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class_10_31 - Statistical Data Mining ORIE 474 Fall 2007...

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Statistical Data Mining ORIE 474 Fall 2007 Tatiyana Apanasovich 10/31/06 Cluster Analysis
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Cluster Analysis (CA): Building Blocks Distance measures We need to specify what we mean by certain points being “closer to” each other Match the CA method to the objective Objectives can be characterized by the kind of cluster or cluster structure ( shape ) we try to detect Ex: Cluster = set of points s.t. the maximum distance between any 2 points in the cluster is minimal Cluster = set of points s.t. each point in the cluster is as close as possible to some other member of the cluster
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Types of Cluster Analysis Algorithms Algorithms can be based on Trying to find the optimal partition into a specified number of clusters Hierarchical methods to discover cluster structure Probabilistic models for underlying clusters
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Partition-Based Clustering Algorithms Task of this DM algorithm: Partition the data set into k disjoint set of points s.t. the points within each class are as homogeneous as possible Given a set of n data points D={x 1 ,…,x n }, the task is to find K clusters C ={C 1 ,…,C K } s.t. each data point x i is assigned to a unique cluster Homogeneity is captured by appropriate score functions
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Partition-Based Clustering (cont’d) Centroid = Average of the points in a cluster Often considered to be a representative point for that cluster In general, there is no explicit statement of what sort of shape the clusters sought should have We will take in detail about Score Functions Basic Algorithms
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A. Score Functions for Partition- Based Clustering Terminology: d(x,y): distance between x, y in D Assumption: d(.,.) is a metric on D Most score functions are build s.t. Clusters are compact Clusters are as far from each other as possible Formulation of these notions Within-cluster variation wc( C ) Between-cluster variation bc( C )
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Score Functions (cont’d) Cluster centers r k for each cluster C 1 ,…,C k Can be designated representative point x i in C k If taking means makes sense, we can use the centroid of the points in cluster C k , that is where n k is the number of data points in cluster C k = k i C x i k k x n r 1
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Within- and Between-Cluster Variation Within-cluster variation If d(x i ,r k ) equals the Euclidean distance, than wc( C ) is also called the within-cluster sum of squares Between-cluster variation Overall quality/score of a clustering C Monotone combination of wc( C ) and bc( C ), e.g. bc( C ) / wc( C ) ∑∑ = = = = K k C x k i K k k k i r x d C wc C wc 1 2 1 ) , ( ) ( ) ( 2 1 ) , ( ) ( < = K l k l k r r d C bw
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K-Means Algorithm for k=1,…,K, let r(k) be a randomly chosen point from D while changes in cluster C k happen do form clusters: for k=1,…,K do C k ={x in D | d(r k ,x)≤d(r j ,x)} for all j=1,…,K, j≠k} end compute new cluster centers: for k=1,…,K do end end = k C i x i x k n k r 1
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K-Means Clustering: Example Source: Wine recognition data from ML Repository at UCI The different colors indicate 3 different classes of wine (3 different cultivars)
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Ex: Wine Recognition Data Raw data
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