By Junjie Wu

Nearly we all know K-means set of rules within the fields of information mining and enterprise intelligence. however the ever-emerging information with tremendous advanced features carry new demanding situations to this "old" set of rules. This e-book addresses those demanding situations and makes novel contributions in developing theoretical frameworks for K-means distances and K-means dependent consensus clustering, deciding on the "dangerous" uniform impression and zero-value predicament of K-means, adapting correct measures for cluster validity, and integrating K-means with SVMs for infrequent category research. This booklet not just enriches the clustering and optimization theories, but in addition offers stable information for the sensible use of K-means, specifically for very important initiatives comparable to community intrusion detection and credits fraud prediction. The thesis on which this publication is predicated has gained the "2010 nationwide very good Doctoral Dissertation Award", the top honor for no more than a hundred PhD theses in line with yr in China.

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42 3 Generalizing Distance Functions for Fuzzy c-Means Clustering Let the solution set = {(U ∗ , V ∗ ) ∈ M f c ×Rcd |Jm (U ∗ , V ∗ ) ≤ Jm (U, V ∗ ) ∀ U ∈ M f c , and Jm (U ∗ , V ∗ ) < Jm (U ∗ , V ) ∀ V = V ∗ }. 3 ([16]) Given the fuzzy c-means problem defined by Eqs. 2), let (U (0) , G(U (0) )) be the starting point of iteration with Tm , where U (0) ∈ M f c . ∞ either terminates at a point in , or Then the iteration sequence ((U (l) , V (l) ))l=0 there is a subsequence converging to a point in .

Extensive experiments on a number of real-world data sets clearly illustrate the uniform effect of K-means and the biased effect of the entropy measure, via the help of the Coefficient of Variation statistic. Most importantly, we unveil the danger induced by the combined use of K-means and the entropy measure. That is, many true clusters will become unidentifiable when applying K-means for highly imbalanced data, but this situation is often disguised by the low values of the entropy measure. References 1.

12), we can know that Eq. 13) is true. Discussion. By Eq. 12), we know that the minimization of the K-means objective function Fk is equivalent to the maximization of the distance function FD(k) , where (k) both Dk and n are constants for a given data set. For FD in Eq. e. e. n 1 = n 2 = · · · = n k = n/k. Note that we have isolated the effect of two components: m i − m j 2 and n i n j here to simplify the discussion. For real-world data sets, however, these two components are interactive. 3 The Relationship between K-means Clustering and the Entropy Measure In this section, we study the relationship between K-means clustering and a widely used clustering validation measure: Entropy (E).