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Data Mining Cluster Analysis: Basic Concepts and Algorithms

Data Mining Cluster Analysis: Basic Concepts and Algorithms
Data Mining Cluster Analysis: Basic Concepts and Algorithms © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1What 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 groups Intercluster Intracluster distances are distances are maximized minimized © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Applications of Cluster Analysis Discovered Clusters Industry Group  Understanding AppliedMatlDOWN,BayNetworkDown,3COMDOWN, CabletronSysDOWN,CISCODOWN,HPDOWN, 1 DSCCommDOWN,INTELDOWN,LSILogicDOWN, Technology1DOWN – Group related documents MicronTechDOWN,TexasInstDown,TellabsIncDown, NatlSemiconductDOWN,OraclDOWN,SGIDOWN, SunDOWN for browsing, group genes AppleCompDOWN,AutodeskDOWN,DECDOWN, ADVMicroDeviceDOWN,AndrewCorpDOWN, and proteins that have 2 ComputerAssocDOWN,CircuitCityDOWN, Technology2DOWN CompaqDOWN, EMCCorpDOWN, GenInstDOWN, MotorolaDOWN,MicrosoftDOWN,ScientificAtlDOWN similar functionality, or FannieMaeDOWN,FedHomeLoanDOWN, MBNACorpDOWN,MorganStanleyDOWN group stocks with similar FinancialDOWN 3 BakerHughesUP,DresserIndsUP,HalliburtonHLDUP, price fluctuations LouisianaLandUP,PhillipsPetroUP,UnocalUP, OilUP 4 SchlumbergerUP  Summarization – Reduce the size of large data sets Clustering precipitation in Australia © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›What 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 identical © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Notion of a Cluster can be Ambiguous How many clusters Six Clusters Two Clusters Four Clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types 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 nonoverlapping subsets (clusters) such that each data object is in exactly one subset  Hierarchical clustering – A set of nested clusters organized as a hierarchical tree © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Partitional Clustering Original Points A Partitional Clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering p1 p3 p4 p2 p1 p2 p3 p4 Traditional Hierarchical Clustering Traditional Dendrogram p1 p3 p4 p2 p1 p2 p3 p4 Nontraditional Hierarchical Clustering Nontraditional Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Other Distinctions Between Sets of Clusters  Exclusive versus nonexclusive – In nonexclusive clusterings, points may belong to multiple clusters. – Can represent multiple classes or „border‟ points  Fuzzy versus nonfuzzy – 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 densities © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types of Clusters  Wellseparated clusters  Centerbased clusters  Contiguous clusters  Densitybased clusters  Property or Conceptual  Described by an Objective Function © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types of Clusters: WellSeparated  WellSeparated 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 wellseparated clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types of Clusters: CenterBased  Centerbased – 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 centerbased clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types of Clusters: ContiguityBased  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 clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types of Clusters: DensityBased  Densitybased – A cluster is a dense region of points, which is separated by lowdensity regions, from other regions of high density. – Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 densitybased clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types of Clusters: Conceptual Clusters  Shared Property or Conceptual Clusters – Finds clusters that share some common property or represent a particular concept. . 2 Overlapping Circles © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Types 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. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›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 clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Characteristics of the Input Data Are Important  Type of proximity or density measure – This is a derived measure, but central to clustering  Sparseness – Dictates type of similarity – Adds to efficiency  Attribute type – Dictates type of similarity  Type of Data – Dictates type of similarity – Other characteristics, e.g., autocorrelation  Dimensionality  Noise and Outliers  Type of Distribution © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Clustering Algorithms  Kmeans and its variants  Hierarchical clustering  Densitybased clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Kmeans 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 very simple © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Kmeans Clustering – Details  Initial centroids are often chosen randomly. – Clusters produced vary from one run to another.  