2 = 1 0 0 0 0 0 0 1 0 2 x1 x2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5 ||x1|| = (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5 = (42) 0.5 = 6.481 ||x2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2)0.5= (6) 0.5 = 2.245 cos( x1, x2 ) = .3150

17. Correlation • Correlation measures the linear relationship between objects • To compute correlation, we standardize data objects, p and q, and then take their dot product

18. Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1.

19. K-means Clustering • Partitional clustering approach • Each cluster is associated with a centroid (center point) • Each point is assigned to the cluster with the closest centroid • Number of clusters, K, must be specified • The basic algorithm is very simple

20. where xn is a vector representing the nth data point and mj is the geometric centroid of the data points in Sj k-means Clustering • An algorithm for partitioning (or clustering) N data points into K disjoint subsets Sj containing Nj data points so as to minimize the sum-of-squares criterion

21. K-means Clustering – Details • Initial centroids are often chosen randomly. • Clusters produced vary from one run to another. • The centroid is (typically) the mean of the points in the cluster. • ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc. • K-means will converge for common distance functions. • Most of the convergence happens in the first few iterations. • Often the stopping condition is changed to ‘Until relatively few points change clusters’ • Complexity is O( n * K * I * d ) • n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

22. Optimal Clustering Sub-optimal Clustering Two different K-means Clusterings Original Points • Importance of choosing initial centroids

23. Evaluating K-means Clusters • 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 Ci and mi is the representative point for cluster Ci • can show that micorresponds 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

24. Solutions to Initial Centroids Problem • Multiple runs • Helps, but probability is not on your side • Sample and use hierarchical clustering to determine initial centroids • Select more than k initial centroids and then select among these initial centroids • Select most widely separated • Postprocessing • Bisecting K-means • Not as susceptible to initialization issues Handling Empty Clusters Basic K-means algorithm can yield empty clusters

25. Pre-processing and Post-processing • Pre-processing • Normalize the data • Eliminate outliers • Post-processing • Eliminate small clusters that may represent outliers • Split ‘loose’ clusters, i.e., clusters with relatively high SSE • Merge clusters that are ‘close’ and that have relatively low SSE

26. Bisecting K-means • Bisecting K-means algorithm • Variant of K-means that can produce a partitional or a hierarchical clustering

27. Bisecting K-means Example

28. Limitations of K-means • K-means has problems when clusters are of differing • Sizes • Densities • Non-globular shapes • K-means has problems when the data contains outliers.

29. Limitations of K-means: Differing Sizes K-means (3 Clusters) Original Points

30. Limitations of K-means: Differing Density K-means (3 Clusters) Original Points

31. Limitations of K-means: Non-globular Shapes Original Points K-means (2 Clusters)

32. Overcoming K-means Limitations Original Points K-means Clusters • One solution is to use many clusters. • Find parts of clusters, but need to put together.

33. Overcoming K-means Limitations Original Points K-means Clusters

34. Variations of the K-Means Method • A few variants of the k-means which differ in • Selection of the initial k means • Dissimilarity calculations • Strategies to calculate cluster means • Handling categorical data: k-modes (Huang’98) • Replacing means of clusters with modes • Using new dissimilarity measures to deal with categorical objects • Using a frequency-based method to update modes of clusters • Handling a mixture of categorical and numerical data: k-prototype method

35. The K-Medoids Clustering Method • Find representative objects, called medoids, in clusters • PAM (Partitioning Around Medoids, 1987) • starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering • PAM works effectively for small data sets, but does not scale well for large data sets • CLARA (Kaufmann & Rousseeuw, 1990) • draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output • CLARANS (Ng & Han, 1994): Randomized sampling • Focusing + spatial data structure (Ester et al., 1995)

36. Hierarchical Clustering • Produces a set of nested clusters organized as a hierarchical tree • Can be visualized as a dendrogram • A tree like diagram that records the sequences of merges or splits

37. Strengths of Hierarchical Clustering • 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 • They may correspond to meaningful taxonomies • Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

38. Hierarchical Clustering • Two main types of hierarchical clustering • Agglomerative: • Start with the points as individual clusters • At each step, merge the closest pair of clusters until only one cluster (or k clusters) left Matlab: Statistics Toolbox: clusterdata, which performs all these steps: pdist, linkage, cluster • Divisive: • Start with one, all-inclusive cluster • At each step, split a cluster until each cluster contains a point (or there are k clusters) • Traditional hierarchical algorithms use a similarity or distance matrix • Merge or split one cluster at a time • Image segmentation mostly uses simultaneous merge/split

39. Agglomerative Clustering Algorithm • More popular hierarchical clustering technique • Basic algorithm is straightforward • Compute the proximity matrix • Let each data point be a cluster • Repeat • Merge the two closest clusters • Update the proximity matrix • Until only a single cluster remains • Key operation is the computation of the proximity of two clusters • Different approaches to defining the distance between clusters distinguish the different algorithms

40. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . Starting Situation • Start with clusters of individual points and a proximity matrix Proximity Matrix

41. C1 C2 C3 C4 C5 C1 C2 C3 C4 C5 Intermediate Situation • After some merging steps, we have some clusters C3 C4 Proximity Matrix C1 C5 C2

42. C1 C2 C3 C4 C5 C1 C2 C3 C4 C5 Intermediate Situation • We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C3 C4 Proximity Matrix C1 C5 C2

43. After Merging C2 U C5 • The question is “How do we update the proximity matrix?” C1 C3 C4 C1 ? ? ? ? ? C2 U C5 C3 C3 ? C4 ? C4 Proximity Matrix C1 C2 U C5

44. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity Similarity? • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function • Ward’s Method uses squared error Proximity Matrix

45. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function • Ward’s Method uses squared error Proximity Matrix

46. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function • Ward’s Method uses squared error Proximity Matrix

47. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function • Ward’s Method uses squared error Proximity Matrix

48. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity   • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function • Ward’s Method uses squared error Proximity Matrix

49. 5 1 5 5 4 1 3 1 4 1 2 2 5 2 5 5 2 1 5 2 5 2 2 2 3 3 6 6 3 6 3 1 6 3 3 1 4 4 4 1 3 4 4 4 Hierarchical Clustering: Comparison MIN MAX Ward’s Method Group Average

50. Hierarchical Clustering: Time and Space requirements • O(N2) space since it uses the proximity matrix. • N is the number of points. • O(N3) time in many cases • There are N steps and at each step the size, N2, proximity matrix must be updated and searched • Complexity can be reduced to O(N2 log(N) ) time for some approaches