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4. Clustering Methods Concepts Partitional (k-Means, k-Medoids) Hierarchical (Agglomerative & Divisive, COBWEB) Density-based (DBSCAN, CLIQUE) Large size data (STING, BIRCH, CURE) Spring 2003 Data Mining by H. Liu, ASU 1 Concepts of Clustering • Clusters • Different ways of representing clusters – – – – – – Division with boundaries Venn diagram or spheres Probabilistic Dendrograms I1 Trees I2 … Rules 1 2 3 0.5 0.2 0.3 In Spring 2003 Data Mining by H. Liu, ASU 2 • About clusters – Inter-clusters distance maximization – Intra-clusters distance minimization • Clustering vs. classification – Which one is more difficult? Why? – Various possible ways of clustering, which way is the best? Spring 2003 Data Mining by H. Liu, ASU 3 Major Categories • Partitioning: Divide into k partitions (k fixed); repartition to get better clustering. • Hierarchical: Divide into different number of partitions in layers - merge (bottom-up) or divide (top-down). • Density-based: Continue to grow a cluster as long as the density of the cluster exceeds a threshold • Grid-based: First divide space into grids, then perform clustering on the grids. Spring 2003 Data Mining by H. Liu, ASU 4 k-Means • Algorithm 1. 2. 3. 4. 5. • Given k Randomly pick k instances as the initial centers Assign the rest instances to closest one of k clusters Recalculate the mean of each cluster Repeat 3 & 4 until means don’t change How good the clusters are – Initial and final clusters – Within-cluster variation diff(x,mean)^2 – Why don’t we consider inter-cluster distance? Spring 2003 Data Mining by H. Liu, ASU 5 Example • For simplicity, 1 dimensional objects and k=2. • Objects: 1, 2, 5, 6,7 • K-means: – Randomly select 5 and 6 as initial centroids; – => Two clusters {1,2,5} and {6,7}; meanC1=8/3, meanC2=6.5 – => {1,2}, {5,6,7}; meanC1=1.5, meanC2=6 – => no change. – Aggregate dissimilarity = 0.5^2 + 0.5^2 + 1^2 + 1^2 = 2.5 Spring 2003 Data Mining by H. Liu, ASU 6 Discussions • Limitations: – – – – Means cannot be defined for categorical attributes; Choice of k; Sensitive to outliers; Crisp clustering • Variants of k-means exist: – Using modes to deal with categorical attributes • How about distance measures • Is it similar to or different from k-NN? – With and without learning Spring 2003 Data Mining by H. Liu, ASU 7 k-Medoids • k-Means algorithm is sensitive to outliers – Is this true? How to prove it? • Medoid – the most centrally located point in a cluster, as a representative point of the cluster. • In contrast, a centroid is not necessarily inside a cluster. • An example Initial Medoids Spring 2003 Data Mining by H. Liu, ASU 8 Partition Around Medoids • PAM: 1. Given k 2. Randomly pick k instances as initial medoids 3. Assign each instance to the nearest medoid x 4. Calculate the objective function • the sum of dissimilarities of all instances to their nearest medoids 5. Randomly select an instance y 6. Swap x by y if the swap reduces the objective function 7. Repeat (3-6) until no change Spring 2003 Data Mining by H. Liu, ASU 9 k-Means and k-Medoids • The key difference lies in how they update means or medoids • Both require distance calculation and reassignment of instances • Time complexity Outlier (100 unit away) – Which one is more costly? • Dealing with outliers Spring 2003 Data Mining by H. Liu, ASU 10 EM (Expectation and Maximization) • Moving away from crisp clusters as in k-Means by allowing an instance to belong to several clusters • Finite mixtures – a statistical clustering model – A mixture is a set of k probability distributions, representing k clusters – The simplest finite mixture: one feature with a Gaussian – When k=2, we need to estimate 5 parameters: 2 pairs of μ and σ and pA, where pB = 1- pA • EM – Estimate using instances – Maximize the overall likelihood that data came from this data set Spring 2003 Data Mining by H. Liu, ASU 11 Agglomerative • Each object is viewed as a cluster (bottom up). • Repeat until the number of clusters is small enough – Choose a closest pair of clusters – Merge the two into one • Defining “closest”: Centroid (mean of cluster) distance, (average) sum of pairwise distance, … – Refer to the Evaluation part • A dendrogram is a tree that shows clustering process. Spring 2003 Data Mining by H. Liu, ASU 12 Dendrogram • Cluster 1, 2, 4, 5, 6, 7 into two clusters (centriod distance) 1 2 4 5 6 7 Spring 2003 Data Mining by H. Liu, ASU 13 An example to show different Links • Single link A B C D E – Merge the nearest clusters measured by the shortest edge between the two – (((A B) (C D)) E) • Complete link – Merge the nearest clusters measured by the longest edge between the two – (((A B) E) (C D)) A 0 1 2 2 3 B 1 0 2 4 3 C 2 2 0 1 5 D 2 4 1 0 3 E 3 3 5 3 0 B A • Average link – Merge the nearest clusters measured byE the average edge length between the two – (((A B) (C D)) E) Spring 2003 Data Mining by H. Liu, ASU C D 14 Divisive • All instances belong to one cluster (top-down) • To find an optimal division at each layer (especially the top one) is computationally prohibitive. • One heuristic method is based on the Minimum Spanning Tree (MST) algorithm – Connecting all instances with MST (O(N2)) – Repeatedly cut out the longest edges at each iteration until some stopping criterion is met or until one instance remains in each cluster. Spring 2003 Data Mining by H. Liu, ASU 15 COBWEB • Building a conceptual hierarchy incrementally • Category Utility: kijP(fi=vij)P(fi=vij|ck)P(ck|fi=vij) – All categories ck, all features fi, all feature values vij • It attempts to maximize both the probability that two objects in the same category have values in common and the probability that objects in different categories will have different property values Spring 2003 Data Mining by H. Liu, ASU 16 • Processing one instance at a time by evaluating – Placing the instance in the best existing category – Adding a new category containing only the instance – Merging of two existing categories into a new one and adding the instance to that category – Splitting of an existing category into two and placing the instance in the best new resulting category Grandparent Grandparent Split Parent Child 1 Spring 2003 Child 2 Merge Child 1 Data Mining by H. Liu, ASU Child 2 17 Density-based • BBSCAN –Density-Based Clustering of Applications with Noise • It grows regions with sufficiently high density into clusters and can discover clusters of arbitrary shape in spatial databases with noise. – Many existing clustering algorithms find spherical shapes of clusters • DEBSCAN defines a cluster as a maximal set of density-connected points. Spring 2003 Data Mining by H. Liu, ASU 18 • Defining density and connection – – – – – -neighborhood of an object x (core object) (M, P, O) MinPts of objects within -neighborhood (say, 3) directly density-reachable (Q from M, M from P) density-reachable (Q from P, P not from Q) [asymmetric] density-connected (O, R, S) [symmetric] for border points • What is the relationship between DR and DC? Q R M S Q Spring 2003 O Data Mining by H. Liu, ASU 19 • Clustering with DBSCAN – Search for clusters by checking the -neighborhood of each instance x – If the -neighborhood of x contains more than MinPts, create a new cluster with x as a core object – Iteratively collect directly density-reachable objects from these core object and merge density-reachable clusters – Terminate when no new point can be add to any cluster • DBSCAN is sensitive to the thresholds of density, but it is many folds faster than CLARANS • Time complexity O(N log N) if a spatial index is used, O(N2) otherwise Spring 2003 Data Mining by H. Liu, ASU 20 Dealing with Large Data • Key ideas – Reducing the number of instances yet to maintain the distribution – Identifying relevant subspaces where clusters possibly exist – Using summarized information to avoid repeated data access • Sampling – CLARA (Clustering LARge Applications) working on samples instead of the whole data – CLARANS (Clustering Large Applications based on RANdomized Search) Spring 2003 Data Mining by H. Liu, ASU 21 • Grid: STING (STatistical INformation Grid) – Statistical parameters of higher-level cells can easily be computed from those of lower-level cells • Attribute-independent: count • Attribute-dependent: mean, standard deviation, min, max • Type of distribution: normal, uniform, exponential, or unknown – Irrelevant cells can be removed Spring 2003 Data Mining by H. Liu, ASU 22 Representatives • BIRCH using Clustering Feature (CF) and CF tree – A cluster feature is a triplet about sub-clusters of instances (N, LS, SS) • N - the number of instances, LS – linear sum, SS – square sum – Two thresholds: branching factor and the max number of children per non-leaf node – Two phases 1. Build an initial in-memory CF tree 2. Apply a clustering algorithm to cluster the leaf nodes in CF tree • CURE (Clustering Using REpresentitives) is another example Spring 2003 Data Mining by H. Liu, ASU 23 • Taking advantage of the property of density – If it’s dense in higher dimensional subspaces, it should be dense in some lower dimensional subspaces – CLIQUE (CLustering In QUEst) • With high dimensional data, there are many void subspaces • Using the property identified, we can start with dense lower dimensional data • CLIQUE is a density-based method that can automatically find subspaces of the highest dimensionality such that high-density clusters exist in those subspaces Spring 2003 Data Mining by H. Liu, ASU 24 Chameleon • A hierarchical Clustering Algorithm Using Dynamic Modeling – Observations on the weakness of CURE and ROCK Spring 2003 Data Mining by H. Liu, ASU 25 Summary • There are many clustering algorithms • Good clustering algorithms maximize inter-cluster dissimilarity and intra-cluster similarity • Without prior knowledge, it is difficult to choose the best clustering algorithm. • Clustering is an important tool for outlier analysis. Spring 2003 Data Mining by H. Liu, ASU 26 Bibliography • I.H. Witten and E. Frank. Data Mining – Practical Machine Learning Tools and Techniques with Java Implementations. 2000. Morgan Kaufmann. • M. Kantardzic. Data Mining – Concepts, Models, Methods, and Algorithms. 2003. IEEE. • J. Han and M. Kamber. Data Mining – Concepts and Techniques. 2001. Morgan Kaufmann. • M. H. Dunham. Data Mining – Introductory and Advanced Topics. Spring 2003 Data Mining by H. Liu, ASU 27