The centroid is (typically) the mean of the points in the cluster.  „Closeness‟ is measured by Euclidean distance, cosine similarity, correlation, etc.  Kmeans will converge for common similarity measures mentioned above.  Most of the convergence happens in the first few iterations. – Often the stopping condition is changed to „Until relatively few points change clusters‟  Complexity 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 4/18/2004 ‹›Two different Kmeans Clusterings 3 2.5 2 Original Points 1.5 1 0.5 0 2 1.5 1 0.5 0 0.5 1 1.5 2 x 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x x Optimal Clustering Suboptimal Clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y yImportance of Choosing Initial Centroids IIIIIItttttte e e e e er r r r r ra a a a a attttttiiiiiio o o o o on n n n n n 1 2 4 5 3 6 3 3 3 3 3 3 2 2 2 2 2 2......5 5 5 5 5 5 2 2 2 2 2 2 1 1 1 1 1 1......5 5 5 5 5 5 1 1 1 1 1 1 0 0 0 0 0 0......5 5 5 5 5 5 0 0 0 0 0 0 2 2 2 2 2 2 1 1 1 1 1 1......5 5 5 5 5 5 1 1 1 1 1 1 0 0 0 0 0 0......5 5 5 5 5 5 0 0 0 0 0 0 0 0 0 0 0 0......5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1 1 1......5 5 5 5 5 5 2 2 2 2 2 2 x x x x x x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y y yImportance of Choosing Initial Centroids Iteration 1 Iteration 2 Iteration 3 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x x x Iteration 4 Iteration 5 Iteration 6 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x x x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y y yEvaluating Kmeans Clusters  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. K 2 SSE dist (m ,x)  i i1 xC i – x is a data point in cluster C and m is the representative point for i i cluster C i  can show that m corresponds to the center (mean) of the cluster i – 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Importance of Choosing Initial Centroids … IIIIIttttte e e e er r r r ra a a a atttttiiiiio o o o on n n n n 4 3 1 2 5 3 3 3 3 3 2 2 2 2 2.....5 5 5 5 5 2 2 2 2 2 1 1 1 1 1.....5 5 5 5 5 1 1 1 1 1 0 0 0 0 0.....5 5 5 5 5 0 0 0 0 0 2 2 2 2 2 1 1 1 1 1.....5 5 5 5 5 1 1 1 1 1 0 0 0 0 0.....5 5 5 5 5 0 0 0 0 0 0 0 0 0 0.....5 5 5 5 5 1 1 1 1 1 1 1 1 1 1.....5 5 5 5 5 2 2 2 2 2 x x x x x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y yImportance of Choosing Initial Centroids … Iteration 1 Iteration 2 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x x Iteration 3 Iteration 4 Iteration 5 3 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x x x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y yProblems with Selecting Initial Points  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 10 – For example, if K = 10, then probability = 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›10 Clusters Example IIIItttte e e er r r ra a a attttiiiio o o on n n n 1 4 3 2 8 8 8 8 6 6 6 6 4 4 4 4 2 2 2 2 0 0 0 0 2 2 2 2 4 4 4 4 6 6 6 6 0 0 0 0 5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 x x x x Starting with two initial centroids in one cluster of each pair of clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y10 Clusters Example Iteration 1 Iteration 2 8 8 6 6 4 4 2 2 0 0 2 2 4 4 6 6 0 5 10 15 20 0 5 10 15 20 x x Iteration 3 Iteration 4 8 8 6 6 4 4 2 2 0 0 2 2 4 4 6 6 0 5 10 15 20 0 5 10 15 20 x x Starting with two initial centroids in one cluster of each pair of clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y10 Clusters Example IIIItttte e e er r r ra a a attttiiiio o o on n n n 3 1 4 2 8 8 8 8 6 6 6 6 4 4 4 4 2 2 2 2 0 0 0 0 2 2 2 2 4 4 4 4 6 6 6 6 0 0 0 0 5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 x x x x Starting with some pairs of clusters having three initial centroids, while other have only one. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y y10 Clusters Example Iteration 1 Iteration 2 8 8 6 6 4 4 2 2 0 0 2 2 4 4 6 6 0 5 10 15 20 0 5 10 15 20 x x Iteration 3 Iteration 4 8 8 6 6 4 4 2 2 0 0 2 2 4 4 6 6 0 5 10 15 20 0 5 10 15 20 x x Starting with some pairs of clusters having three initial centroids, while other have only one. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y ySolutions to Initial Centroids Problem  Multiple runs – Helps, but probability is not on your 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 widely separated  Postprocessing  Bisecting Kmeans – Not as susceptible to initialization issues © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Handling Empty Clusters  Basic Kmeans algorithm can yield empty 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 empty clusters, the above can be repeated several times. