Hierarchical Clustering. Clustering Overview

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1 lustering Overview Last lecture What is clustering Partitional algorithms: K-means Today s lecture Hierarchical algorithms ensity-based algorithms: SN Techniques for clustering large databases IRH UR ata Mining: lustering Hierarchical lustering Produces a set of nested clusters organized as a hierarchical tree an be visualized as a dendrogram tree like diagram that records the sequences of merges or splits ata Mining: lustering 6 Strengths of Hierarchical lustering We do not have to assume (or provide) any particular number of clusters ny desired number of clusters can be obtained by cutting the dendogram at the proper level They may correspond to meaningful taxonomies xamples in biological sciences (e.g., animal kingdom, phylogeny reconstruction, ) natural languages Hierarchical lustering Two main types of hierarchical clustering gglomerative Start with the points as individual clusters t each step, merge the closest pair of clusters until there is only one cluster (or k clusters) left ivisive Start with one, all-inclusive cluster t 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 ata Mining: lustering ata Mining: lustering gglomerative lustering lgorithm More popular hierarchical clustering technique asic algorithm is straightforward. ompute the proximity matrix. Let each data point be a cluster. Repeat. Merge the two closest clusters. Update the proximity matrix 6. Until only a single cluster remains Key operation is the computation of the proximity of two clusters ifferent approaches to defining the distance between clusters distinguish the different algorithms ata Mining: lustering 6 Starting Situation Start with clusters of individual points and a proximity matrix p p p p p.. p p p p p.... Proximity Matrix... p p p p p9 p p p ata Mining: lustering 7

2 Intermediate Situation Intermediate Situation fter some merging steps, we get some clusters We want to merge the two closest clusters ( and ) and update the proximity matrix. Proximity Matrix Proximity Matrix... p p p p p9 p p p ata Mining: lustering 8... p p p p p9 p p p ata Mining: lustering 9 fter Merging The question is How do we update the proximity matrix? U U??????? Proximity Matrix istance etween lusters Single Link: smallest distance between points omplete Link: largest distance between points verage Link: average distance between points entroid: distance between centroids U... p p p p p9 p p p ata Mining: lustering 6 ata Mining: lustering 6 Hierarchical lustering lgorithms Single Link MST (Minimum Spanning Tree) Single Link omplete Link verage Link luster Similarity: MIN or Single Link Similarity of two clusters is based on the two most similar (closest) points in the different clusters etermined by one pair of points, i.e., by one link in the proximity graph. I I I I I I I I I I ata Mining: lustering 6 ata Mining: lustering 6

3 gglomerative MIN lustering xample Hierarchical lustering: MIN Threshold of ata Mining: lustering 6 Nested lusters endrogram ata Mining: lustering 6 Strength of MIN Limitations of MIN Two lusters Two lusters an handle non-elliptical shapes Sensitive to noise and outliers ata Mining: lustering 66 ata Mining: lustering 67 luster Similarity: MX or omplete Linkage Hierarchical lustering: MX Similarity of two clusters is based on the two least similar (most distant) points in the different clusters etermined by all pairs of points in the two clusters I I I I I I I I I I Nested lusters endrogram ata Mining: lustering 68 ata Mining: lustering 69

4 Strength of MX Limitations of MX Two lusters Two lusters Less susceptible to noise and outliers ata Mining: lustering 7 Tends to break large clusters iased towards globular clusters ata Mining: lustering 7 luster Similarity: Group verage Hierarchical lustering: Group verage Proximity of two clusters is the average of pairwise proximity between points in the two clusters. pi lusteri p luster proximity(p,p) j j proximity(lusteri,lusterj) = luster luster Need to use average connectivity for scalability since total proximity favors large clusters I I I I I I I I I I ata Mining: lustering 7 i i j j Nested lusters endrogram ata Mining: lustering 7 Hierarchical lustering: Group verage ompromise between Single and omplete Link Strengths Less susceptible to noise and outliers Limitations iased towards globular clusters ata Mining: lustering 7 luster Similarity: Ward s Method Similarity of two clusters is based on the increase in squared error when two clusters are merged Similar to group average if distance between points is distance squared Less susceptible to noise and outliers iased towards globular clusters Hierarchical analogue of K-means an be used to initialize K-means ata Mining: lustering 7

5 Hierarchical lustering: omparison Hierarchical lustering: Time & Space Requirements 6 6 MIN MX Ward s Method Group verage 6 6 O(N ) space since it uses the proximity matrix. N is the number of points. O(N ) time in many cases There are N steps and at each step the size, N, proximity matrix must be updated and searched omplexity can be reduced to O(N log(n) ) time for some approaches ata Mining: lustering 76 ata Mining: lustering 77 Hierarchical lustering: Problems and Limitations Once a decision is made to combine two clusters, it cannot be undone No objective function is directly minimized ifferent schemes have problems with one or more of the following: Sensitivity to noise and outliers ifficulty handling different sized clusters and convex shapes reaking large clusters ata Mining: lustering 78 Partitional lustering (reminder) Nonhierarchical Usually deals with static sets reates clusters in one step as opposed to several steps Since only one set of clusters is output, the user normally has to input the desired number of clusters, k Need some metric/criterion that determines the goodness of each proposed solution ata Mining: lustering 79 Some Partitional lustering lgorithms MST xample MST: Minimum Spanning Tree Squared rror K-Means Nearest Neighbor PM: Partitioning round Medoids LR: lustering LRge pplications LRNS: lustering Large pplications based on RNdomized Search ata Mining: lustering 8 ata Mining: lustering 8

