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clustering - Clustering Algorithms Applications...

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1 Clustering Algorithms Applications Hierarchical Clustering k  -Means Algorithms CURE Algorithm
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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.
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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 Clustering in two dimensions looks  easy. Clustering small amounts of data looks  easy. And in most cases, looks are  not   deceiving.
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5 The Curse of Dimensionality Many applications involve not 2, but 10  or 10,000 dimensions. High-dimensional spaces look different:  almost all pairs of points are at about  the same distance.
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6 Example : Curse of Dimensionality Assume random points within a  bounding box, e.g., values between 0  and 1 in each dimension. In 2 dimensions: a variety of distances  between 0 and 1.41. In 10,000 dimensions, the difference in  any one dimension is distributed as a  triangle.
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7 Example  – Continued The law of large numbers applies. Actual distance between two random  points is the sqrt of the sum of squares  of essentially the same set of  differences.
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8 Example  High-Dimension  Application: SkyCat A catalog of 2 billion “sky objects”  represents objects by their radiation in 7  dimensions (frequency bands). Problem : cluster into similar objects,  e.g., galaxies, nearby stars, quasars,  etc. Sloan Sky Survey is a newer, better  version.
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9 Example : Clustering CD’s  (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 CD’s have similar sets of  customers, and vice-versa.
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10 The Space of CD’s Think of a space with one dimension for  each customer. Values in a dimension may be 0 or 1 only. A CD’s point in this space is             ( x 1 x 2 ,…,  x k ), where  x i  = 1 iff the  th   customer bought the CD. Compare with boolean matrix: rows =  customers; cols. = CD’s.
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11 Space of CD’s – (2) For Amazon, the dimension count is  tens of millions. An alternative : use minhashing/LSH to  get Jaccard similarity between “close”  CD’s. 1 minus Jaccard similarity can serve as  a (non-Euclidean) distance.
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12 Example : Clustering Documents Represent a document by a vector    ( x 1 x 2 ,…,  x k ), where  x i  = 1 iff the  th  word (in  some order) appears in the document.
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