Cluster Analysis: Agglomerate Hierarchical Clustering

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1 Cluster Analysis: Agglomerate Hierarchical Clustering Yonghee Lee Department of Statistics, The University of Seoul Oct 29, 2015

2 Contents 1 Cluster Analysis Introduction Distance matrix Agglomerative Hierarchical Methods Example: Nearest neighbor linkage Example: Complete neighbor linkage Example: Group average linkage Remarks on Hierarchical Methods Nonhierarchical Clustering Methods

3 Classification vs Clustering Classification The known number of groups and group memberships for the training data Want to assign new observations to one of the groups Supervised learning (learning with teacher) Clustering No assumption for the number of groups and memberships No assumption for the structure of groups Unsupervised learning (learning without teacher) Use the values of the variables to devise an algorithm for grouping the objects into groups Similar s are in the same group Measure for distance (similarity)

4 Distance Measure Two p-dimensional vectors: x = (x 1, x 2,, x p ) and y = (y 1, y 2,, y p ) Euclidean distance d(x, y) = p (x y) (x y) = (x i y i ) 2 Statistical distance i=1 d(x, y) = (x y) A(x y) where usually A = S 1 and S is sample covariance matrix of x and y

5 Hierarchical Clustering Methods Definition (Hierarchical Methods) First, distances of all pairs of s are calculated Then groups are formed by a process of agglomeration or division Agglomerative Hierarchical method A series of successive merges Divisive Hierarchical method A series of successive divisions Dendrogram (Tree diagram): Two dimensional graph illustrating merges and divisions Agglomerative Hierarchical method: linkage methods Nearest neighbor linkage Complete neighbor linkage Group average linkage

6 Distance matrix of s Table : A distance matrix of five s sub

7 Agglomerative Hierarchical Clustering Methods Definition (Algorithm) 1 Start with N cluster containing one item and find N N symmetric matrix of distances D = {d ij } 2 Search the distance matrix for the nearest pair of clusters Let U and V be two nearest clusters 3 Merge the two clusters U and V and update the matrix of distances D 4 Repeat steps 2 and 3 a total of N 1 times Record the order of merges of clusters and their levels (distances)

8 Agglomerative Hierarchical Clustering Methods Linkage Methods: How to determine the distance of two clusters? Suppose two clusters U and V are merged and how to define new distance with the cluster W Nearest neighbor linkage: minimum distance d (UV )W = min(d U,W, d V,W ) Complete neighbor linkage: maximum distance d (UV )W = max(d U,W, d V,W ) Group average linkage: average distance d (UV )W = d ij /(N (UV ) N W ) i where object i belongs to (UV ) and j belongs to W k

9 Example: Nearest neighbor linkage Table : Nearest neighbor linkage: A distance matrix of five s sub Step 1 The minimum distance is 2 of two s 3 and 5 So s 3 and 5 are merged Recalculate the distance matrix such that d (3,5),1 = min(d (3,1), d (5,1) ) = min(3, 11) = 3 d (3,5),2 = min(d (3,2), d (5,2) ) = min(7, 10) = 7

10 Example: Nearest neighbor linkage Table : Nearest neighbor linkage - A distance matrix: after step 1 sub (3,5) (3,5) Step 2 The minimum distance is 3 between the 1 and the cluster (3,5) So the 1 and the cluster (3,5) are merged Recalculate the distance matrix such that d (1,3,5),2 = min(d (3,5),2, d (1,2) ) = min(7, 9) = 7 d (1,3,5),4 = min(d (3,5),4, d (1,4) ) = min(8, 6) = 6

11 Example: Nearest neighbor linkage Table : Nearest neighbor linkage - A distance matrix: after step 2 sub (1,3,5) 2 4 (1,3,5) Step 3 The minimum distance is 5 between the s 2 and 4 So the s 2 and 4 are merged Recalculate the distance matrix such that d (1,3,5),(2,4) = min(d (1,3,5),2, d (1,3,5),2 ) = min(7, 6) = 6

12 Example: Nearest neighbor linkage Table : Nearest neighbor linkage - A distance matrix: after step 3 sub (1,3,5) (2,4) (1,3,5) - (2,4) 6 - Finally, the cluster (1,3,5) and the cluster (2,4) are merged

13 Example: Complete neighbor linkage Table : Complete neighbor linkage: A distance matrix of five s sub Step 1 The minimum distance is 2 of two s 3 and 5 So s 3 and 5 are merged Recalculate the distance matrix such that d (3,5),1 = max(d (3,1), d (5,1) ) = max(3, 11) = 11 d (3,5),2 = max(d (3,2), d (5,2) ) = max(7, 10) = 10

14 Example: Complete neighbor linkage Table : Complete neighbor linkage - A distance matrix: after step 1 sub (3,5) (3,5) Step 2 The minimum distance is 5 between the s 2 and 4 So the s 2 and 4 are merged Recalculate the distance matrix such that d (2,4),(3,5) = max(d 2,(3,5), d 4,(3,5) ) = max(10, 9) = 10 d (2,4),1 = max(d 2,1, d 4,1 ) = max(9, 6) = 9

15 Example: Complete neighbor linkage Table : Complete neighbor linkage - A distance matrix: after step 2 sub (3,5) (2,4) 1 (3,5) - (2,4) Step 3 The minimum distance is 9 between the 1 and the cluster (2,4) So the 1 and the cluster (2,4) are merged Recalculate the distance matrix such that d (1,2,4),(3,5) = max(d 1,(3,5), d (2,4),(3,4) ) = max(11, 10) = 11

16 Example: Complete neighbor linkage Table : Complete neighbor linkage - A distance matrix: after step 3 sub (1,2,4) (3,5) (1,2,4) - (3,5) 11 - Finally, the cluster (1,2,4) and the cluster (3,5) are merged

17 Example: Group average linkage Table : Group average linkage - A distance matrix of five s sub Step 1 The minimum distance is 2 of two s 3 and 5 So s 3 and 5 are merged Recalculate the distance matrix such that d (3,5),1 = mean(d (3,1), d (5,1) ) = mean(3, 11) = (11 + 3)/2 = 7 d (3,5),2 = mean(d (3,2), d (5,2) ) = mean(7, 10) = (7 + 10)/2 = 85

18 Example: Group average linkage Table : Group average linkage - A distance matrix: after step 1 sub (3,5) (3,5) Step 2 The minimum distance is 5 between the s 2 and 4 So the s 2 and 4 are merged Recalculate the distance matrix such that d (2,4),(3,5) = mean(d 2,3, d 2,5, d 4,3, d 4,5 ) = mean(7, 10, 9, 8) = ( )/85 Calculations continue

19 Remarks on Hierarchical Methods Divisive Hierarchical methods have been used less often than Agglomerative Hierarchical Methods There are many (many) algorithms for cluster analysis, but there is no generally accepted best method Performance of cluster analysis depends on the shape of clusters, especially how clusters are overlapped (see pp 129 of textbook)

20 Nonhierarchical Clustering Methods The distance matrix D is not required The number of clusters is determined before or during the procedure Example: K- means method Algorithm for K-means method 1 Partition the item into K initial clusters, equivalently into K centroids (means) 2 Proceed through the list of items, assigning an item to the cluster whose centroid is nearest (example Euclidean distance) Recalculate the centroids for clusters losing and receiving the item 3 Repeat steps 2 until no more reassignings take place

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