6 MST lgorithm Sum Squared rror (SS) ata Mining: lustering 8 ata Mining: lustering 8 Squared rror lgorithm Nearest Neighbor lustering Items are iteratively merged into the existing clusters that are closest Incremental and serial algorithm Threshold, t, used to determine if items are added to existing clusters or whether a new cluster should be created ata Mining: lustering 8 ata Mining: lustering 8 Nearest Neighbor lustering lgorithm PM: Partitioning round Medoids (K-Medoids) Handles outliers well Ordering of input does not impact results omputationally complex - does not scale well ach cluster represented by one item, called the medoid Initial set of k medoids is randomly chosen and all items which are not medoids are examined to see if they should replace an existing medoid. ata Mining: lustering 86 ata Mining: lustering 87 6

7 PM ost alculation PM lgorithm t each step in algorithm, medoids are changed if the overall cost is improved. jih cost change for an item t j associated with swapping medoid t i with non-medoid t h. ata Mining: lustering 88 ata Mining: lustering 89 PM xample LR and LRNS LR: lustering LRge pplications. etermine set of medoids M by using a sample of the database, where <<. o clustering of based on the set of medoids M LRNS: lustering Large pplications based on RNdomized Search Improved LR clustering algorithm which uses several randomly picked samples instead of only one ata Mining: lustering 9 ata Mining: lustering 9 SN: ensity ased Spatial lustering of pplications with Noise SN xample ased on the notion of density Outliers will not effect creation of clusters Two parameters which are input: MinPts minimum number of points in cluster ps for each point in cluster there must be another point in it less than this distance away The number of clusters, k, is not input but is determined by the algorithm itself. ata Mining: lustering 9 ata Mining: lustering 9 7

8 SN xample SN xample ata Mining: lustering 9 ata Mining: lustering 9 SN SN is a density-based algorithm ensity = number of points within a specified radius (ps) point is a core point if it has more than a specified number of points (MinPts) within ps These are points that are at the interior of a cluster border point has fewer than MinPts within ps, but is in the neighborhood of a core point noise point is any point that is not a core point or a border point. ata Mining: lustering 96 SN ensity oncepts ps-neighborhood: Points within ps distance of a point. ore point: Point whose ps-neighborhood is dense enough (MinPts) and forms the main portion of some cluster irectly density-reachable: point p is directly density-reachable from a point q if the distance between them is small (ps) and q is a core point. ensity-reachable: point is density-reachable form another point if there is a path from one to the other consisting of only core points. ata Mining: lustering 97 ensity oncepts used in SN SN lgorithm ps Minpts = Minpts = ps liminate noise points Perform clustering on the remaining points p q ps-neighborhood ore points & order points ensity reachable ata Mining: lustering 98 ata Mining: lustering 99 8

9 SN: ore, order and Noise Points When SN Works Well ps =, MinPts = Point types: core, border and noise ata Mining: lustering lusters Resistant to Noise an handle clusters of different shapes and sizes ata Mining: lustering When SN oes NOT Work Well SN: etermining PS and MinPts (MinPts=, ps=9.7). Idea is that for points in a cluster, their kth nearest neighbors are at roughly the same distance Noise points have the kth nearest neighbor at farther distance So, plot sorted distance of every point to its kth nearest neighbor Varying densities High-dimensional data (MinPts=, ps=9.9) ata Mining: lustering ata Mining: lustering lustering Large atabases Most clustering algorithms assume a large data structure which is memory resident lustering may be performed first on a sample of the database, then applied to the entire database lgorithms IRH UR lustering Large atabases: esired Features One scan (or less) of Online ble to report clustering status while running Suspendable, stopable, resumable Incremental ble to handle dynamic updates ble to work with limited main memory ifferent techniques to scan the (e.g. use sampling) Process each tuple once ata Mining: lustering ata Mining: lustering 9

10 IRH: alanced Iterative Reducing and lustering using Hierarchies Incremental, hierarchical, one scan Saves clustering information in a balanced tree ach entry in the tree contains summary information about one cluster New nodes inserted in closest entry in tree dapts to main memory size by changing the threshold value Larger threshold Smaller tree lustering Feature F Triple: (N,LS,SS) N: Number of points in cluster LS: Sum of points in the cluster SS: Sum of squares of points in the cluster F Tree alanced search tree Node has F triple for each child Leaf node represents cluster and has F value for each subcluster in it Subcluster has maximum diameter ata Mining: lustering 6 ata Mining: lustering 7 IRH lgorithm IRH: Improving lusters. reate initial F tree using lgorithm. If there is insufficient memory to construct the F tree with a given threshold, the threshold value is increased and a new smaller F tree is constructed.. pply another global clustering approach applied to the leaf nodes in the F tree. Here each leaf node is treated as a single point for clustering.. The last phase (which is optional) re-clusters all points by placing them in the cluster which has the closest centroid. ata Mining: lustering 8 ata Mining: lustering 9 UR: lustering Using Rpresentatives UR pproach Use many points to represent a cluster instead of only one Representative points need to be well scattered Handles outliers well removes clusters which ither grow very slowly Or contain very few points ata Mining: lustering ata Mining: lustering

11 UR lgorithm UR for Large atabases. Obtain a sample of the database. Partition the sample into p partitions. Partially cluster the points in each partition. Remove outliers based on size of cluster. ompletely cluster all data in the samples (representatives) 6. luster entire database on disk using c points to represent each cluster. n item in the database is placed in the cluster which has the closest representative point to it. ata Mining: lustering ata Mining: lustering omparison of lustering Techniques lustering Validity For supervised classification we have a variety of measures to evaluate how good our model is ccuracy, precision, recall For cluster analysis, the analogous question is how to evaluate the goodness of the resulting clusters? ut clusters are in the eye of the beholder! ata Mining: lustering ata Mining: lustering

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