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Unformatted text preview: 1 Clustering Distance Measures Hierarchical Clustering k Means Algorithms 2 The Problem of Clustering Given a set of points, with a notion of distance between points, group the points into some number of clusters , so that members of a cluster are in some sense as close to each other as possible. 3 Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x 4 Problems With Clustering Clustering in two dimensions looks easy. Clustering small amounts of data looks easy. And in most cases, looks are not deceiving. 5 The Curse of Dimensionality Many applications involve not 2, but 10 or 10,000 dimensions. Highdimensional spaces look different: almost all pairs of points are at about the same distance. Assuming random points within a bounding box, e.g., values between 0 and 1 in each dimension. 6 Example : SkyCat A catalog of 2 billion sky objects represented objects by their radiation in 9 dimensions (frequency bands). Problem : cluster into similar objects, e.g., galaxies, nearby stars, quasars, etc. Sloan Sky Survey is a newer, better version. 7 Example : Clustering CDs (Collaborative Filtering) Intuitively: music divides into categories, and customers prefer a few categories. But what are categories really? Represent a CD by the customers who bought it. Similar CDs have similar sets of customers, and viceversa. 8 The Space of CDs Think of a space with one dimension for each customer. Values in a dimension may be 0 or 1 only. A CDs point in this space is ( x 1 , x 2 ,, x k ), where x i = 1 iff the i th customer bought the CD. Compare with the correlated items matrix: rows = customers; cols. = CDs. 9 Example : Clustering Documents Represent a document by a vector ( x 1 , x 2 ,, x k ), where x i = 1 iff the i th word (in some order) appears in the document. It actually doesnt matter if k is infinite; i.e., we dont limit the set of words. Documents with similar sets of words may be about the same topic. 10 Example : Protein Sequences Objects are sequences of {C,A,T,G}. Distance between sequences is edit distance , the minimum number of inserts and deletes needed to turn one into the other. Note there is a distance, but no convenient space in which points live. 11 Distance Measures Each clustering problem is based on some kind of distance between points....
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This document was uploaded on 01/25/2012.
 Spring '09
 Algorithms

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