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Updating Centers Incrementally  In the basic Kmeans 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 expensive – Introduces an order dependency – Never get an empty cluster – Can use “weights” to change the impact © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Preprocessing and Postprocessing  Preprocessing – Normalize the data – Eliminate outliers  Postprocessing – 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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Bisecting Kmeans  Bisecting Kmeans algorithm – Variant of Kmeans that can produce a partitional or a hierarchical clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Bisecting Kmeans Example © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Limitations of Kmeans  Kmeans has problems when clusters are of differing – Sizes – Densities – Nonglobular shapes  Kmeans has problems when the data contains outliers. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Limitations of Kmeans: Differing Sizes Kmeans (3 Clusters) Original Points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Limitations of Kmeans: Differing Density Kmeans (3 Clusters) Original Points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Limitations of Kmeans: Nonglobular Shapes Original Points Kmeans (2 Clusters) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Overcoming Kmeans Limitations Original Points Kmeans Clusters One solution is to use many clusters. Find parts of clusters, but need to put together. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Overcoming Kmeans Limitations Original Points Kmeans Clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Overcoming Kmeans Limitations Original Points Kmeans Clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›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 5 6 0.2 4 3 4 2 0.15 5 2 0.1 1 0.05 1 3 0 1 3 2 5 4 6 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Strengths of Hierarchical Clustering  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  They may correspond to meaningful taxonomies – Example in biological sciences (e.g., animal kingdom, phylogeny reconstruction, …) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering  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, allinclusive cluster  At each step, split a cluster until each cluster contains a point (or there are k clusters)  Traditional hierarchical algorithms use a similarity or distance matrix – Merge or split one cluster at a time © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Agglomerative Clustering Algorithm  More popular hierarchical clustering technique  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  Key operation is the computation of the proximity of two clusters – Different approaches to defining the distance between clusters distinguish the different algorithms © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Starting Situation  Start with clusters of individual points and a proximity matrix p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5 . . . Proximity Matrix ... p1 p2 p3 p4 p9 p10 p11 p12 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Intermediate Situation  After some merging steps, we have some clusters C1 C2 C3 C4 C5 C1 C2 C3 C3 C4 C4 C5 Proximity Matrix C1 C5 C2 ... p1 p2 p3 p4 p9 p10 p11 p12 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Intermediate Situation  We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C2 C3 C4 C5 C1 C2 C3 C3 C4 C4 C5 Proximity Matrix C1 C5 C2 ... p1 p2 p3 p4 p9 p10 p11 p12 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›After Merging  The question is “How do we update the proximity matrix” C2 U C1 C5 C3 C4 C1 C2 U C5 C3 C3 C4 C4 Proximity Matrix C1 C2 U C5 ... p1 p2 p3 p4 p9 p10 p11 p12 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Define InterCluster Similarity p1 p2 p3 p4 p5 . . . p1 Similarity p2 p3 p4 p5  MIN .  MAX .  Group Average . Proximity Matrix  Distance Between Centroids  Other methods driven by an objective function – Ward‟s Method uses squared error © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Define InterCluster Similarity p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5  MIN .  MAX .  Group Average . Proximity Matrix  Distance Between Centroids  Other methods driven by an objective function – Ward‟s Method uses squared error © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Define InterCluster Similarity p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5  MIN .  MAX .  Group Average . Proximity Matrix  Distance Between Centroids  Other methods driven by an objective function – Ward‟s Method uses squared error © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Define InterCluster Similarity p1 p2 p3 p4 p5 . . . p1 p2 p3 p4 p5  MIN .  