Tan,Steinbach, Kumar Introduction to Data Mining 4/18/ Tan,Steinbach, Kumar Introduction to Data Mining 4/18/

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1 Data Mining Cluster Analsis: Basic Concepts and Algorithms Lecture Notes for Chapter Introduction to Data Mining b Tan, Steinbach, Kumar What is Cluster Analsis? 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 Intra-cluster distances are minimized Inter-cluster distances are maimized Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Applications of Cluster Analsis What is not Cluster Analsis? Understanding Group related documents for browsing, group genes and proteins that have similar functionalit, or group stocks with similar price fluctuations Summarization Reduce the size of large data sets Discovered Clusters Applied-Matl-DOWN,Ba-Network-Down,-COM-DOWN, Cabletron-Ss-DOWN,CISCO-DOWN,HP-DOWN, DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN, Micron-Tech-DOWN,Teas-Inst-Down,Tellabs-Inc-Down, Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN, Sun-DOWN Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN, ADV-Micro-Device-DOWN,Andrew-Corp-DOWN, Computer-Assoc-DOWN,Circuit-Cit-DOWN, Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN, Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN Fannie-Mae-DOWN,Fed-Home-Loan-DOWN, MBNA-Corp-DOWN,Morgan-Stanle-DOWN Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP, Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP, Schlumberger-UP Industr Group Technolog-DOWN Technolog-DOWN Financial-DOWN Oil-UP Supervised classification Have class label information. Clustering is also called unsupervised classification. Simple segmentation Dividing students into different registration groups alphabeticall, b last name. Clustering precipitation in Australia Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Notion of a Cluster can be Ambiguous Tpes of Clusterings How man clusters? Two Clusters Si Clusters Four Clusters A clustering is a set of clusters Important distinction between hierarchical and partitional sets of clusters Partitional Clustering A division of data objects into non-overlapping subsets (clusters) such that each data object is in eactl one subset Hierarchical clustering A set of nested clusters organized as a hierarchical tree Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //

2 Partitional Clustering Hierarchical Clustering p p p p p p p p Hierarchical Clustering Dendrogram A Partitional Clustering Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining // Other Distinctions Between Sets of Clusters Eclusive versus non-eclusive In non-eclusive clusterings, points ma belong to multiple clusters. Can represent multiple classes or border points Fuzz versus non-fuzz In fuzz clustering, a point belongs to ever cluster with some weight between and Weights must sum to Probabilistic clustering has similar characteristics Partial versus complete In some cases, we onl want to cluster some of the data Heterogeneous versus homogeneous Cluster of widel different sizes, shapes, and densities Tpes of Clusters Well-separated clusters Center-based clusters Contiguous clusters Densit-based clusters Propert or Conceptual Described b an Objective Function Tan,Steinbach, Kumar Introduction to Data Mining // 9 Tan,Steinbach, Kumar Introduction to Data Mining // Tpes of Clusters: Well-Separated Well-Separated Clusters: A cluster is a set of points such that an point in a cluster is closer (or more similar) to ever other point in the cluster than to an point not in the cluster. Tpes 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 an 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 well-separated clusters center-based clusters Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //

3 Tpes of Clusters: Contiguit-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 an point not in the cluster. Tpes of Clusters: Densit-Based Densit-based A cluster is a dense region of points, which is separated b low-densit regions, from other regions of high densit. Used when the clusters are irregular or intertwined, and when noise and outliers are present. contiguous clusters densit-based clusters Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Tpes of Clusters: Conceptual Clusters Shared Propert or Conceptual Clusters Finds clusters that share some common propert or represent a particular concept.. Overlapping Rings Tpes of Clusters: Objective Function Clusters Defined b an Objective Function Finds clusters that minimize or maimize an objective function. Enumerate all possible was of dividing the points into clusters and evaluate the qualit of each potential set of clusters b using the given objective function. (NP Hard) Can have global or local objectives. Hierarchical clustering algorithms tpicall have local objectives Partitioning algorithms tpicall 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. Miture models assume that the data is a miture' of a number of probabilit distributions. Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Clustering Algorithms K-means and its variants Hierarchical clustering Densit-based clustering K-means Clustering Partitioning approach Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest centroid Number of clusters, K, must be specified The basic algorithm is ver simple Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining //

4 K-means Clustering Details Eample: Old Faithful Initial centroids are often chosen randoml. Clusters produced var from one run to another. The centroid is (tpicall) the mean of the points in the cluster. Closeness is measured b Euclidean distance, Manhattan distance, cosine similarit, correlation, etc. K-means will converge for common similarit measures mentioned above. Most of the convergence happens in the first few iterations. Often the stopping condition is changed to Until relativel few points change clusters Compleit is O( n * K * I * d ) n = number of points, K = number of clusters, I = number of iterations, d = number of attributes Tan,Steinbach, Kumar Introduction to Data Mining // 9 Tan,Steinbach, Kumar Introduction to Data Mining // Eample: Old Faithful Two different K-means Clusterings (a) (b) (c) (d)... (e) (f) (g) (h) (i) Optimal Clustering Sub-optimal Clustering Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Importance of Choosing Initial Centroids Importance of Choosing Initial Centroids. Iteration. Iteration. Iteration Iteration Iteration Iteration Iteration Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //

