CISC 4631 Data Mining

1. CISC 4631Data Mining Lecture 08: • Clustering Theses slides are based on the slides by • Tan, Steinbach and Kumar (textbook authors) • EamonnKoegh(UC Riverside) • Raymond Mooney (UT Austin)

2. Inter-cluster distances are maximized Intra-cluster distances are minimized What is Clustering? • Finding groups of objects such that objects in a group will be similar to one another and different from the objects in other groups • Also called unsupervised learning, sometimes called classification by statisticians and sorting by psychologists and segmentation by people in marketing

3. What is a natural grouping among these objects? Clustering is subjective 3 Simpson's Family Females Males School Employees

4. Similarity is Subjective 4

5. Intuitions behind desirable distance measure properties • D(A,B) = D(B,A)Symmetry • Otherwise you could claim “Alex looks like Bob, but Bob looks nothing like Alex.” • D(A,A) = 0Constancy of Self-Similarity • Otherwise you could claim “Alex looks more like Bob, than Bob does.” • D(A,B) = 0 IIf A=B Positivity (Separation) • Otherwise there are objects in your world that are different, but you cannot tell apart. • D(A,B)  D(A,C) + D(B,C)Triangular Inequality • Otherwise you could claim “Alex is very like Bob, and Alex is very like Carl, but Bob is very unlike Carl.”

6. Applications of Cluster Analysis • Understanding • Group related documents for browsing, group genes and proteins that have similar functionality, group stocks with similar price fluctuations, or customers that have similar buying habits • Summarization • Reduce the size of large data sets Clustering precipitation in Australia

7. How many clusters? Six Clusters Two Clusters Four Clusters Notion of a Cluster can be Ambiguous So tell me how many clusters do you see?

8. Types of Clusterings • A clustering is a set of clusters • Important distinction between hierarchical and partitionalsets of clusters • Partitional Clustering • A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset • Hierarchical clustering • A set of nested clusters organized as a hierarchical tree

9. A Partitional Clustering Partitional Clustering Original Points

10. Hierarchical Clustering Traditional Hierarchical Clustering Traditional Dendrogram Simpsonian Dendrogram

11. Other Distinctions Between Sets of Clusters • Exclusive versus non-exclusive • In non-exclusive clusterings points may belong to multiple clusters • Can represent multiple classes or ‘border’ points • Fuzzy versus non-fuzzy • In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1 • Weights must sum to 1 • Probabilistic clustering has similar characteristics • Partial versus complete • In some cases, we only want to cluster some of the data

12. Types of Clusters • Well-separated clusters • Center-based clusters (our main emphasis) • Contiguous clusters • Density-based clusters • Described by an Objective Function

13. Types of Clusters: Well-Separated • Well-Separated Clusters: • 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

14. Types of Clusters: Center-Based • Center-based • 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 • The center of a cluster is often a centroid, the average of all the points in the cluster (assuming numerical attributes), or a medoid, the most “representative” point of a cluster (used if there are categorical features) 4 center-based clusters

15. 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

16. 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

17. Types of Clusters: Objective Function • Clusters Defined by an Objective Function • Finds 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) • Example: Sum of squares of distances to cluster center

18. Clustering Algorithms • K-means and its variants • Hierarchical clustering • Density-based clustering

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 • K-means tutorial available from http://maya.cs.depaul.edu/~classes/ect584/WEKA/k-means.html

20. K-means Clustering • Ask user how many clusters they’d like. (e.g. k=3) • Randomly guess k cluster Center locations • Each datapoint finds out which Center it’s closest to. • Each Center finds the centroid of the points it owns… • …and jumps there • …Repeat until terminated! 5 4 3 2 1 0 0 1 2 3 4 5

