Download Algorithms for Fuzzy Clustering: Methods in c-Means by Sadaaki Miyamoto PDF

By Sadaaki Miyamoto

The major topic of this booklet is the bushy c-means proposed by means of Dunn and Bezdek and their adaptations together with fresh stories. a first-rate this is why we be aware of fuzzy c-means is that the majority technique and alertness reviews in fuzzy clustering use fuzzy c-means, and consequently fuzzy c-means may be thought of to be a massive means of clustering as a rule, regardless no matter if one is attracted to fuzzy equipment or now not. not like such a lot reviews in fuzzy c-means, what we emphasize during this publication is a relatives of algorithms utilizing entropy or entropy-regularized equipment that are much less recognized, yet we contemplate the entropy-based solution to be one other beneficial approach to fuzzy c-means. all through this e-book considered one of our intentions is to discover theoretical and methodological transformations among the Dunn and Bezdek conventional process and the entropy-based approach. We do word declare that the entropy-based approach is healthier than the normal process, yet we think that the tools of fuzzy c-means develop into complete via including the entropy-based approach to the tactic through Dunn and Bezdek, considering that we will realize natures of the either equipment extra deeply via contrasting those two.

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

Example text

Such misclassifications arise in many real applications. Therefore a new method is necessary to overcome such a difficulty. A natural idea is to use an additional variable that controls cluster sizes, in other words, cluster volumes. We consider the following two objective functions for this purpose [68, 111]. 15) N (αi )1−m (uki )m D(xk , vi ). 8 Fig. 2. Output from FCM The variable A = (α1 , . . , αc ) controls cluster sizes. The constraint for A is c A = { A = (α1 , . . , αc ) : αj = 1 ; αi ≥ 0, 1 ≤ i ≤ c }.

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) ). 61), this iterative calculation procedure ¯ , V¯ ) is is a variation of fixed point iteration. 62). Note that uki ∈ [0, 1] and vi is in the convex hull with the vertices of X, the solution is in a rectangle of Rc×N +c×p . 62) has a fixed point from Brouwer’s theorem (cf. 1). 2). 62) is not necessarily convergent. However, experiences in many simulations and real applications tell us that such algorithms as described above actually converge, and even in worst cases when we observe divergent solutions we can change parameters and initial values and try again.

2. Observe that Upose (x; vi ) approaches zero more rapidly than Upos (x; vi ) when x → ∞, since it is easy to see that Upos (x; vi ) → ∞, Upose (x; vi ) as x → ∞ from the Taylor expansion of the exponential function. 11) i=1 k=1 where we should note the obvious constraint uki ≥ 0. The same technique to put 2 wki = uki is useful and we derive the solution u ¯ki = 1− 0 D(xk ,¯ vi ) ν ( D(xνk ,¯vi ) < 1), (otherwise). 12) N with the same formula for the cluster centers: v¯i = N u¯ki xk k=1 u ¯ki .

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