By Sadaaki Miyamoto

The major topic of this ebook is the bushy *c*-means proposed through Dunn and Bezdek and their diversifications together with contemporary stories. a prime for the reason that we pay attention to fuzzy *c*-means is that almost all method and alertness reviews in fuzzy clustering use fuzzy *c*-means, and therefore fuzzy *c*-means might be thought of to be a tremendous means of clustering regularly, regardless no matter if one is attracted to fuzzy tools or now not. not like so much reports in fuzzy *c*-means, what we emphasize during this ebook is a kinfolk of algorithms utilizing entropy or entropy-regularized equipment that are much less identified, yet we examine the entropy-based strategy to be one other important approach to fuzzy *c*-means. all through this e-book one in every of our intentions is to discover theoretical and methodological adjustments among the Dunn and Bezdek conventional technique and the entropy-based technique. We do word declare that the entropy-based approach is healthier than the conventional procedure, yet we think that the tools of fuzzy *c*-means develop into *complete* by way of including the entropy-based solution to the strategy via Dunn and Bezdek, because we will be able to become aware of natures of the either tools extra deeply by way of contrasting those two.

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**Additional resources for Algorithms for Fuzzy Clustering: Methods in c-Means Clustering with Applications**

**Example text**

FLC5. Update m1 (t), . . 69) for l = 1, 2, . . , c. End FLC. 69), H : [0, 1] → [0, 1] is either linear or a sigmoid function such that H(0) = 0 and H(1) = 1. There are variations of FLC. For example, the step FLC5 can be replaced by FLC5’: Let = arg max ukj . Then 1≤j≤c m (t + 1) = m (t) + α(t)H(uk )[x(t) − m (t)], mi (t + 1) = mi (t), i = . As many other variations of the competitive learning algorithms have been proposed, the corresponding fuzzy learning methods can be derived without diﬃculty.

The formulation by the calculus of variations in this section thus justiﬁes the use of these functions in the fuzzy learning and the ﬁxed point iterations. 12 Mixture Density Model and the EM Algorithm Although this book mainly discusses fuzzy clustering, a statistical model is closely related to the methods of fuzzy c-means. In this section we overview the mixture density model that is frequently employed for both supervised and unsupervised classiﬁcation [25, 98, 131]. For this purpose we use terms in probability and statistics in this section.

61) When a ﬁxed point x ˜ exists and we expect the iterative solution converges to the ﬁxed point, the iterative calculation is called ﬁxed point iteration. 24) respectively by ¯ , V ). When the number of iterations is represented by ¯ = T1 (U, V¯ ), V¯ = T2 (U U Heuristic Algorithms of Fixed Point Iterations 31 n = 1, 2, . . and the solution of the n-th iteration is expressed as U (n) and V (n) , then the above form is rewritten as U (n+1) = T1 (U (n) , V (n) ), V (n+1) = T2 (U (n+1) , V (n) ).