Examples

The FRITs were used for the approximation of the function

z=.

Fig. 1 Surface representing the relation between the inputs and outputs of the modeled phenomena

In course of the experimentation two fuzzy systems were created, each one applying a sparse rule base with six rules. The rules describe the minima and maxima of the output as well as the output corresponding to the four points bounding the input space.

The first system contains only trapezoidal shaped linguistic terms in the antecedent as well as in the consequent side. [Download system 1]

The second system contains trapezoidal shaped linguistic terms in the antecedent side and singleton shaped sets in the consequent side. [Download system 2]

In case of both systems two observations were used a triangle shaped ([Download observation 1]) and a singleton one ([Download observation 2]). The rule antecedents are the same in case of the two methods. Figures 2 and 3 present the antecedent space of the two systems and the observations. Figures 4 and 5 present the antecedent partitions of the two systems and the observations (drawn by bold lines).

Fig. 2 Antecedent universe and triangle shaped observation

Fig. 3 Antecedent universe and singleton observation

Fig. 4 Antecedent partitions and triangle shaped observation

Fig. 5 Antecedent partitions and singleton shaped observation

The experiments were done by the help of a script [Download script] and the toolbox.

Experimenting the first system (trapezoidal shaped consequent linguistic terms)

KH


Fig. 6 Triangle shaped observation	Fig. 7 Singleton observation
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Fig. 8	Fig. 9

KH stabilized


Fig. 10 Triangle shaped observation	Fig. 11 Singleton observation

MACI


Fig. 12 Triangle shaped observation	Fig. 13 Singleton observation

IMUL


Fig. 14 Triangle shaped observation	Fig. 15 Singleton observation

CRF


Fig. 16 Triangle shaped observation	Fig. 17 Singleton observation

FIVE


Fig. 18 Triangle shaped observation	Fig. 19 Singleton observation

VKK


Fig. 20 Triangle shaped observation	Fig. 21 Singleton observation

GM with SCM+FPL


Fig. 22 Triangle shaped observation	Fig. 23 Singleton observation

FRIPOC


Fig. 24 Triangle shaped observation	Fig. 25 Singleton observation

LESFRI


Fig. 26 Triangle shaped observation	Fig. 27 Singleton observation

VEIN


Fig. 28 Triangle shaped observation	Fig. 29 Singleton observation

Experimenting the second system (singleton shaped consequent linguistic terms)

KH


Fig. 30 Triangle shaped observation	Fig. 31 Singleton observation