By Sadaaki Miyamoto

The major topic of this publication is the bushy c-means proposed through Dunn and Bezdek and their diversifications together with fresh stories. a first-rate this is why we be aware of fuzzy c-means is that almost all method and alertness experiences in fuzzy clustering use fuzzy c-means, and for that reason fuzzy c-means will be thought of to be a tremendous means of clustering mostly, regardless even if one is attracted to fuzzy equipment or no longer. in contrast to so much reviews in fuzzy c-means, what we emphasize during this booklet is a family members of algorithms utilizing entropy or entropy-regularized equipment that are much less recognized, yet we think about the entropy-based option to be one other beneficial approach to fuzzy c-means. all through this publication certainly one of our intentions is to discover theoretical and methodological transformations among the Dunn and Bezdek conventional approach and the entropy-based process. We do word declare that the entropy-based process is best than the conventional technique, yet we think that the tools of fuzzy c-means turn into complete by means of including the entropy-based option to the tactic through Dunn and Bezdek, seeing that we will become aware of natures of the either tools extra deeply via contrasting those two.

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Represented by i wki We hence define the Lagrangian N L= k=1 N c 1 2 wki Dki + ν 2 i=1 N c c 4 wki − k=1 i=1 2 wki − 1) μk ( i=1 k=1 where Dki = D(xk , vi ). From 1 ∂L 2 = wki (Dki + νwki − μk ) = 0, 2 ∂wki 2 we have wki = 0 or wki = ν −1 (μk − Dki ). Using uki , uki = 0 or uki = ν −1 (μk − Dki ). 43) Notice that uki = ν −1 (μk − Dki ) ≥ 0. The above solution has been derived from the necessary condition for optimality. 43). Let us simplify the problem in order to find the optimal solution. 44) i=1 N J (k) and each J (k) can independently be minimized from other Then, Jqfc = k=1 J (k ) (k = k).

Pi ) are vectors, and Σi = (σij ) (1 ≤ j, ≤ p) is the covariance matrix; |Σi | is the determinant of Σi . 88), while the solutions for μi and Σi are as follows [131]. μi = Σi = 1 Ψi 1 Ψi N ψik xk , i = 1, . . 91) k=1 N ψik (xk − μi )(xk − μi ) , i = 1, . . , m. 92). Readers who are uninterested in mathematical details may skip the proof. Let jth component of vector μi be μji or (μi )j , and (i, ) component of matrix Σi be σij or (Σi )j . A matrix of which (i, j) component is f ij is denoted by [f ij ].

3. Assume that an estimate Φ for Φ is given. 85) where E(log f |x, Φ ) is the conditional expectation given x and Φ . Let us assume that k(y|x, Φ ) is the conditional probability function of y given x and Φ . It then follows that Q(Φ|Φ ) = k(y|x, Φ ) log f (y|Φ). 86) y∈χ−1 (x) We are now ready to describe the EM algorithm. The EM algorithm (O) Set an initial estimate Φ(0) for Φ. Let (M) until convergence. (E) (Expectation Step) Calculate Q(Φ|Φ( ) ). (M) (Maximization Step) Find the maximizing solution = 0.

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