Data Mining Cluster Analsis: Basic Concepts and Algorithms Tan,Steinbach, Kumar Introduction to Data Mining /8/ 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
Applications of 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 Summarization Reduce the size of large data sets Clustering precipitation in Australia What is not Cluster Analsis? Supervised classification Have class label information Simple segmentation Dividing students into different registration groups alphabeticall, b last name Results of a quer Groupings are a result of an eternal specification Graph partitioning Some mutual relevance and snerg, but areas are not identical
Notion of a Cluster can be Ambiguous How man clusters? Si Clusters Two Clusters Four Clusters Tpes 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 eactl one subset Hierarchical clustering A set of nested clusters organized as a hierarchical tree
Partitional Clustering Original Points A Partitional Clustering Hierarchical Clustering p p p p p p p p Traditional Hierarchical Clustering Traditional Dendrogram p p p p p p p p Non-traditional Hierarchical Clustering Non-traditional Dendrogram
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 Conceptual Clusters
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. well-separated clusters 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 center-based clusters
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. 8 contiguous clusters 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. 6 densit-based clusters
Tpes of Clusters: Conceptual Clusters Shared Propert or Conceptual Clusters Finds clusters that share some common propert or represent a particular concept.. Overlapping Circles Characteristics of the Input Data Are Important Tpe of proimit or densit measure This is a derived measure, but central to clustering Sparseness Dictates tpe of similarit Adds to efficienc Attribute tpe Dictates tpe of similarit Tpe of data Dictates tpe of similarit Other characteristics, e.g., autocorrelation Dimensionalit Noise and outliers
Clustering Algorithms K-means and its variants Hierarchical clustering Densit-based clustering 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 b user The basic algorithm is ver simple
K-means: Eample 9 8 7 6 Assign each objects 6 7 8 9 to most similar center K= Arbitraril choose K object as initial cluster center 9 8 7 6 6 7 8 9 9 8 7 6 6 7 8 9 Update the cluster means Update the cluster means 9 8 7 6 6 7 8 9 9 8 7 6 reassign 6 7 8 9 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. 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
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 SSE = K i= C i dist ( m, ) is a data point in cluster C i and m i is the representative point for cluster C i Given two clusters, choose the one with the smallest error One eas wa to reduce SSE is to increase K, the number of clusters u A good clustering with smaller K can have a lower SSE than a poor clustering with higher K i Two different K-means Clusterings.. Original Points. - -. - -......... - -. - -... Optimal Clustering - -. - -... Sub-optimal Clustering
Importance of Choosing Initial Centroids Iteration Iteration Iteration......... - -. - -... - -. - -... - -. - -... Iteration Iteration Iteration 6......... - -. - -... - -. - -... - -. - -... Importance of Choosing Initial Centroids Iteration Iteration...... - -. - -... - -. - -... Iteration Iteration Iteration......... - -. - -... - -. - -... - -. - -...
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 relativel small when K is large If clusters are the same size, n, then For eample, if K =, then probabilit =!/ =.6 Sometimes the initial centroids will readjust themselves in right wa, and sometimes the don t Solutions to Initial Centroids Problem Multiple runs Helps, but probabilit is not on our side Sample and use hierarchical clustering to determine initial centroids Select more than k initial centroids and then select among these initial centroids Select most widel separated Postprocessing Bisecting K-means Not as susceptible to initialization issues
Handling Empt Clusters Basic K-means algorithm can ield empt clusters Several strategies Choose the point that contributes most to SSE Choose a point from the cluster with the highest SSE If there are several empt clusters, the above can be repeated several times. Updating Centers Incrementall In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid An alternative is to update the centroids after each assignment (incremental approach) Each assignment updates zero or two centroids More epensive Introduces an order dependenc Never get an empt cluster Can use weights to change the impact
Pre-processing and Post-processing Pre-processing Normalize the data Eliminate outliers Post-processing Eliminate small clusters that ma represent outliers Split loose clusters, i.e., clusters with relativel high SSE Merge clusters that are close and that have relativel low SSE Can use these steps during the clustering process Bisecting K-means Bisecting K-means algorithm Variant of K-means that can produce a partitional or a hierarchical clustering
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.