MAX .  Group Average . Proximity Matrix  Distance Between Centroids  Other methods driven by an objective function – Ward‟s Method uses squared error © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›How to Define InterCluster Similarity p1 p2 p3 p4 p5 . . . p1  p2 p3 p4 p5  MIN .  MAX .  Group Average . Proximity Matrix  Distance Between Centroids  Other methods driven by an objective function – Ward‟s Method uses squared error © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Cluster Similarity: MIN or Single Link  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. I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0.20 I2 0.90 1.00 0.70 0.60 0.50 I3 0.10 0.70 1.00 0.40 0.30 I4 0.65 0.60 0.40 1.00 0.80 I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: MIN 5 1 3 0.2 5 2 1 0.15 2 3 6 0.1 0.05 4 0 4 3 6 2 5 4 1 Nested Clusters Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Strength of MIN Original Points Two Clusters • Can handle nonelliptical shapes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Limitations of MIN Original Points Two Clusters • Sensitive to noise and outliers © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Cluster Similarity: MAX or Complete Linkage  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 I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0.20 I2 0.90 1.00 0.70 0.60 0.50 I3 0.10 0.70 1.00 0.40 0.30 I4 0.65 0.60 0.40 1.00 0.80 I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: MAX 4 1 0.4 2 5 0.35 5 0.3 2 0.25 0.2 3 6 0.15 3 1 0.1 0.05 4 0 3 6 4 1 2 5 Nested Clusters Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Strength of MAX Original Points Two Clusters • Less susceptible to noise and outliers © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Limitations of MAX Original Points Two Clusters •Tends to break large clusters •Biased towards globular clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Cluster Similarity: Group Average  Proximity of two clusters is the average of pairwise proximity between points in the two clusters. proximity(p ,p )  i j pCluster i i pCluster j j proximity(Cluster ,Cluster ) i j Cluster Cluster i j  Need to use average connectivity for scalability since total proximity favors large clusters I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0.20 I2 0.90 1.00 0.70 0.60 0.50 I3 0.10 0.70 1.00 0.40 0.30 I4 0.65 0.60 0.40 1.00 0.80 I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: Group Average 5 4 1 0.25 2 0.2 5 2 0.15 3 6 0.1 1 0.05 4 0 3 6 4 1 2 5 3 Nested Clusters Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: Group Average  Compromise between Single and Complete Link  Strengths – Less susceptible to noise and outliers  Limitations – Biased towards globular clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Cluster Similarity: Ward’s Method  Similarity 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  Biased towards globular clusters  Hierarchical analogue of Kmeans – Can be used to initialize Kmeans © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: Comparison 5 1 4 1 3 2 5 5 5 2 1 2 MIN MAX 2 3 6 3 6 3 1 4 4 4 5 5 1 4 1 2 2 5 Ward’s Method 5 2 2 Group Average 3 3 6 6 3 1 1 4 4 4 3 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: Time and Space requirements 2  O(N ) space since it uses the proximity matrix. – N is the number of points. 3  O(N ) time in many cases 2 – There are N steps and at each step the size, N , proximity matrix must be updated and searched 2 – Complexity can be reduced to O(N log(N) ) time for some approaches © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Hierarchical Clustering: Problems and Limitations  Once a decision is made to combine two clusters, it cannot be undone  No objective function is directly minimized  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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›MST: Divisive Hierarchical Clustering  Build MST (Minimum Spanning Tree) – Start with a tree that consists of any point – In successive steps, look for the closest pair of points (p, q) such that one point (p) is in the current tree but the other (q) is not – Add q to the tree and put an edge between p and q © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›MST: Divisive Hierarchical Clustering  Use MST for constructing hierarchy of clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›DBSCAN  DBSCAN is a densitybased algorithm. – Density = 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 any point that is not a core point or a border point. