### Table 1 Four validation functionals for the fuzzy c-mean Validity index Functional description Optimal cluster number

in Abstract

1997

"... In PAGE 3: ... and the uniform data functional Windham, 1982 . Table1 lists a number of cluster validation in- dices, which are evaluated in our study. The func- tional partition coefficient V and the partition en- PC tropy V use only the membership values u of a PE ik fuzzy partition of data set X.... ..."

### Table 1: Cluster validity measures: fuzzy hypervolume

1996

"... In PAGE 4: ... The cluster validity measures are presented in Table 1. It is evident from the Table1 that the best per- formance is achieved for segmentation of image into 7 clusters where the partition density shows a clear max- imum in all three cases. The second method has shown to be the fastest of the three tested methods.... ..."

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### Table 1 A brief summary of four selected validity functions Validity function Functional description

2004

"... In PAGE 4: ... However, if we consider all fuzzy partitions (including of the crisp ones) for a data set, the good partition should meet that (A) the objective function converges, and (B) its fuzziness is as small as possible. Table1 is a brief summary of 4 selected cluster validity functions which will be used in Section 4 to evaluate the performance of FCM clustering. It is noted that in some cases these validity functions cannot get their optimal values simultaneously.... ..."

### Table 1: Cluster validity measures: fuzzy hypervolume (fhv[103]) and partition density (pd[10?3]).

1996

"... In PAGE 3: ... The cluster validity measures are presented in Table 1. It is evident from the Table1 that the best perform- ance is achieved for segmentation of image into 7 clusters where the partition density shows a clear maximum in all three cases. The second method has shown to be the fastest of the three tested methods.... ..."

Cited by 3

### Table 1: Cluster validity measures: fuzzy hypervolume (fhv[103]) and partition density (pd[10?3]).

"... In PAGE 4: ... The cluster validity measures are presented in Table 1. It is evident from the Table1 that the best per- formance is achieved for segmentation of image into 7 clusters where the partition density shows a clear max- imum in all three cases. The second method has shown to be the fastest of the three tested methods.... ..."

### Table 2.1: Cluster validity measures: fuzzy hypervolume (fhv) and partition density (pd). to compare the proposed techniques with respect to convergence speed and validity meas- ures. Experimental results demonstrate that the proposed techniques for new cluster center selection are useful for application in unsupervised fuzzy K-means clustering algorithms.

### Table 1: Cluster validitymeasures: fuzzy hypervolume

1996

"... In PAGE 3: ...The cluster validity measures are presented in Table 1. It is evident from the Table1 thatthe best perform- ance is achieved for segmentation of image into 7 clusters where the partition densityshows a clear maximumin all three cases. The secondmethod has shown tobe the fastest of thethree tested methods.... ..."

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### Table 1 Nine validity indices for the Fuzzy Clustering. Validity index Functional description

2006

"... In PAGE 6: ...of comparison, we have chosen the normalized invariant criterion VN INV . This index measures the trace of the product matrix between the inverse of the within-cluster scatter matrix and the between-cluster scatter matrix normalized by the number of clusters c2, as presented in Table1 . The cluster number that maximizes VN INV is considered to be the optimal value for the number of clusters present in the data.... In PAGE 6: ... The cluster number that maximizes VAPD is considered to be the optimal value for the number of clusters present in the data. Table1 lists all of these validity indices. 3.... ..."

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### Table 1 Fuzzy rule bases

2006

"... In PAGE 3: ... The rules reflect an initial strategy for combining the different forecast values that has been suggested by a user. For example, if the same level of trust is given to the customer forecast and expert forecast, the rules in Rule Base 1 can have the form as given in Table1 (a). Rule Base 2 and Rule Base 3 are defined in a similar way (see Table 1 (a) and (b), respectively).... In PAGE 3: ... For example, if the same level of trust is given to the customer forecast and expert forecast, the rules in Rule Base 1 can have the form as given in Table 1 (a). Rule Base 2 and Rule Base 3 are defined in a similar way (see Table1 (a) and (b), respectively). However, the proposed DSS_DF includes a learning mechanism that modifies and improves the initial rule bases... ..."

### Table 2. Prototypes Easy, Medium and Difficult to understand - Formal Representation of conceptual prototypes. The prototypes have been represented as fuzzy numbers, which are going to allow us to obtain a degree of membership in the concept. For the sake of simplicity in the model, they have been represented by triangular fuzzy numbers. Therefore, in order to construct the prototypes (triangular fuzzy numbers) we only need to know their centrepoints ( centre of the prototype ), which are obtained by normalising and aggregating the metric values corresponding to the class diagrams of each of the prototypes (see figure 4).

2002

"... In PAGE 5: ... The selected algorithm for data mining process was summarise functions (calculating factors such as medium, minimum and maximum time spent for maintaining each diagram, and finding for each one the average values). Table2 shows the parametric definition of the prototypes. These parameters will be modified taking into account the degree of affinity of a new class diagram with the prototypes.... ..."

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