1. Machine Learning CS 165B Spring 2012 1

2. Course outline Introduction (Ch. 1) Concept learning (Ch. 2) Decision trees (Ch. 3) Ensemble learning Neural Networks (Ch. 4) Linear classifiers Support Vector Machines Bayesian Learning (Ch. 6) Genetic Algorithms (Ch. 9) Instance-based Learning (Ch. 8) Clustering Computational learning theory (Ch. 7) 2

3. Organizing data into classes such that there is high intra-class similarity low inter-class similarity Finding the class labels and the number of classes directly from the data (in contrast to classification). More informally, finding natural groupings among objects. What is clustering? 3 Remember Fischer’s discriminant

4. What is a natural grouping among these objects? 4

5. School Employees Simpson's Family MalesFemales Clustering is subjective What is a natural grouping among these objects? 5

6. What is similarity? The quality or state of being similar; likeness; resemblance; as, a similarity of features. Similarity is hard to define, but… “We know it when we see it” The real meaning of similarity is a philosophical question. We will take a more pragmatic approach. Webster's Dictionary 6

7. Defining distance measures Definition: Let O 1 and O 2 be two objects from the universe of possible objects. The distance (dissimilarity) between O 1 and O 2 is a real number denoted by D(O 1,O 2 ) 0.233342.7 PeterPiotr 7

8. What properties should a distance measure have? D(A,B) = D(B,A)Symmetry D(A,B) = 0 iff A= B Reflexive D(A,B)  D(A,C) + D(B,C)Triangle Inequality PeterPiotr 3 d('', '') = 0 d(s, '') = d('', s) = |s| -- i.e. length of s d(s1+ch1, s2+ch2) = min( d(s1, s2) + if ch1=ch2 then 0 else 1 fi, d(s1+ch1, s2) + 1, d(s1, s2+ch2) + 1 ) When we peek inside one of these black boxes, we see some function on two variables. These functions might very simple or very complex. In either case it is natural to ask, what properties should these functions have? 8

9. Two types of clusteringHierarchical Partitional algorithms: Construct various partitions and then evaluate them by some criterion Hierarchical algorithms: Create a hierarchical decomposition of the set of objects using some criterion Partitional 9

10. Desirable Properties of clustering algorithm Scalability (in terms of both time and space) Ability to deal with different data types Minimal requirements for domain knowledge to determine input parameters Able to deal with noise and outliers Insensitive to order of input records Incorporation of user-specified constraints Interpretability and usability 10

11. Summarizing similarity measurements In order to better appreciate and evaluate the examples given in the early part of this talk, we will now introduce the dendrogram. The similarity between two objects in a dendrogram is represented as the height of the lowest internal node they share. 11

12. Pedro (Portuguese) Petros (Greek), Peter (English), Piotr (Polish), Peadar (Irish), Pierre (French), Peder (Danish), Peka (Hawaiian), Pietro (Italian), Piero (Italian Alternative), Petr (Czech), Pyotr (Russian) Cristovao (Portuguese) Christoph (German), Christophe (French), Cristobal (Spanish), Cristoforo (Italian), Kristoffer (Scandinavian), Krystof (Czech), Christopher (English) Miguel (Portuguese) Michalis (Greek), Michael (English), Mick (Irish!) Hierarchical clustering using string edit distance Piotr Pyotr Petros Pietro Pedro Pierre Piero Peter Pede r Peka Peadar Michalis Michael Miguel Mick Cristovao Christopher Christophe Christoph Crisdean Cristobal Cristoforo Kristoffer Krystof 12

13. (How-to) Hierarchical clustering The number of dendrograms with n leafs = (2n -3)!/[(2 (n -2) ) (n -2)!] Number Number of Possible of LeafsDendrograms 213 415 5105...… 10 34,459,425 Since we cannot test all possible trees we will have to use heuristics: Bottom-Up (agglomerative): Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. Top-Down (divisive): Starting with all the data in a single cluster, consider every possible way to divide the cluster into two. Choose the best division and recursively operate on both sides. 13

14. 08877 0244 033 01 0 D(, ) = 8 D(, ) = 1 We begin with a distance matrix which contains the distances between every pair of objects in our database. 14 Distance matrix

15. Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best 15 Bottom-Up (agglomerative)

16. Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best Consider all possible merges… … Choose the best 16 Bottom-Up (agglomerative)

17. Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best Consider all possible merges… … Choose the best Consider all possible merges… Choose the best … 17 Bottom-Up (agglomerative)

18. Starting with each item in its own cluster, find the best pair to merge into a new cluster. Repeat until all clusters are fused together. … Consider all possible merges… Choose the best Consider all possible merges… … Choose the best Consider all possible merges… Choose the best … 18 Bottom-Up (agglomerative)

19. We know how to measure the distance between two objects, but defining the distance between an object and a cluster, or defining the distance between two clusters is not obvious. Single linkage (nearest neighbor): Single linkage (nearest neighbor): In this method the distance between two clusters is determined by the distance of the two closest objects (nearest neighbors) in the different clusters. Complete linkage (farthest neighbor): Complete linkage (farthest neighbor): In this method, the distances between clusters are determined by the greatest distance between any two objects in the different clusters (i.e., by the "furthest neighbors"). Group average linkage Group average linkage: In this method, the distance between two clusters is calculated as the average distance between all pairs of objects in the two different clusters. 19 Extending distance measure to clusters

20. No need to specify the number of clusters in advance. Hierarchal nature maps nicely onto human intuition for some domains They do not scale well: time complexity of at least O(n 2 ), where n is the number of total objects. Like any heuristic search algorithms, local optima are a problem. 20 Summary of hierarchal clustering

21. Partitional clustering Nonhierarchical, each instance is placed in exactly one of K nonoverlapping clusters. Since only one set of clusters is output, the user normally has to input the desired number of clusters K. 21

22. 1. Decide on a value for k. 2. Initialize the k cluster centers (randomly, if necessary). 3. Decide the class memberships of the N objects by assigning them to the nearest cluster center. 4. Re-estimate the k cluster centers, by assuming the memberships found above are correct. 5. If none of the N objects changed membership in the last iteration, exit. Otherwise goto 3. 22 Algorithm k-means

23. 0 1 2 3 4 5 012345 K-means clustering: step 1 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3 23

24. 0 1 2 3 4 5 012345 K-means clustering: step 2 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3 24

25. 0 1 2 3 4 5 012345 K-means clustering: step 3 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3 25

26. 0 1 2 3 4 5 012345 K-means clustering: step 4 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3 26

27. K-means clustering: step 5 Algorithm: k-means, Distance Metric: Euclidean Distance k1k1 k2k2 k3k3 27

28. Evaluation of K-means Strength –Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. –Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms Weakness –Applicable only when mean is defined, then what about categorical data? –Need to specify k, the number of clusters, in advance –Unable to handle noisy data and outliers –Not suitable for clusters with non-convex shapes 28

29. Other clustering methods Density based clustering Subspace clustering EM clustering Gaussian Mixture Models Other concepts Dimensionality reduction PCA 29