5 Importance of Choosing Initial Centroids Importance of Choosing Initial Centroids. Iteration. Iteration Iteration Iteration Iteration Iteration Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Evaluating K-means Clusters K-means: Eample 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. SSE = K i= C i dist ( m, ) 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 clusterings, we choose the one with the smallest error One eas wa 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 (due to local minima). i obj - - C () m () = m () = C () + C () + = = Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining // K-means: Eample K-means: Eample d (,m ) = (-) + (-) = +9 = d (,m ) = (-) + (-) = +9 = d (,m ) = (-) + (--) = +9 = d (,m ) = (-) + (--) = +9 = 9 d (,m ) = (--) + (-) = d (,m ) = (--) + (-) = Assign to cluster Assign to cluster Assign to cluster obj - - C () C () C () d (,m ) = (-) + (-) = d (,m ) = (-) + (-) = Assign to cluster m () = + = m () = + = Tan,Steinbach, Kumar Introduction to Data Mining // 9 Tan,Steinbach, Kumar Introduction to Data Mining //

6 K-means: Eample Solution obtained obj - - C () C () C () Algorithm has converged because no reassignment have taken place Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // K-means: Eample Problems with Selecting Initial Points obj - - C () C () C () If there are K real clusters then the chance of selecting one centroid from each cluster is small. Chance is relativel small when K is large If clusters are the same size, n, then What happens if we start from this initial assignment? Calculate m () and m () and the resulting C (). For eample, if K =, then probabilit =!/ =. Sometimes the initial centroids will readjust themselves in right wa, and sometimes the don t Consider an eample of five pairs of clusters Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Clusters Eample Clusters Eample Iteration Iteration Iteration Iteration Iteration Starting with two initial centroids in one cluster of each pair of clusters Tan,Steinbach, Kumar Introduction to Data Mining // Starting with two initial centroids in one cluster of each pair of clusters Tan,Steinbach, Kumar Introduction to Data Mining //

7 Clusters Eample Clusters Eample Iteration Iteration Iteration Iteration Iteration Starting with some pairs of clusters having three initial centroids, while other have onl one. Tan,Steinbach, Kumar Introduction to Data Mining // 7 Starting with some pairs of clusters having three initial centroids, while other have onl one. Tan,Steinbach, Kumar Introduction to Data Mining // Solutions to Initial Centroids Problem Multiple runs Helps, but probabilit is not on our side Sample and use hierarchical clustering to determine initial centroids Select more than k initial centroids and then select among these initial centroids Select most widel separated Postprocessing Bisecting K-means Not as susceptible to initialization issues Handling Empt Clusters Basic K-means algorithm can ield empt clusters Several strategies Choose the point that contributes most to SSE Choose a point from the cluster with the highest SSE If there are several empt clusters, the above can be repeated several times. Tan,Steinbach, Kumar Introduction to Data Mining // 9 Tan,Steinbach, Kumar Introduction to Data Mining // Updating Centers Incrementall In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid An alternative is to update the centroids after each assignment (incremental approach) Each assignment updates zero or two centroids More epensive Introduces an order dependenc Never get an empt cluster Can use weights to change the impact Pre-processing and Post-processing Pre-processing Normalize the data Eliminate outliers Post-processing Eliminate small clusters that ma represent outliers Split loose clusters, i.e., clusters with relativel high SSE Merge clusters that are close and that have relativel low SSE Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //

8 Bisecting K-means Bisecting K-means Eample Bisecting K-means algorithm Variant of K-means that can produce a partitional or a hierarchical clustering Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Limitations of K-means Limitations of K-means: Differing Sizes K-means has problems when clusters are of differing Sizes Densities Non-globular shapes K-means has problems when the data contains outliers. K-means ( Clusters) Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Limitations of K-means: Differing Densit Limitations of K-means: Non-globular Shapes K-means ( Clusters) K-means ( Clusters) Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining //

9 Overcoming K-means Limitations Overcoming K-means Limitations K-means Clusters K-means Clusters One solution is to use man clusters. Find parts of clusters, but need to put together. Tan,Steinbach, Kumar Introduction to Data Mining // 9 Tan,Steinbach, Kumar Introduction to Data Mining // Overcoming K-means Limitations K-means Clusters Tan,Steinbach, Kumar Introduction to Data Mining //