# Lecture 10 Clustering 1 Preview Introduction Partitioning methods

Lecture 10 Clustering 1 Preview Introduction Partitioning methods Hierarchical methods Model-based methods Density-based methods 2 Examples of Clustering Applications

Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with a high average claim cost Urban planning: Identifying groups of houses according to their house type, value, and geographical location Seismology: Observed earth quake epicenters should be clustered along continent faults 4 What Is a Good Clustering? A good clustering method will produce clusters with

High intra-class similarity Low inter-class similarity Precise definition of clustering quality is difficult Application-dependent Ultimately subjective 5 Requirements for Clustering in Data Mining Scalability Ability to deal with different types of attributes Discovery of clusters with arbitrary shape

Minimal domain knowledge required to determine input parameters Ability to deal with noise and outliers Insensitivity to order of input records Robustness wrt high dimensionality Incorporation of user-specified constraints Interpretability and usability 6 Similarity and Dissimilarity Between Objects Same we used for IBL (e.g, Lp norm)

Euclidean distance (p = 2): d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j2 ip jp Properties of a metric d(i,j): d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j) 7 Major Clustering Approaches

Partitioning: Construct various partitions and then evaluate them by some criterion Hierarchical: Create a hierarchical decomposition of the set of objects using some criterion Model-based: Hypothesize a model for each cluster and find best fit of models to data Density-based: Guided by connectivity and density functions 8 Partitioning Algorithms Partitioning method: Construct a partition of a database D of n objects into a set of k clusters Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Global optimal: exhaustively enumerate all partitions

Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen, 1967): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw, 1987): Each cluster is represented by one of the objects in the cluster 9 K-Means Clustering Given k, the k-means algorithm consists of four steps: Select initial centroids at random. Assign each object to the cluster with the nearest centroid. Compute each centroid as the mean of the objects assigned to it. Repeat previous 2 steps until no change. 10 K-Means Clustering (contd.)

Example 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3

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6 7 8 9 10 11 Comments on the K-Means Method Strengths Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Often terminates at a local optimum. The global optimum may be found using techniques such as simulated annealing and genetic algorithms Weaknesses Applicable only when mean is defined (what about categorical data?) Need to specify k, the number of clusters, in advance Trouble with noisy data and outliers Not suitable to discover clusters with non-convex shapes 12

Hierarchical Clustering Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 a b Step 1 Step 2 Step 3 Step 4 ab abcde c cde d de e Step 4 agglomerative (AGNES)

Step 3 Step 2 Step 1 Step 0 divisive (DIANA) 13 AGNES (Agglomerative Nesting) Produces tree of clusters (nodes) Initially: each object is a cluster (leaf) Recursively merges nodes that have the least dissimilarity Criteria: min distance, max distance, avg distance, center distance Eventually all nodes belong to the same cluster (root) 10

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5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1

2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6

7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 14 A Dendrogram Shows How the

Clusters are Merged Hierarchically Decompose data objects into several levels of nested partitioning (tree of clusters), called a dendrogram. A clustering of the data objects is obtained by cutting the dendrogram at the desired level. Then each connected component forms a cluster. 15 DIANA (Divisive Analysis) Inverse order of AGNES Start with root cluster containing all objects Recursively divide into subclusters Eventually each cluster contains a single object 10 10 10 9

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3 3 3 2 2 2 1 1 1 0 0 1 2 3 4 5 6

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0 1 2 3 4 5 6 7 8 9 10 16 Other Hierarchical Clustering Methods Major weakness of agglomerative clustering methods

Do not scale well: time complexity of at least O(n2), where n is the number of total objects Can never undo what was done previously Integration of hierarchical with distance-based clustering BIRCH: uses CF-tree and incrementally adjusts the quality of sub-clusters CURE: selects well-scattered points from the cluster and then shrinks them towards the center of the cluster by a specified fraction 17 BIRCH BIRCH: Balanced Iterative Reducing and Clustering using Hierarchies (Zhang, Ramakrishnan & Livny, 1996) Incrementally construct a CF (Clustering Feature) tree Parameters: max diameter, max children Phase 1: scan DB to build an initial in-memory CF tree (each node: #points, sum, sum of squares) Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree

Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans Weaknesses: handles only numeric data, sensitive to order of data records. 18 Clustering Feature Vector Clustering Feature: CF = (N, LS, SS) N: Number of data points LS: Ni=1 Xi CF = (5, (16,30),(54,190)) SS: Ni=1 Xi2 10 9 8 7 6 5 4 3 2 1 0 0 1 2

3 4 5 6 7 8 9 10 (3,4) (2,6) (4,5) (4,7) (3,8) 19 CF Tree Root B=7 CF1 CF2 CF3

CF6 L=6 child1 child2 child3 child6 CF1 Non-leaf node CF2 CF3 CF5 child1 child2 child3 child5 Leaf node prev CF1 CF2 CF6 next Leaf node prev

CF1 CF2 CF4 next 20 CURE (Clustering Using REpresentatives) CURE: non-spherical clusters, robust wrt outliers Uses multiple representative points to evaluate the distance between clusters Stops the creation of a cluster hierarchy if a level consists of k clusters 21 Drawbacks of Distance-Based Method Drawbacks of square-error-based clustering method Consider only one point as representative of a cluster

