cs345-cl - Clustering Distance Measures Hierarchical...

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1 Clustering Distance Measures Hierarchical Clustering k -Means Algorithms
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2 The Problem of Clustering rhombus6 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.
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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
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4 Problems With Clustering rhombus6 Clustering in two dimensions looks easy. rhombus6 Clustering small amounts of data looks easy. rhombus6 And in most cases, looks are not deceiving.
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5 The Curse of Dimensionality rhombus6 Many applications involve not 2, but 10 or 10,000 dimensions. rhombus6 High-dimensional spaces look different: almost all pairs of points are at about the same distance. rhombus4 Assuming random points within a bounding box, e.g., values between 0 and 1 in each dimension.
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6 Example : SkyCat rhombus6 A catalog of 2 billion “sky objects” represented objects by their radiation in 9 dimensions (frequency bands). rhombus6 Problem : cluster into similar objects, e.g., galaxies, nearby stars, quasars, etc. rhombus6 Sloan Sky Survey is a newer, better version.
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7 Example : Clustering CD’s (Collaborative Filtering) rhombus6 Intuitively: music divides into categories, and customers prefer a few categories. rhombus4 But what are categories really? rhombus6 Represent a CD by the customers who bought it. rhombus6 Similar CD’s have similar sets of customers, and vice-versa.
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8 The Space of CD’s rhombus6 Think of a space with one dimension for each customer. rhombus4 Values in a dimension may be 0 or 1 only. rhombus6 A CD’s point in this space is ( x 1 , x 2 ,…, x k ), where x i = 1 iff the i th customer bought the CD. rhombus4 Compare with the “correlated items” matrix: rows = customers; cols. = CD’s.
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9 Example : Clustering Documents rhombus6 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. rhombus4 It actually doesn’t matter if k is infinite; i.e., we don’t limit the set of words. rhombus6 Documents with similar sets of words may be about the same topic.
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10 Example : Protein Sequences rhombus6 Objects are sequences of {C,A,T,G}. rhombus6 Distance between sequences is edit distance , the minimum number of inserts and deletes needed to turn one into the other. rhombus6 Note there is a “distance,” but no convenient space in which points “live.”
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11 Distance Measures rhombus6 Each clustering problem is based on some kind of “distance” between points. rhombus6 Two major classes of distance measure: 1. Euclidean 2. Non-Euclidean
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12 Euclidean Vs. Non-Euclidean rhombus6 A Euclidean space has some number of real-valued dimensions and “dense” points. rhombus4 There is a notion of “average” of two points.
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