Data Mining Cluster Analysis Lecture Notes PDF

Data Mining Cluster Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar...

Data Mining Cluster Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1  What 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 Inter-cluster Intra-cluster distances are distances are maximized minimized © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 2  Applications of Cluster Analysis  Understanding Discovered Clusters Industry Group 1 Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN, Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN, – Group related documents DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN, Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down, Technology1-DOWN Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN, for browsing, group genes Sun-DOWN 2 Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN, and proteins that have ADV-Micro-Device-DOWN,Andrew-Corp-DOWN, Computer-Assoc-DOWN,Circuit-City-DOWN, Technology2-DOWN Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN, similar functionality, or Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN group stocks with similar 3 Fannie-Mae-DOWN,Fed-Home-Loan-DOWN, MBNA-Corp-DOWN,Morgan-Stanley-DOWN Financial-DOWN price fluctuations 4 Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP, Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP, Oil-UP Schlumberger-UP  Summarization – Reduce the size of large data sets Clustering precipitation in Australia © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 3  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 4  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 5  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 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 tree © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 6  Partitional Clustering Original Points A Partitional Clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 7  Hierarchical Clustering p1 p3 p4 p2 p1 p2 p3 p4 Traditional Hierarchical Clustering Traditional Dendrogram p1 p3 p4 p2 p1 p2 p3 p4 Non-traditional Hierarchical Clustering Non-traditional Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 8  Other 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 densities © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 9  Types of Clusters  Well-separated clusters  Center-based clusters  Contiguous clusters  Density-based clusters  Property or Conceptual  Described by an Objective Function © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 10  Types 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 clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 11  Types 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 clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 12  Types 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 clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 13  Types 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 clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 14  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 15  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 16  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 17  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 18  Clustering Algorithms  K-means and its variants  Hierarchical clustering  Density-based clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 19  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 very simple © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 20  K-means 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.  K-means 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 21  Two different K-means Clusterings 3 2.5 2 Original Points 1.5 y 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 y y 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 Sub-optimal Clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 22  Importance of Choosing Initial Centroids Iteration 6 1 2 3 4 5 3 2.5 2 1.5 y 1 0.5 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 23  Importance 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 y y y 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 y y y 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 24  Evaluating K-means 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 SSE   dist 2 ( mi , x ) i 1 xCi – x is a data point in cluster Ci and mi is the representative point for cluster Ci  can show that mi corresponds to the center (mean) of the cluster – 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 25  Importance of Choosing Initial Centroids … Iteration 5 1 2 3 4 3 2.5 2 1.5 y 1 0.5 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 26  Importance of Choosing Initial Centroids … Iteration 1 Iteration 2 3 3 2.5 2.5 2 2 1.5 1.5 y y 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 y y y 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 27  Problems 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 – For example, if K = 10, then probability = 10!/1010 = 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 28  10 Clusters Example Iteration 4 1 2 3 8 6 4 2 0 y -2 -4 -6 0 5 10 15 20 x Starting with two initial centroids in one cluster of each pair of clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 29  10 Clusters Example Iteration 1 Iteration 2 8 8 6 6 4 4 2 2 0 0 y y -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 y -2 y -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 30  10 Clusters Example Iteration 4 1 2 3 8 6 4 2 0 y -2 -4 -6 0 5 10 15 20 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 31  10 Clusters Example Iteration 1 Iteration 2 8 8 6 6 4 4 2 2 0 0 y y -2 -2 -4 -4 -6 -6 0 5 10 15 20 0 5 10 15 20 Iteration x 3 Iteration x 4 8 8 6 6 4 4 2 2 0 0 y y -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 32  Solutions 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 K-means – Not as susceptible to initialization issues © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 33  Handling Empty Clusters  Basic K-means 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 34  Updating Centers Incrementally  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 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 35  Pre-processing and Post- processing  Pre-processing – Normalize the data – Eliminate outliers  Post-processing – 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 36  Bisecting K-means  Bisecting K-means algorithm – Variant of K-means that can produce a partitional or a hierarchical clustering © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 37  Bisecting K-means Example © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 38  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 4/18/2004 39  Limitations of K-means: Differing Sizes Original Points K-means (3 Clusters) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 40  Limitations of K-means: Differing Density Original Points K-means (3 Clusters) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 41 Limitations of K-means: Non-globular Shapes Original Points K-means (2 Clusters) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 42 Overcoming K-means Limitations Original Points K-means 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 43  Overcoming K-means Limitations Original Points K-means Clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 44 Overcoming K-means Limitations Original Points K-means Clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 45  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 6 5 0.2 4 3 4 0.15 2 5 0.1 2 1 0.05 3 1 0 1 3 2 5 4 6 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 46  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 47  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, 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 