By Sadaaki Miyamoto
The major topic of this ebook is the bushy c-means proposed by means of Dunn and Bezdek and their diversifications together with contemporary experiences. a prime this is why we pay attention to fuzzy c-means is that almost all technique and alertness reports in fuzzy clustering use fuzzy c-means, and for this reason fuzzy c-means will be thought of to be an important means of clustering quite often, regardless even if one is attracted to fuzzy equipment or no longer. not like such a lot reports in fuzzy c-means, what we emphasize during this ebook is a relatives of algorithms utilizing entropy or entropy-regularized tools that are much less identified, yet we think of the entropy-based technique to be one other beneficial approach to fuzzy c-means. all through this ebook one in every of our intentions is to discover theoretical and methodological variations among the Dunn and Bezdek conventional process and the entropy-based technique. We do notice declare that the entropy-based procedure is best than the normal technique, yet we think that the tools of fuzzy c-means turn into complete by means of including the entropy-based technique to the strategy by means of Dunn and Bezdek, due to the fact we will realize natures of the either equipment extra deeply through contrasting those two.
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Additional info for Algorithms for Fuzzy Clustering: Methods in c-Means Clustering with Applications
We brieﬂy (and rather roughly) note a general method of ﬁxed point iteration. Let S be a compact subset of Rp and T be a mapping deﬁned on S into S (T : S → S). An element x ∈ S is said to be a ﬁxed point of T if and only if T (x) = x. 1 below). Suppose that we start from an initial value x(1) and iterate x(n+1) = T (x(n) ), n = 1, 2, . . 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 ).
FC4. [Test convergence:] If U End FC. Thus FC(J0 , Ub ) is employed for crisp c-means; FC(Jfcm , Uf ) for fuzzy c-means. We consider if there are other ways to fuzzify the crisp c-means. As the standard method by Dunn and Bezdek introduces nonlinearity (uki )m , we should consider the use of another type of nonlinearity. The method of Dunn and Bezdek has another feature: it smoothes the crisp solution into a diﬀerentiable one. Moreover the fuzzy solution approximates the crisp one in the sense that the fuzzy solution converges to the crisp solution as m → 1.
After that we show the solutions for multivariate normal distributions. For the univariate normal distributions, 2 (x−μi ) 1 − pi (x|φi ) = √ e 2σi 2 , 2πσi i = 1, . . , m where φi = (μi , σi ). For the optimal solution we should minimize m J= N 2 (x −μ ) 1 − k2σ 2i i ψik log √ e . 2πσi i=1 k=1 From ∂J =− ∂μi we have μi = 1 Ψi N ψik k=1 xk − μi = 0, σi2 N ψik xk , i = 1, . . , m. 89) k=1 In the same manner, from ∂J = ∂σi N ψik k=1 (xk − μi )2 − σi3 N ψik k=1 1 = 0, σi We have σi2 = 1 Ψi N ψik (xk − μi )2 = k=1 1 Ψi N ψik x2k − μ2i , i = 1, .