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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 What is Cluster Analysis? l 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 Inter-cluster distances are maximized Intra-cluster distances are minimized

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 2 Applications of Cluster Analysis l Understanding –Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations l Summarization –Reduce the size of large data sets Clustering precipitation in Australia

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 3 What is not Cluster Analysis? l Supervised classification –Have class label information l Simple segmentation –Dividing students into different registration groups alphabetically, by last name l Results of a query –Groupings are a result of an external specification l Graph partitioning –Some mutual relevance and synergy, but areas are not identical

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 4 Notion of a Cluster can be Ambiguous How many clusters? Four ClustersTwo Clusters Six Clusters

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 5 Types of Clusterings l A clustering is a set of clusters l Important distinction between hierarchical and partitional sets of clusters l Partitional Clustering –A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset l Hierarchical clustering –A set of nested clusters organized as a hierarchical tree

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 6 Partitional Clustering Original Points A Partitional Clustering

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 7 Hierarchical Clustering Traditional Hierarchical Clustering Non-traditional Hierarchical ClusteringNon-traditional Dendrogram Traditional Dendrogram

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 8 Other Distinctions Between Sets of Clusters l Exclusive versus non-exclusive –In non-exclusive clusterings, points may belong to multiple clusters. –Can represent multiple classes or ‘border’ points l 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

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 9 Types of Clusters: Center-Based l 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, or a medoid, the most “representative” point of a cluster 4 center-based clusters

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 10 Clustering Algorithms l K-means and its variants l Hierarchical clustering

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 11 K-means Clustering l Partitional clustering approach l Each cluster is associated with a centroid (center point) l Each point is assigned to the cluster with the closest centroid l Number of clusters, K, must be specified l The basic algorithm is very simple

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 12 K-means Clustering – Details l Initial centroids are often chosen randomly. –Clusters produced vary from one run to another. l The centroid is (typically) the mean of the points in the cluster. l ‘Closeness’ is measured by Euclidean distance (or other norms) l K-means will converge for common similarity measures mentioned above. l Most of the convergence happens in the first few iterations. –Often the stopping condition is changed to ‘Until relatively few points change clusters’ l Complexity is O( n * K * I * d ) –n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 13 Two different K-means Clusterings Sub-optimal ClusteringOptimal Clustering Original Points

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 14 Importance of Choosing Initial Centroids

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 15 Evaluating K-means Clusters l 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 C i and m i is the representative point for cluster C i can show that m i corresponds 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

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 16 Problems with Selecting Initial Points l If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small. –Chance is relatively small when K is large –If clusters are the same size, n, then –For example, if K = 10, then probability = 10!/10 10 = 0.00036 –Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t –Consider an example of five pairs of clusters

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 17 Solutions to Initial Centroids Problem l Multiple runs –Helps, but probability is not on your side l Sample and use hierarchical clustering to determine initial centroids l Select more than k initial centroids and then select among these initial centroids –Select most widely separated l Postprocessing

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 18 Handling Empty Clusters l Basic K-means algorithm can yield empty clusters l Several strategies –Choose the point that contributes most to SSE –Choose a point from the cluster with the highest SSE –If there are several empty clusters, the above can be repeated several times.

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 19 Updating Centers Incrementally l In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid l An alternative is to update the centroids after each assignment (incremental approach) –Each assignment updates zero or two centroids –More expensive –Introduces an order dependency –Never get an empty cluster –Can use “weights” to change the impact

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 20 Pre-processing and Post-processing l Pre-processing –Normalize the data –Eliminate outliers l 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 –Can use these steps during the clustering process ISODATA

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 21 Bisecting K-means l Bisecting K-means algorithm –Variant of K-means that can produce a partitional or a hierarchical clustering

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 22 Limitations of K-means l K-means has problems when clusters are of differing –Sizes –Densities –Non-globular shapes l K-means has problems when the data contains outliers.

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 23 Limitations of K-means: Differing Sizes Original Points K-means (3 Clusters)

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 24 Limitations of K-means: Differing Density Original Points K-means (3 Clusters)

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 25 Limitations of K-means: Non-globular Shapes Original Points K-means (2 Clusters)

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 26 Overcoming K-means Limitations Original PointsK-means Clusters One solution is to use many clusters. Find parts of clusters, but need to put together.

