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 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 Simple segmentation Dividing students into different registration groups alphabeticall, b last name Summarization Reduce the size of large data sets Clustering precipitation in Australia Results of a quer Groupings are a result of an eternal specification Graph partitioning Some mutual relevance and snerg, but areas are not identical 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 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 //
Partitional Clustering Hierarchical Clustering p p p p p p p p Traditional Hierarchical Clustering Traditional Dendrogram p p p A Partitional Clustering p Non-traditional Hierarchical Clustering p p p p Non-traditional Dendrogram 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 //
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 Clustering Algorithms K-means and its variants Hierarchical clustering Densit-based clustering Overlapping Circles Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 9 K-means Clustering Partitional clustering 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 // K-means Clustering Details 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, 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 //
Two different K-means Clusterings Importance of Choosing Initial Centroids Iteration - - - - - - - - - - - - Optimal Clustering - - - - Sub-optimal Clustering Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Importance of Choosing Initial Centroids Evaluating K-means Clusters Iteration Iteration Iteration 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 - - - - Iteration - - - - - - - - Iteration - - - - - - - - Iteration - - - - SSE = K i= Ci 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 clusters, we can 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 i 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 - - - - - - - - - - - - - - - - Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 7
Problems with Selecting Initial Points Clusters Eample 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 Iteration 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 // - - - Starting with two initial centroids in one cluster of each pair of clusters Tan,Steinbach, Kumar Introduction to Data Mining // 9 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 some pairs of clusters having three initial centroids, while other have onl one Tan,Steinbach, Kumar Introduction to Data Mining // Clusters Eample Solutions to Initial Centroids Problem Iteration - - - Iteration - - - Iteration - - - Iteration - - - 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 Starting with some pairs of clusters having three initial centroids, while other have onl one Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //
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 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 Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Pre-processing and Post-processing Pre-processing Normalize the data Eliminate outliers Bisecting K-means Bisecting K-means algorithm Variant of K-means that can produce a partitional or a hierarchical clustering Post-processing Eliminate small clusters that ma represent outliers Split loose clusters, ie, clusters with relativel high SSE Merge clusters that are close and that have relativel low SSE Can use these steps during the clustering process Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 7 Bisecting K-means Eample Limitations of K-means K-means has problems when clusters are of differing Sizes Densities Non-globular shapes K-means has problems when the data contains outliers Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 9
Limitations of K-means: Differing Sizes Limitations of K-means: Differing Densit K-means ( Clusters) K-means ( Clusters) Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Limitations of K-means: Non-globular Shapes 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 // Tan,Steinbach, Kumar Introduction to Data Mining // Overcoming K-means Limitations Overcoming K-means Limitations K-means Clusters K-means Clusters Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //
Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree Can be visualized as a dendrogram A tree like diagram that records the sequences of merges or splits Strengths of Hierarchical Clustering Do not have to assume an particular number of clusters An desired number of clusters can be obtained b cutting the dendogram at the proper level The ma correspond to meaningful taonomies Eample in biological sciences (eg, animal kingdom, phlogen reconstruction, ) Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 7 Hierarchical Clustering Two main tpes of hierarchical clustering Agglomerative: Start with the points as individual clusters At each step, merge the closest pair of clusters until onl 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) Traditional hierarchical algorithms use a similarit or distance matri Merge or split one cluster at a time Agglomerative Clustering Algorithm More popular hierarchical clustering technique Basic algorithm is straightforward Compute the proimit matri Let each data point be a cluster Repeat Merge the two closest clusters Update the proimit matri Until onl a single cluster remains Ke operation is the computation of the proimit of two clusters Different approaches to defining the distance between clusters distinguish the different algorithms Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 9 Starting Situation Start with clusters of individual points and a proimit matri p p p p p p p p p p Proimit Matri Intermediate Situation After some merging steps, we have some clusters C C C C C C C C C C C C C Proimit Matri C C Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //
Intermediate Situation After Merging We want to merge the two closest clusters (C and C) and update the proimit matri C C C C C C C C C C C C C Proimit Matri The question is How do we update the proimit matri? C C C C C U C C C C C U C??????? C C Proimit Matri C C C U C Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // How to Define Inter-Cluster Similarit How to Define Inter-Cluster Similarit p p p p p p p p p p Similarit? p p p p p p p p MIN MAX Group Average Distance Between Centroids Other methods driven b an objective function Ward s Method uses squared error p Proimit Matri MIN MAX Group Average Distance Between Centroids Other methods driven b an objective function Ward s Method uses squared error p Proimit Matri Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // How to Define Inter-Cluster Similarit How to Define Inter-Cluster Similarit MIN MAX Group Average Distance Between Centroids Other methods driven b an objective function Ward s Method uses squared error p p p p p p p p p p Proimit Matri MIN MAX Group Average Distance Between Centroids Other methods driven b an objective function Ward s Method uses squared error p p p p p p p p p p Proimit Matri Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 7
