Cluster AnalysisWhat is Cluster Analysis?Slide 3Clustering: Rich Applications and Multidisciplinary EffortsQuality: What Is Good Clustering?Similarity and Dissimilarity Between ObjectsSimilarity and Dissimilarity Between Objects (Cont.)Major Clustering ApproachesSome Other Major Clustering ApproachesTypical Alternatives to Calculate the Distance between ClustersSlide 11Clustering ApproachesPartitioning Algorithms: Basic ConceptSlide 14The K-Means Clustering MethodK-means ClusteringSlide 17Slide 18Slide 19Slide 20Slide 21ExampleSlide 23Slide 24In class PracticeSlide 26Slide 27Comments on the K-Means MethodWhat Is the Problem of the K-Means Method?Fuzzy C-means ClusteringSlide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Fuzzy C-means Clustering http://home.dei.polimi.it/matteucc/Clustering/tutorial_html/cmeans.htmlSlide 39Slide 40Slide 41Slide 42Slide 43Cluster Analysis Dr. Bernard Chen Ph.D.Assistant ProfessorDepartment of Computer Science University of Central ArkansasFall 2010What is Cluster Analysis?Cluster: a collection of data objectsSimilar to one another within the same clusterDissimilar to the objects in other clustersCluster analysisFinding similarities between data according to the characteristics found in the data and grouping similar data objects into clustersWhat is Cluster Analysis?Clustering analysis is an important human activityEarly in childhood, we learn how to distinguish between cats and dogsUnsupervised learning: no predefined classesTypical applicationsAs a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithmsClustering: Rich Applications and Multidisciplinary Efforts Pattern RecognitionSpatial Data Analysis Create thematic maps in GIS by clustering feature spacesDetect spatial clusters or for other spatial mining tasksImage ProcessingEconomic Science (especially market research)WWWDocument classificationCluster Weblog data to discover groups of similar access patternsQuality: What Is Good Clustering?A good clustering method will produce high quality clusters withhigh intra-class similarity (Similar to one another within the same cluster)low inter-class similarity(Dissimilar to the objects in other clusters)The quality of a clustering method is also measured by its ability to discover some or all of the hidden patternsSimilarity and Dissimilarity Between ObjectsDistances are normally used to measure the similarity or dissimilarity between two data objectsSome popular ones include: Minkowski distance:where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integerIf q = 1, d is Manhattan distanceqqppqqjxixjxixjxixjid )||...|||(|),(2211||...||||),(2211 ppjxixjxixjxixjid Similarity and Dissimilarity Between Objects (Cont.)If q = 2, d is Euclidean distance:Also, one can use weighted distance, parametric Pearson correlation, or other disimilarity measures)||...|||(|),(2222211 ppjxixjxixjxixjid Major Clustering ApproachesPartitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of square errorsTypical methods: k-means, k-medoids, CLARANSHierarchical approach: Create a hierarchical decomposition of the set of data (or objects) using some criterionTypical methods: Hierarchical, Diana, Agnes, BIRCH, ROCK, CAMELEONDensity-based approach: Based on connectivity and density functionsTypical methods: DBSACN, OPTICS, DenClueSome Other Major Clustering ApproachesGrid-based approach: based on a multiple-level granularity structureTypical methods: STING, WaveCluster, CLIQUEModel-based: A model is hypothesized for each of the clusters and tries to find the best fit of that model to each otherTypical methods: EM, SOM, COBWEBFrequent pattern-based:Based on the analysis of frequent patternsTypical methods: pClusterUser-guided or constraint-based: Clustering by considering user-specified or application-specific constraintsTypical methods: COD (obstacles), constrained clusteringTypical Alternatives to Calculate the Distance between ClustersSingle link: smallest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = min(tip, tjq)Complete link: largest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = max(tip, tjq)Average: avg distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = avg(tip, tjq)Typical Alternatives to Calculate the Distance between ClustersCentroid: distance between the centroids of two clusters, i.e., dis(Ki, Kj) = dis(Ci, Cj)Centroid: the “middle” of a cluster Medoid: distance between the medoids of two clusters, i.e., dis(Ki, Kj) = dis(Mi, Mj)Medoid: one chosen, centrally located object in the clusterNtNiipmC)(1Clustering Approaches1. Partitioning Methods2. Hierarchical Methods3. Density-Based MethodsPartitioning Algorithms: Basic ConceptPartitioning method: Construct a partition of a database D of n objects into a set of k clusters, s.t., min sum of squared distance21)(mimKmtkmtCmiPartitioning Algorithms: Basic ConceptGiven a k, find a partition of k clusters that optimizes the chosen partitioning criterionGlobal optimal: exhaustively enumerate all partitionsHeuristic methods: k-means and k-medoids algorithmsk-means (MacQueen’67): Each cluster is represented by the center of the clusterk-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the clusterThe K-Means Clustering Method Given k, the k-means algorithm is implemented in four steps:1. Partition objects into k nonempty subsets2. Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster)3. Assign each object to the cluster with the nearest seed point 4. Go back to Step 2, stop when no more new assignmentK-means ClusteringK-means ClusteringK-means ClusteringK-means ClusteringK-means ClusteringThe K-Means Clustering Method 0123456789100 1 2 3 4 5 6 7 8 9 100123456789100 1 2 3 4 5 6 7 8 9 100123456789100 1 2 3 4 5 6 7 8 9 100123456789100 1 2 3 4 5 6 7 8 9 100123456789100 1 2 3 4 5 6 7 8 9