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 27 Hierarchical Clustering l Produces a set of nested clusters organized as a hierarchical tree l Can be visualized as a dendrogram –A tree like diagram that records the sequences of merges or splits

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 28 Strengths of Hierarchical Clustering l 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 l They may correspond to meaningful taxonomies –Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …)

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 29 Hierarchical Clustering l 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) l Traditional hierarchical algorithms use a similarity or distance matrix –Merge or split one cluster at a time

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 30 Agglomerative Clustering Algorithm l More popular hierarchical clustering technique l Basic algorithm is straightforward 1.Compute the proximity matrix 2.Let each data point be a cluster 3.Repeat 4.Merge the two closest clusters 5.Update the proximity matrix 6.Until only a single cluster remains l Key operation is the computation of the proximity of two clusters –Different approaches to defining the distance between clusters distinguish the different algorithms

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 31 Starting Situation l Start with clusters of individual points and a proximity matrix p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 32 Intermediate Situation l After some merging steps, we have some clusters C1 C4 C2 C5 C3 C2C1 C3 C5 C4 C2 C3C4C5 Proximity Matrix

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 33 Intermediate Situation l We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C4 C2 C5 C3 C2C1 C3 C5 C4 C2 C3C4C5 Proximity Matrix

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 34 After Merging l The question is “How do we update the proximity matrix?” C1 C4 C2 U C5 C3 ? ? ? ? ? C2 U C5 C1 C3 C4 C2 U C5 C3C4 Proximity Matrix

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 35 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Similarity? l MIN l MAX l Group Average l Distance Between Centroids Proximity Matrix

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 36 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 37 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 38 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 39 How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1p2p3p4p5......... Proximity Matrix l MIN l MAX l Group Average l Distance Between Centroids

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 40 Cluster Similarity: MIN or Single Link l 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. 12345

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 41 Hierarchical Clustering: MIN Nested ClustersDendrogram 1 2 3 4 5 6 1 2 3 4 5

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 42 Strength of MIN Original Points Two Clusters Can handle non-elliptical shapes

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 43 Limitations of MIN Original Points Two Clusters Sensitive to noise and outliers

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 44 Cluster Similarity: MAX or Complete Linkage l 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 12345

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 45 Hierarchical Clustering: MAX Nested ClustersDendrogram 1 2 3 4 5 6 1 2 5 3 4

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 46 Strength of MAX Original Points Two Clusters Less susceptible to noise and outliers

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 47 Limitations of MAX Original Points Two Clusters Tends to break large clusters Biased towards globular clusters

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 48 Cluster Similarity: Group Average l Proximity of two clusters is the average of pairwise proximity between points in the two clusters. l Need to use average connectivity for scalability since total proximity favors large clusters 12345

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 49 Hierarchical Clustering: Group Average Nested ClustersDendrogram 1 2 3 4 5 6 1 2 5 3 4

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 50 Hierarchical Clustering: Group Average l Compromise between Single and Complete Link l Strengths –Less susceptible to noise and outliers l Limitations –Biased towards globular clusters

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 51 Hierarchical Clustering: Comparison Group Average Ward’s Method 1 2 3 4 5 6 1 2 5 3 4 MINMAX 1 2 3 4 5 6 1 2 5 3 4 1 2 3 4 5 6 1 2 5 3 4 1 2 3 4 5 6 1 2 3 4 5

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 52 Hierarchical Clustering: Time and Space requirements l O(N 2 ) space since it uses the proximity matrix. –N is the number of points. l O(N 3 ) time in many cases –There are N steps and at each step the size, N 2, proximity matrix must be updated and searched –Complexity can be reduced to O(N 2 log(N) ) time for some approaches

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 53 Hierarchical Clustering: Problems and Limitations l Once a decision is made to combine two clusters, it cannot be undone l No objective function is directly minimized l 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

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© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 54 Cluster Validity l For supervised classification we have a variety of measures to evaluate how good our model is –Accuracy, precision, recall l For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters? l But “clusters are in the eye of the beholder”! l Then why do we want to evaluate them? –To avoid finding patterns in noise –To compare clustering algorithms –To compare two sets of clusters –To compare two clusters

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