Good only for convex clusters, of similar size and density, and if k can be reasonably estimated 22 Cure: The Algorithm Draw random sample s Partition sample to p partitions with size s/ p Partially cluster partitions into s/pq clusters Cluster partial clusters, shrinking representatives towards centroid Label data on disk 23 Data Partitioning and Clustering

s = 50 p=2 s/p = 25 s/pq = 5 y y y x y y x x x x 24 Cure: Shrinking Representative Points y y x

x Shrink the multiple representative points towards the gravity center by a fraction of . Multiple representatives capture the shape of the cluster 25 Model-Based Clustering Basic idea: Clustering as probability estimation One model for each cluster Generative model: Probability of selecting a cluster Probability of generating an object in cluster Find max. likelihood or MAP model Missing information: Cluster membership Use EM algorithm

Quality of clustering: Likelihood of test objects 26 Mixtures of Gaussians Cluster model: Normal distribution (mean, covariance) Assume: diagonal covariance, known variance, same for all clusters Max. likelihood: mean = avg. of samples But what points are samples of a given cluster? Estimate prob. that point belongs to cluster Mean = weighted avg. of points, weight = prob. But to estimate probs. we need model Chicken and egg problem: use EM algorithm 27 EM Algorithm for Mixtures

Initialization: Choose means at random E step: For all points and means, compute Prob(point| mean) Prob(mean|point) = Prob(mean) Prob(point|mean) / Prob(point) M step: Each mean = Weighted avg. of points Weight = Prob(mean|point) Repeat until convergence 28 EM Algorithm (contd.) Guaranteed to converge to local optimum K-means is special case 29 AutoClass

Developed at NASA (Cheeseman & Stutz, 1988) Mixture of Nave Bayes models Variety of possible models for Prob(attribute| class) Missing information: Class of each example Apply EM algorithm as before Special case of learning Bayes net with missing values Widely used in practice 30 COBWEB Grows tree of clusters (Fisher, 1987) Each node contains: P(cluster), P(attribute|cluster) for each attribute Objects presented sequentially Options: Add to node, new node; merge, split Quality measure: Category utility: Increase in predictability of attributes/#Clusters

31 A COBWEB Tree 32 Neural Network Approaches Neuron = Cluster = Centroid in instance space Layer = Level of hierarchy Several competing sets of clusters in each layer Objects sequentially presented to network Within each set, neurons compete to win object Winning neuron is moved towards object Can be viewed as mapping from low-level features to high-level ones 33 Competitive Learning 34

Self-Organizing Feature Maps Clustering is also performed by having several units competing for the current object The unit whose weight vector is closest to the current object wins The winner and its neighbors learn by having their weights adjusted SOMs are believed to resemble processing that can occur in the brain Useful for visualizing high-dimensional data in 2- or 3-D space 35 Density-Based Clustering Clustering based on density (local cluster criterion), such as density-connected points

Major features: Discover clusters of arbitrary shape Handle noise One scan Need density parameters as termination condition Representative algorithms: DBSCAN (Ester et al., 1996) DENCLUE (Hinneburg & Keim, 1998) 36 Definitions (I) Two parameters: Eps: Maximum radius of neighborhood MinPts: Minimum number of points in an Epsneighborhood of a point NEps(p) ={q D | dist(p,q) <= Eps} Directly density-reachable: A point p is directly

density-reachable from a point q wrt. Eps, MinPts iff 1) p belongs to NEps(q) 2) q is a core point: p q MinPts = 5 Eps = 1 cm |NEps (q)| >= MinPts 37 Definitions (II) Density-reachable: Density-connected p A point p is density-reachable

from a point q wrt. Eps, MinPts if there is a chain of points p1, , pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts. p1 q p q o 38 DBSCAN: Density Based Spatial Clustering of Applications with Noise Relies on a density-based notion of cluster: A cluster is defined as a maximal set of densityconnected points Discovers clusters of arbitrary shape in spatial databases with noise

Outlier Border Core Eps = 1cm MinPts = 5 39 DBSCAN: The Algorithm Arbitrarily select a point p Retrieve all points density-reachable from p wrt Eps and MinPts. If p is a core point, a cluster is formed. If p is a border point, no points are densityreachable from p and DBSCAN visits the next point of the database. Continue the process until all of the points have been processed. 40 DENCLUE: Using Density Functions

DENsity-based CLUstEring (Hinneburg & Keim, 1998) Major features Good for data sets with large amounts of noise Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets Significantly faster than other algorithms (faster than DBSCAN by a factor of up to 45) But needs a large number of parameters 41 DENCLUE Uses grid cells but only keeps information about grid cells that do actually contain data points and manages these cells in a tree-based access structure. Influence function: describes the impact of a data

point within its neighborhood. Overall density of the data space can be calculated as the sum of the influence function of all data points. Clusters can be determined mathematically by identifying density attractors. Density attractors are local maxima of the overall density function. 42 Influence Functions Example f Gaussian ( x , y ) e f D Gaussian f d ( x , y )2

2 2 N ( x ) i 1 e D Gaussian N d ( x , xi ) 2 2 2 ( x, xi ) i 1 ( xi x) e d ( x , xi ) 2 2 2 43 Density Attractors 44 Center-Defined & Arbitrary Clusters 45

Clustering: Summary Introduction Partitioning methods Hierarchical methods Model-based methods Density-based methods 46

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