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›DBSCAN: Core, Border, and Noise Points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›DBSCAN Algorithm  Eliminate noise points  Perform clustering on the remaining points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›DBSCAN: Core, Border and Noise Points Original Points Point types: core, border and noise Eps = 10, MinPts = 4 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›When DBSCAN Works Well Original Points Clusters • Resistant to Noise • Can handle clusters of different shapes and sizes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›When DBSCAN Does NOT Work Well (MinPts=4, Eps=9.75). Original Points • Varying densities • Highdimensional data (MinPts=4, Eps=9.92) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›DBSCAN: Determining EPS and MinPts th  Idea is that for points in a cluster, their k nearest neighbors are at roughly the same distance th  Noise points have the k nearest neighbor at farther distance th  So, plot sorted distance of every point to its k nearest neighbor © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Cluster Validity  For supervised classification we have a variety of measures to evaluate how good our model is – Accuracy, precision, recall  For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters  But “clusters are in the eye of the beholder”  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 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Clusters found in Random Data 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 Random 0.6 DBSCAN Points 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x 1 1 0.9 0.9 0.8 0.8 Kmeans Complete 0.7 0.7 Link 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y y y yDifferent Aspects of Cluster Validation 1. Determining the clustering tendency of a set of data, i.e., distinguishing whether nonrandom structure actually exists in the data. 2. Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels. 3. Evaluating how well the results of a cluster analysis fit the data without reference to external information. Use only the data 4. Comparing the results of two different sets of cluster analyses to determine which is better. 5. Determining the „correct‟ number of clusters. For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Measures of Cluster Validity  Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types. – External Index: Used to measure the extent to which cluster labels match externally supplied class labels.  Entropy – Internal Index: Used to measure the goodness of a clustering structure without respect to external information.  Sum of Squared Error (SSE) – Relative Index: Used to compare two different clusterings or clusters.  Often an external or internal index is used for this function, e.g., SSE or entropy  Sometimes these are referred to as criteria instead of indices – However, sometimes criterion is the general strategy and index is the numerical measure that implements the criterion. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Measuring Cluster Validity Via Correlation  Two matrices – Proximity Matrix – “Incidence” Matrix  One row and one column for each data point  An entry is 1 if the associated pair of points belong to the same cluster  An entry is 0 if the associated pair of points belongs to different clusters  Compute the correlation between the two matrices – Since the matrices are symmetric, only the correlation between n(n1) / 2 entries needs to be calculated.  High correlation indicates that points that belong to the same cluster are close to each other.  Not a good measure for some density or contiguity based clusters. © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Measuring Cluster Validity Via Correlation  Correlation of incidence and proximity matrices for the Kmeans clusterings of the following two data sets. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x Corr = 0.9235 Corr = 0.5810 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y yUsing Similarity Matrix for Cluster Validation  Order the similarity matrix with respect to cluster labels and inspect visually. 1 1 10 0.9 0.9 20 0.8 0.8 30 0.7 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 0 0 20 40 60 80 100 Similarity 0 0.2 0.4 0.6 0.8 1 Points x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y PointsUsing Similarity Matrix for Cluster Validation  Clusters in random data are not so crisp 1 1 10 0.9 0.9 20 0.8 0.8 30 0.7 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 0 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Similarity Points x DBSCAN © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› Points yUsing Similarity Matrix for Cluster Validation  Clusters in random data are not so crisp 1 1 10 0.9 0.9 20 0.8 0.8 30 0.7 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 0 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Similarity x Points Kmeans © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› Points yUsing Similarity