similarity or distance matrix – Merge or split one cluster at a time © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 48  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 49  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 50  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 C2 C5... p1 p2 p3 p4 p9 p10 p11 p12 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 51  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 C2 C5... p1 p2 p3 p4 p9 p10 p11 p12 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 52  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 53  How to Define Inter-Cluster 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 54  How to Define Inter-Cluster 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 55  How to Define Inter-Cluster 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 56  How to Define Inter-Cluster 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 57  How to Define Inter-Cluster 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 58  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 59  Hierarchical Clustering: MIN 5 1 3 5 0.2 2 1 0.15 2 3 6 0.1 0.05 4 4 0 3 6 2 5 4 1 Nested Clusters Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 60  Strength of MIN Original Points Two Clusters Can handle non-elliptical shapes © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 61  Limitations of MIN Original Points Two Clusters Sensitive to noise and outliers © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 62  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 63  Hierarchical Clustering: MAX 4 1 2 5 0.4 0.35 5 2 0.3 0.25 3 6 0.2 3 0.15 1 0.1 4 0.05 0 3 6 4 1 2 5 Nested Clusters Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 64  Strength of MAX Original Points Two Clusters Less susceptible to noise and outliers © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 65  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 66  Cluster Similarity: Group Average  Proximity of two clusters is the average of pairwise proximity between points in the two clusters.  proximity(p ,p ) piCluster i j i pjCluster proximity( Cluster i , Cluster j)  j |Clusteri | |Clusterj |  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 67  Hierarchical Clustering: Group Average 5 4 1 0.25 2 5 0.2 2 0.15 3 6 0.1 1 0.05 4 0 3 3 6 4 1 2 5 Nested Clusters Dendrogram © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 68  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 69  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 K-means – Can be used to initialize K-means © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 70  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 1 5 4 1 2 2 5 Ward’s Method 5 2 2 3 6 Group Average 3 6 3 4 1 1 4 4 3 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 71  Hierarchical Clustering: Time and Space requirements  O(N2) space since it uses the proximity matrix. – N is the number of points.  O(N3) time in many cases – There are N steps and at each step the size, N2, proximity matrix must be updated and searched – Complexity can be reduced to O(N2 log(N) ) time for some approaches © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 72  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 73  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 74  MST: Divisive Hierarchical Clustering  Use MST for constructing hierarchy of clusters © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 75  DBSCAN  DBSCAN is a density-based 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 76  DBSCAN: Core, Border, and Noise Points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 77  DBSCAN Algorithm  Eliminate noise points  Perform clustering on the remaining points © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 78  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 79  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 80  When DBSCAN Does NOT Work Well (MinPts=4, Eps=9.75). Original Points Varying densities High-dimensional data (MinPts=4, Eps=9.92) © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 81  DBSCAN: Determining EPS and MinPts  Idea is that for points in a cluster, their kth nearest neighbors are at roughly the same distance  Noise points have the kth nearest neighbor at farther distance  So, plot sorted distance of every point to its kth nearest neighbor © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 82  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 83  Clusters found in Random Data 1 1 0.9 0.9 0.8 0.8 0.7 0.7 Random 0.6 0.6 DBSCAN Points 0.5 0.5 y y 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 K-means Complete 0.8 0.8 0.7 0.7 Link 0.6 0.6 0.5 0.5 y y 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 84  Different Aspects of Cluster Validation 1. Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random 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 85  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 86  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(n-1) / 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 87  Measuring Cluster Validity Via Correlation  Correlation of incidence and proximity matrices for the K-means 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 y y 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 88 Using 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 Points 50 0.5 0.5 y 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 89  Using 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 Points 50 0.5 0.5 y 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 DBSCAN © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 90  Using 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 Points 50 0.5 0.5 y 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 K-means © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 91  Using 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 Points 50 0.5 0.5 y 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 Complete Link © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 92  Using Similarity Matrix for Cluster Validation 1 0.9 1 500 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 93  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 6 9 8 4 7 2 6 SSE 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 94  Internal Measures: SSE  SSE curve for a more complicated data set 1 2 6 3 4 5 7 SSE of clusters found using K-means © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 95  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 96  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 Count 0.5 y 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 97 Statistical Framework for Correlation  Correlation of incidence and proximity matrices for the K-means 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 y y 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 98  Internal 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) WSS   ( x  mi )2 i xC i – Separation is measured by the between cluster sum of squares BSS  Ci ( m  mi ) 2 i – Where |Ci| is the size of cluster i © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 99  Internal Measures: Cohesion and Separation  Example: SSE – BSS + WSS = constant m    1 m1 2 3 4 m2 5 K=1 cluster: WSS (1  3) 2  ( 2  3) 2  ( 4  3) 2  (5  3) 2 10 BSS4 (3  3) 2 0 Total 10  0 10 K=2 clusters: WSS (1  1.5) 2  ( 2  1.5) 2  ( 4  4.5) 2  (5  4.5) 2 1 BSS2 (3  1.5) 2  2 ( 4.5  3) 2 9 Total 1  9 10 © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 100  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 101  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 102  External Measures of Cluster Validity: Entropy and Purity © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 103  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 104