What is Cluster Analysis?Applications of Cluster AnalysisWhat is not Cluster Analysis?Notion of a Cluster can be AmbiguousTypes of ClusteringsPartitional ClusteringHierarchical ClusteringOther Distinctions Between Sets of ClustersTypes of ClustersTypes of Clusters: Well-SeparatedTypes of Clusters: Center-BasedTypes of Clusters: Contiguity-BasedTypes of Clusters: Density-BasedTypes of Clusters: Conceptual ClustersTypes of Clusters: Objective FunctionTypes of Clusters: Objective Function …Characteristics of the Input Data Are ImportantData StructuresType of data in clustering analysisInterval-valued variablesSimilarity and Dissimilarity Between ObjectsSimilarity and Dissimilarity Between Objects (Cont.)Binary VariablesDissimilarity between Binary VariablesNominal VariablesOrdinal VariablesRatio-Scaled VariablesVariables of Mixed TypesVector ObjectsClustering AlgorithmsK-means ClusteringK-means Clustering – DetailsTwo different K-means ClusteringsImportance of Choosing Initial CentroidsSlide 35Evaluating K-means ClustersImportance of Choosing Initial Centroids …Slide 38Problems with Selecting Initial Points10 Clusters ExampleSlide 41Slide 42Slide 43Solutions to Initial Centroids ProblemHandling Empty ClustersUpdating Centers IncrementallyPre-processing and Post-processingBisecting K-meansBisecting K-means ExampleComments on the K-Means MethodWhat is Cluster Analysis?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 groupsInter-cluster distances are maximizedIntra-cluster distances are minimizedApplications of Cluster AnalysisUnderstanding–Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuationsSummarization–Reduce the size of large data setsClustering precipitation in AustraliaWhat is not Cluster Analysis?Supervised classification–Have class label informationSimple segmentation–Dividing students into different registration groups alphabetically, by last nameResults of a query–Groupings are a result of an external specificationGraph partitioning–Some mutual relevance and synergy, but areas are not identicalNotion of a Cluster can be AmbiguousHow many clusters?Four Clusters Two Clusters Six ClustersTypes of ClusteringsA clustering is a set of clustersImportant distinction between hierarchical and partitional sets of clusters Partitional Clustering–A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subsetHierarchical clustering–A set of nested clusters organized as a hierarchical treePartitional ClusteringOriginal Points A Partitional ClusteringHierarchical Clusteringp4p1p3p2 p4 p1 p3 p2 p4p1 p2p3p4p1 p2p3Traditional Hierarchical ClusteringNon-traditional Hierarchical Clustering Non-traditional DendrogramTraditional DendrogramOther Distinctions Between Sets of ClustersExclusive versus non-exclusive–In non-exclusive clusterings, points may belong to multiple clusters.–Can represent multiple classes or ‘border’ points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Partial versus complete–In some cases, we only want to cluster some of the dataHeterogeneous versus homogeneous–Cluster of widely different sizes, shapes, and densitiesTypes of Clusters Well-separated clusters Center-based clusters Contiguous clusters Density-based clustersProperty or ConceptualDescribed by an Objective FunctionTypes of Clusters: Well-SeparatedWell-Separated Clusters: –A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 3 well-separated clustersTypes of Clusters: Center-Based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 clustersTypes of Clusters: Contiguity-BasedContiguous Cluster (Nearest neighbor or Transitive)–A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.8 contiguous clustersTypes of Clusters: Density-BasedDensity-based–A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. –Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clustersTypes of Clusters: Conceptual ClustersShared Property or Conceptual Clusters–Finds clusters that share some common property or represent a particular concept. . 2 Overlapping CirclesTypes of Clusters: Objective FunctionClusters Defined by an Objective Function–Finds clusters that minimize or maximize an objective function. –Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard)– Can have global or local objectives. Hierarchical clustering algorithms typically have local objectives Partitional algorithms typically have global objectives–A variation of the global objective function approach is to fit the data to a parameterized model.  Parameters for the model are determined from the data.  Mixture models assume that the data is a ‘mixture' of a number of statistical distributions.Types of Clusters: Objective Function …Map the clustering problem to a different domain and solve a related problem in that domain–Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points– Clustering is equivalent to breaking the graph into connected components, one for each cluster. –Want to minimize the edge weight between clusters and maximize the edge weight within clustersCharacteristics of the Input Data Are ImportantType of proximity or density measure–This is a derived measure,