By János Abonyi, Balázs Feil

The objective of this e-book is to demonstrate that complex fuzzy clustering algorithms can be utilized not just for partitioning of the information. it could actually even be used for visualisation, regression, class and time-series research, consequently fuzzy cluster research is an efficient method of clear up complicated info mining and method identity difficulties. This publication is orientated to undergraduate and postgraduate and is definitely fitted to instructing purposes.

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A specific situation arises when the regression functions fi are linear in the parameters θi , fi (xk ; θi ) = xTi,k θ i , where xi,k is a known arbitrary function of xk . In this case, the parameters can be obtained as a solution of a set of weighted least-squares problem where the membership degrees of the fuzzy partition matrix U serve as the weights. 36 Chapter 1. 1 (Fuzzy c-Regression Models). • Initialization Given a set of data Z = {(x1 , y1 ), . . 56). Choose a weighting exponent m > 1 and a termination tolerance ǫ > 0.

0 0 .. ··· µi,N ⎤ ⎥ ⎥ ⎥. 59) In the original paper that has introduced the fuzzy c-regression models [113], six issues were discussed for possible future research. Two of them were the following: • Determining the number of clusters (the number of regression models). • Avoiding local trap states. The next section gives solutions to these problems by incorporating prior knowledge into the clustering procedure presented in this section. Constrained Prototype based FCRM This section deals with prototypes linear in the parameters (see also [1, 7]).

Some of them attempt to select a good initial partition so that the algorithm is more likely to find the global minimum value. A problem accompanying the use of a partitional algorithm is the choice of the number of desired output clusters. A seminal paper [76] provides guidance on this key design decision. The partitional techniques usually produce clusters by optimizing a criterion function defined either locally (on a subset of the patterns) or globally (defined over all of the patterns). Combinatorial search of the set of possible labelings for an optimum value of a criterion is clearly computationally prohibitive.