Matrix for Cluster Validation  Clusters in random data are not so crisp 1 1 10 0.9 0.9 20 0.8 0.8 30 0.7 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 0 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Similarity Points x Complete Link © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› Points yUsing Similarity Matrix for Cluster Validation 1 0.9 500 1 0.8 2 6 0.7 1000 3 0.6 4 1500 0.5 0.4 2000 0.3 5 0.2 2500 0.1 7 3000 0 500 1000 1500 2000 2500 3000 DBSCAN © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Internal Measures: SSE  Clusters in more complicated figures aren‟t well separated  Internal Index: Used to measure the goodness of a clustering structure without respect to external information – SSE  SSE is good for comparing two clusterings or two clusters (average SSE).  Can also be used to estimate the number of clusters 10 9 6 8 4 7 6 2 5 0 4 2 3 2 4 1 6 0 2 5 10 15 20 25 30 5 10 15 K © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› SSEInternal Measures: SSE  SSE curve for a more complicated data set 1 2 6 3 4 5 7 SSE of clusters found using Kmeans © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Framework for Cluster Validity  Need a framework to interpret any measure. – For example, if our measure of evaluation has the value, 10, is that good, fair, or poor  Statistics provide a framework for cluster validity – The more “atypical” a clustering result is, the more likely it represents valid structure in the data – Can compare the values of an index that result from random data or clusterings to those of a clustering result.  If the value of the index is unlikely, then the cluster results are valid – These approaches are more complicated and harder to understand.  For comparing the results of two different sets of cluster analyses, a framework is less necessary. – However, there is the question of whether the difference between two index values is significant © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Statistical Framework for SSE  Example – Compare SSE of 0.005 against three clusters in random data – Histogram shows SSE of three clusters in 500 sets of random data points of size 100 distributed over the range 0.2 – 0.8 for x and y values 1 50 0.9 45 0.8 40 0.7 35 0.6 30 0.5 25 0.4 20 0.3 15 0.2 10 0.1 5 0 0 0.2 0.4 0.6 0.8 1 0 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034 x SSE © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y CountStatistical Framework for Correlation  Correlation of incidence and proximity matrices for the Kmeans clusterings of the following two data sets. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x Corr = 0.9235 Corr = 0.5810 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹› y yInternal Measures: Cohesion and Separation  Cluster Cohesion: Measures how closely related are objects in a cluster – Example: SSE  Cluster Separation: Measure how distinct or well separated a cluster is from other clusters  Example: Squared Error – Cohesion is measured by the within cluster sum of squares (SSE) 2 WSS(x m ) i i xC i – Separation is measured by the between cluster sum of squares 2 BSS C (mm )  i i i – Where C is the size of cluster i i © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Internal Measures: Cohesion and Separation  Example: SSE – BSS + WSS = constant m  1 m 2 3 4 m 5 1 2 2 2 2 2 K=1 cluster: WSS (1 3) (2 3) (4 3) (5 3) 10 2 BSS 4 (3 3) 0 Total 10 0 10 2 2 2 2 K=2 clusters: WSS (11.5) (21.5) (4 4.5) (5 4.5) 1 2 2 BSS 2 (31.5) 2 (4.5 3) 9 Total 1 9 10 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Internal Measures: Cohesion and Separation  A proximity 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. cohesion separation © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Internal Measures: Silhouette Coefficient  Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings  For an individual point, i – Calculate a = average distance of i to the points in its cluster – Calculate b = min (average distance of i to points in another cluster) – The silhouette coefficient for a point is then given by s = 1 – a/b if a b, (or s = b/a 1 if a  b, not the usual case) b – Typically between 0 and 1. a – The closer to 1 the better.  Can calculate the Average Silhouette width for a cluster or a clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›External Measures of Cluster Validity: Entropy and Purity © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›Final Comment on Cluster Validity “The validation of clustering structures is the most difficult and frustrating part of cluster analysis. Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.” Algorithms for Clustering Data, Jain and Dubes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 ‹›
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