Unsupervised learning & Cluster Analysis: Basic Concepts and Algorithms

1. Unsupervised learning &Cluster Analysis: Basic Concepts and Algorithms Assaf Gottlieb Some of the slides are taken form Introduction to data mining, by Tan, Steinbach, and Kumar

2. What is unsupervised learning & Cluster Analysis ? • Learning without a priori knowledge about the classification of samples; learning without a teacher. Kohonen (1995), “Self-Organizing Maps” • “Cluster analysis is a set of methods for constructing a (hopefully) sensible and informative classification of an initially unclassified set of data, using the variable values observed on each individual.” B. S. Everitt (1998), “The Cambridge Dictionary of Statistics”

3. What do we cluster? Features/Variables Samples/Instances

4. Applications of Cluster Analysis • UnderstandingGroup related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations • Data Exploration • Get insight into data distribution • Understand patterns in the data • SummarizationReduce the size of large data setsA preprocessing step

5. Objectives of 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 Competing objectives Intra-cluster distances are minimized Inter-cluster distances are maximized

6. How many clusters? Six Clusters Two Clusters Four Clusters Notion of a Cluster can be Ambiguous Depends on “resolution” !

7. Prerequisites • Understand the nature of your problem, the type of features, etc. • The metric that you choose for similarity (for example, Euclidean distance or Pearson correlation) often impacts the clusters you recover.

8. Similarity/Distance measures • Euclidean Distance • Highly depends on scaleof features may require normalization • City Block

9. deuc=0.5846 deuc=1.1345 These examples of Euclidean distance match our intuition of dissimilarity pretty well… deuc=2.6115

10. deuc=1.41 deuc=1.22 …But what about these? What might be going on with the expression profiles on the left? On the right?

11. Similarity/Distance measures • Cosine • Pearson Correlation • Invariant to scaling (Pearson also to addition) • Spearman correlation for ranks

12. Similarity/Distance measures • Jaccard similarity • When interested in intersection size X U Y X X ∩ Y Y

13. Types of Clusterings • 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

14. A Partitional Clustering Partitional Clustering Original Points

15. Hierarchical Clustering Dendrogram 1 Dendrogram 2

16. Other Distinctions Between Sets of Clustering methods • 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 • Partial versus complete • In some cases, we only want to cluster some of the data • Heterogeneous versus homogeneous • Cluster of widely different sizes, shapes, and densities

17. Clustering Algorithms • Hierarchical clustering • K-means • Bi-clustering

18. 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

19. 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, …)

20. Hierarchical Clustering • Two main types of hierarchical clustering • Agglomerative (bottom up): • 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 (top down): • 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

21. 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

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

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

24. 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 C1 Proximity Matrix C5 C2

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

26. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity Similarity? • MIN • MAX • Group Average • Distance Between Centroids • Ward’s method (not discussed) Proximity Matrix

27. 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

28. 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

29. 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

30. p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . How to Define Inter-Cluster Similarity   • MIN • MAX • Group Average • Distance Between Centroids Proximity Matrix

31. 1 2 3 4 5 Cluster Similarity: MIN or Single Link • 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.

32. 5 1 3 5 2 1 2 3 6 4 4 Hierarchical Clustering: MIN Nested Clusters Dendrogram

33. Two Clusters Strength of MIN Original Points • Can handle non-elliptical shapes

34. Two Clusters Limitations of MIN Original Points • Sensitive to noise and outliers

35. 1 2 3 4 5 Cluster Similarity: MAX or Complete Linkage • 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

36. 4 1 2 5 5 2 3 6 3 1 4 Hierarchical Clustering: MAX Nested Clusters Dendrogram

37. Two Clusters Strength of MAX Original Points • Less susceptible to noise and outliers

38. Two Clusters Limitations of MAX Original Points • Tends to break large clusters • Biased towards globular clusters

39. 1 2 4 5 3 Cluster Similarity: Group Average • Proximity of two clusters is the average of pairwise proximity between points in the two clusters. • Need to use average connectivity for scalability since total proximity favors large clusters

40. 5 4 1 2 5 2 3 6 1 4 3 Hierarchical Clustering: Group Average Nested Clusters Dendrogram

41. Hierarchical Clustering: Group Average • Compromise between Single and Complete Link • Strengths • Less susceptible to noise and outliers • Limitations • Biased towards globular clusters

42. 5 1 5 3 1 4 1 2 5 2 5 2 1 5 5 2 2 2 3 6 3 6 3 6 3 3 1 4 4 1 4 4 4 Hierarchical Clustering: Comparison MAX MIN Group Average

43. Hierarchical Clustering: Problems and Limitations • Once a decision is made to combine two clusters, it cannot be undone • Different schemes have problems with one or more of the following: • Sensitivity to noise and outliers • Difficulty handling different sized clusters and convex shapes • Breaking large clusters (divisive) • Dendrogram correspond to a given hierarchical clustering is not unique, since for each merge one needs to specify which subtree should go on the left and which on the right • They impose structure on the data, instead of revealing structure in these data. • How many clusters? (some suggestions later)

44. 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

45. 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 mostly by Euclidean distance, cosine similarity, 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’ • Complexity is O( n * K * I * d ) • n = number of points, K = number of clusters, I = number of iterations, d = number of attributes Typical choice

46. 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

47. Issues and Limitations for K-means • How to choose initial centers? • How to choose K? • How to handle Outliers? • Clusters different in • Shape • Density • Size • Assumes clusters are spherical in vector space • Sensitive to coordinate changes

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

49. Importance of Choosing Initial Centroids

50. Importance of Choosing Initial Centroids