Limitations of K-means: Differing Sizes Original Points K-means ( Clusters) Limitations of K-means: Differing Densit Original Points K-means ( Clusters)
Limitations of K-means: Non-globular Shapes Original Points K-means ( Clusters) Overcoming K-means Limitations Original Points K-means Clusters One solution is to use man clusters. Find parts of clusters, then put parts together.
Overcoming K-means Limitations Original Points K-means Clusters Overcoming K-means Limitations Original Points K-means Clusters
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. 6 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 (e.g., animal kingdom, phlogen reconstruction, )
Hierarchical Clustering Two main tpes of hierarchical clustering Agglomerative: u Start with the points as individual clusters u At each step, merge the closest pair of clusters until onl one cluster (or k clusters) left Divisive: u Start with one, all-inclusive cluster u 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 6. 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
Starting Situation Start with clusters of individual points and a proimit matri p p p p p.. p p p p p.... Proimit Matri... p p p p p9 p p p 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... p p p p p9 p p p
Intermediate Situation 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 C C... p p p p p9 p p p After Merging The question is How do we update the proimit matri? C C U C C C C? C C C C U C???? C? C? Proimit Matri C U C... p p p p p9 p p p
How to Define Inter-Cluster Similarit p p p p p... Similarit? 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 How to Define Inter-Cluster Similarit p 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
How to Define Inter-Cluster Similarit p 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 How to Define Inter-Cluster Similarit p 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
How to Define Inter-Cluster Similarit p 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 Cluster Similarit: MIN or Single Link Similarit of two clusters is based on the two most similar (closest) points in the different clusters Determined b one pair of points, i.e., b one link in the proimit graph. I I I I I I..9..6. I.9..7.6. I..7... I.6.6...8 I....8.
Hierarchical Clustering: MIN 6.... 6 Nested Clusters Dendrogram Strength of MIN Original Points Two Clusters Can handle non-elliptical shapes
Limitations of MIN Original Points Two Clusters Sensitive to noise and outliers 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 I I I I I I..9..6. I.9..7.6. I..7... I.6.6...8 I....8.
Hierarchical Clustering: MAX 6........ 6 Nested Clusters Dendrogram Strength of MAX Original Points Two Clusters Less susceptible to noise and outliers
Limitations of MAX Original Points Two Clusters Tends to break large clusters Biased towards globular clusters Cluster Similarit: Group Average Proimit of two clusters is the average of pairwise proimit between points in the two clusters. proimit(p i,pj) pi Clusteri p Cluster j j 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..6. I.9..7.6. I..7... I.6.6...8 I....8. i j
Hierarchical Clustering: Group Average. 6.... 6 Nested Clusters Dendrogram Cluster Similarit: Ward s Method 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 Biased towards globular clusters Hierarchical analogue of K-means Can be used to initialize K-means
Hierarchical Clustering: Comparison 6 MIN MAX 6 6 Group Average Ward s Method 6 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
Hierarchical Clustering: Problems and Limitations Once a decision is made to combine two clusters, it cannot be undone No objective function is directl minimized Different schemes have problems with one or more of the following: Sensitivit to noise and outliers Difficult handling different sized clusters and conve shapes Breaking large clusters 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 u 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.
