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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.
 Fall '11
 Dr.Robin

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