21. k1 k2 k3 K-means Clustering: Step 1 5 4 3 2 1 0 0 1 2 3 4 5

22. k1 k2 k3 K-means Clustering 5 4 3 2 1 0 0 1 2 3 4 5

23. k1 k2 k3 K-means Clustering 5 4 3 2 1 0 0 1 2 3 4 5

24. k1 k2 k3 K-means Clustering 5 4 3 2 1 0 0 1 2 3 4 5

25. k1 k2 k3 K-means Clustering

26. 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, correlation, etc. • K-means will converge for common similarity measures mentioned above. • Most of the convergence happens in the first few iterations. • Often the stopping condition is changed to ‘Until relatively few points change clusters’

27. 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. • We can show that to minimize SSE the best update strategy is to use the center 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

28. Optimal Clustering Sub-optimal Clustering Two different K-means Clusterings Original Points

29. Importance of Choosing Initial Centroids If you happen to choose good initial centroids, then you will get this after 6 iterations

30. Importance of Choosing Initial Centroids Good clustering

31. Importance of Choosing Initial Centroids … Bad Clustering

32. 10 Clusters Example Starting with two initial centroids in one cluster of each pair of clusters

33. 10 Clusters Example Starting with some pairs of clusters having three initial centroids, while other have only one.

34. 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

35. Limitations of K-means • K-means has problems when clusters are of differing • Sizes (biased toward the larger clusters) • Densities • Non-globular shapes • K-means has problems when the data contains outliers.

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

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

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

39. 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.

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

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

42. 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

43. South Georgia & South Sandwich Islands Serbia & Montenegro (Yugoslavia) St. Helena & Dependencies U.K. AUSTRALIA ANGUILLA FRANCE NIGER INDIA IRELAND BRAZIL • Hierarchal clustering can sometimes show patterns that are meaningless or spurious • For example, in this clustering, the tight grouping of Australia, Anguilla, St. Helena etc is meaningful, since all these countries are former UK colonies. • However the tight grouping of Niger and India is completely spurious, there is no connection between the two.

44. South Georgia & South Sandwich Islands Serbia & Montenegro (Yugoslavia) St. Helena & Dependencies U.K. AUSTRALIA ANGUILLA FRANCE NIGER INDIA IRELAND BRAZIL • The flag of Niger is orange over white over green, with an orange disc on the central white stripe, symbolizing the sun. The orange stands the Sahara desert, which borders Niger to the north. Green stands for the grassy plains of the south and west and for the River Niger which sustains them. It also stands for fraternity and hope. White generally symbolizes purity and hope. • The Indian flag is a horizontal tricolor in equal proportion of deep saffron on the top, white in the middle and dark green at the bottom. In the center of the white band, there is a wheel in navy blue to indicate the Dharma Chakra, the wheel of law in the Sarnath Lion Capital. This center symbol or the 'CHAKRA' is a symbol dating back to 2nd century BC. The saffron stands for courage and sacrifice; the white, for purity and truth; the green for growth and auspiciousness.

45. We can look at the dendrogram to determine the “correct” number of clusters. In this case, the two highly separated subtrees are highly suggestive of two clusters. (Things are rarely this clear cut, unfortunately)

46. One potential use of a dendrogram is to detect outliers The single isolated branch is suggestive of a data point that is very different to all others Outlier

47. Hierarchical Clustering Build a tree-based hierarchical taxonomy (dendrogram) from a set of unlabeled examples. Recursive application of a standard clustering algorithm can produce a hierarchical clustering. animal vertebrate invertebrate fish reptile amphib. mammal worm insect crustacean

48. 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, …)

49. There is only one dataset that can be perfectly clustered using a hierarchy… (Bovine:0.69395, (Spider Monkey 0.390, (Gibbon:0.36079,(Orang:0.33636,(Gorilla:0.17147,(Chimp:0.19268, Human:0.11927):0.08386):0.06124):0.15057):0.54939);

50. 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 • Divisive: • Start with one, all-inclusive cluster • At each step, split a cluster until each cluster contains a point (or there are k clusters) • Agglomerative is most common