Differential Equations Solutions 50

Differential Equations Solutions 50 - 60 Chapter 11....

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Unformatted text preview: 60 Chapter 11. Solutions: Case Study: Classified Information 1.5 1 0.5 R(c) D(c) 0 −1.5 −1 −0.5 0 c 0.5 1 1.5 Figure 11.1. The functions R and D for a particular dataset. (c) We want to minimize the function n xi − c D(c) = 2 i=1 over all choices of c. Since there is only one center c, this function is convex and differentiable everywhere, and the solution must be a zero of the gradient. Differentiating with respect to c we obtain n 2(xi − c) = 0, i=1 so c= 1 n n xi . i=1 It is easy to verify that this is a minimizer, not a maximizer or a stationary point, so the solution is to choose c to be the centroid (mean value) of the data points. (d) As one data point moves from the others, eventually a center will “follow” it, giving it its own cluster. So the clustering algorithm will fit k − 1 clusters to the remaining n − 1 data points. CHALLENGE 11.3. A sample program is given on the website. Note that when using a general purpose optimization function, it is important to match the ...
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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