How to Define Inter-Cluster Similarit Cluster Similarit: MIN or Single Link MIN MAX Group Average Distance Between Centroids Other methods driven b an objective function Ward s Method uses squared error p p p p p p p p p p Proimit Matri Similarit of two clusters is based on the two most similar (closest) points in the different clusters Determined b one pair of points, ie, b one link in the proimit graph I I I I I I 9 I 9 7 I 7 I I Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 9 Hierarchical Clustering: MIN Strength of MIN Two Clusters Nested Clusters Dendrogram Can handle non-elliptical shapes Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Limitations of MIN Cluster Similarit: MAX or Complete Linkage Similarit of two clusters is based on the two least similar (most distant) points in the different clusters Determined b all pairs of points in the two clusters Sensitive to noise and outliers Two Clusters I I I I I I 9 I 9 7 I 7 I I Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining //
Hierarchical Clustering: MAX Strength of MAX Two Clusters Nested Clusters Dendrogram Less susceptible to noise and outliers Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // Limitations of MAX Cluster Similarit: Group Average Tends to break large clusters Biased towards globular clusters Two Clusters Proimit of two clusters is the average of pairwise proimit between points in the two clusters proimit(p i,p j) pi Clusteri pj Clusterj proimit(clusteri,clusterj) = Cluster Cluster Need to use average connectivit for scalabilit since total proimit favors large clusters I I I I I I 9 I 9 7 I 7 I I i j Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 7 Hierarchical Clustering: Group Average Hierarchical Clustering: Group Average Compromise between Single and Complete Link Strengths Less susceptible to noise and outliers Limitations Biased towards globular clusters Nested Clusters Dendrogram Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 9
Cluster Similarit: Ward s Method Hierarchical Clustering: Comparison Similarit 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 MIN MAX Biased towards globular clusters Hierarchical analogue of K-means Can be used to initialize K-means Ward s Method Group Average Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining // 7 Hierarchical Clustering: Time and Space requirements O(N ) space since it uses the proimit matri N is the number of points O(N ) time in man cases There are N steps and at each step the size, N, proimit matri must be updated and searched Compleit can be reduced to O(N log(n) ) time for some approaches DBSCAN DBSCAN is a densit-based algorithm Densit = number of points within a specified radius (Eps) A point is a core point if it has more than a specified number of points (MinPts) within Eps These are points that are at the interior of a cluster A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point A noise point is an point that is not a core point or a border point Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining // 7 DBSCAN: Core, Border, and Noise Points DBSCAN Algorithm Eliminate noise points Perform clustering on the remaining points Tan,Steinbach, Kumar Introduction to Data Mining // 7 Tan,Steinbach, Kumar Introduction to Data Mining // 7
DBSCAN: Core, Border and Noise Points When DBSCAN Works Well Clusters Eps =, MinPts = Point tpes: core, border and noise Resistant to Noise Can handle clusters of different shapes and sizes Tan,Steinbach, Kumar Introduction to Data Mining // 77 Tan,Steinbach, Kumar Introduction to Data Mining // 7 When DBSCAN Does NOT Work Well DBSCAN: Determining EPS and MinPts (MinPts=, Eps=97) Idea is that for points in a cluster, their k th nearest neighbors are at roughl the same distance Noise points have the k th nearest neighbor at farther distance So, plot sorted distance of ever point to its k th nearest neighbor Varing densities High-dimensional data (MinPts=, Eps=99) Tan,Steinbach, Kumar Introduction to Data Mining // 79 Tan,Steinbach, Kumar Introduction to Data Mining // Measures of Cluster Validit Numerical measures that are applied to judge various aspects of cluster validit, are classified into the following three tpes Eternal Inde: Used to measure the etent to which cluster labels match eternall supplied class labels Entrop Internal Inde: Used to measure the goodness of a clustering structure without respect to eternal information Sum of Squared Error (SSE) Relative Inde: Used to compare two different clusterings or clusters Often an eternal or internal inde is used for this function, eg, SSE or entrop Sometimes these are referred to as criteria instead of indices However, sometimes criterion is the general strateg and inde is the numerical measure that implements the criterion Tan,Steinbach, Kumar Introduction to Data Mining // Internal Measures: Cohesion and Separation Cluster Cohesion: Measures how closel related are objects in a cluster Eample: SSE Cluster Separation: Measure how distinct or wellseparated a cluster is from other clusters Eample: Squared Error Cohesion is measured b the within cluster sum of squares (SSE) WSS = ( mi ) i C Separation is measured b the between cluster sum of squares BSS = C ( m i m i i Where C i is the size of cluster i i Tan,Steinbach, Kumar Introduction to Data Mining // )
Internal Measures: Cohesion and Separation A proimit graph based approach can also be used for cohesion and separation Cluster cohesion is the sum of the weight of all links within a cluster Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster Final Comment on Cluster Validit The validation of clustering structures is the most difficult and frustrating part of cluster analsis Without a strong effort in this direction, cluster analsis will remain a black art accessible onl to those true believers who have eperience and great courage Algorithms for Clustering Data, Jain and Dubes cohesion separation Tan,Steinbach, Kumar Introduction to Data Mining // Tan,Steinbach, Kumar Introduction to Data Mining // 7