DBSCAN: Core, Border, and Noise Points DBSCAN Algorithm Eliminate noise points Perform clustering on the remaining points
DBSCAN: Core, Border and Noise Points Original Points Point tpes: core, border and noise Eps =, MinPts = When DBSCAN Works Well Original Points Clusters Resistant to Noise Can handle clusters of different shapes and sizes
When DBSCAN Does NOT Work Well Original Points (MinPts=, Eps=9.7). Varing densities High-dimensional data (MinPts=, Eps=9.9) DBSCAN: Determining EPS and MinPts 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
Cluster Validit For supervised classification we have a variet of measures to evaluate how good our model is Accurac, precision, recall For cluster analsis, the analogous question is how to evaluate the goodness of the resulting clusters? But clusters are in the ee of the beholder! Then wh 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 Clusters found in Random Data Random Points.9.8.7.6......9...6.8.9.8.7.6........6.8.9 DBSCAN K-means.8.7.6.8.7.6 Complete Link.............6.8...6.8
Different Aspects of Cluster Validation. Determining the clustering tendenc of a set of data, i.e., distinguishing whether non-random structure actuall eists in the data.. Comparing the results of a cluster analsis to eternall known results, e.g., to eternall given class labels.. Evaluating how well the results of a cluster analsis fit the data without reference to eternal information. - Use onl the data. Comparing the results of two different sets of cluster analses to determine which is better.. Determining the correct number of clusters. For,, and, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters. Framework for Cluster Validit Need a framework to interpret an measure. For eample, if our measure of evaluation has the value,, is that good, fair, or poor? Statistics provide a framework for cluster validit The more atpical a clustering result is, the more likel it represents valid structure in the data Can compare the values of an inde that result from random data or clusterings to those of a clustering result. u If the value of the inde is unlikel, 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 analses, a framework is less necessar. However, there is the question of whether the difference between two inde values is significant
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. u Entrop Internal Inde: Used to measure the goodness of a clustering structure without respect to eternal information. u Sum of Squared Error (SSE) Relative Inde: Used to compare two different clusterings or clusters. u Often an eternal or internal inde is used for this function, e.g., 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. Internal Measures: SSE Internal Inde: Used to measure the goodness of a clustering structure without respect to eternal information SSE is good for comparing two clusterings or two clusters (average SSE). Can also be used to estimate the number of clusters 6 - - -6 SSE 9 8 7 6 K
Internal Measures: SSE SSE curve for a more complicated data set 6 7 SSE of clusters found using K-means Internal Measures: Cohesion and Separation Cluster Cohesion: Measures how closel related are objects in a cluster 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 = i C ( m ) i Separation is measured b the between cluster sum of squares BSS = Ci ( m m i ) i Where C i is the size of cluster i i
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. cohesion separation Internal Measures: Correlation Two matrices Proimit Matri Incidence Matri u u u One row and one column for each data point An entr is if the associated pair of points belong to the same cluster An entr is if the associated pair of points belongs to different clusters Compute the correlation between the two matrices Since the matrices are smmetric, onl the correlation between n(n-) / 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 densit or contiguit based clusters.
Internal Measures: Correlation Correlation of incidence and proimit matrices for the K-means clusterings of the following two data sets..9.8.7.6........6.8.9.8.7.6........6.8 Corr =.9 Corr =.8 Internal Measures: Correlation Order the similarit matri with respect to cluster labels and inspect visuall..9.8.7.6........6.8 Points 6 7 8 9.9.8.7.6..... 6 8 Similarit Points
Internal Measures: Correlation Clusters in random data are not so crisp.9.8.7.9.8.7.6.. Points 6.6........6.8 7 8 9... 6 8 Similarit Points Internal Measures: Correlation.9 6.8.7.6.. 7... DBSCAN
Significance of Measures: SSE Compare three clusters (SSE=.) with random data Histogram shows SSE of three clusters in sets of random data each contains data points range over..8 for and values.9.8.7.6........6.8 Count.6.8....6.8... SSE Significance of Measures: Correlation Correlation of incidence and proimit matrices for the K-means clusterings of the following two data sets..9.8.7.6........6.8.9.8.7.6........6.8 Corr =.9 Corr =.8
Eternal Measures of Cluster Validit: Entrop and Purit 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