Generating Complete, Distinguishable, Consistent and Compact
Fuzzy Systems Using Evolutionary Algorithms
Completeness of fuzzy systems consists of two main factors: completeness
of fuzzy partitions and completeness of fuzzy rule structure. Example of
complete/incomplete fuzzy partion, as well as complete/incomplete fuzzy rule
structure are shown in Fig.1 and Fig. 2.
Distinguishability is one of the most important aspect for interpretability
of fuzzy systems. It is possible to assign a linguistic term to a fuzzy
subset only of this condition is satisfied. Fig. 2 shows a distinguishable and
an indistinguishable fuzzy partiton.
The completeness and distinguishability of a fuzzy partition can be
guaranteed by imposing a constraint on the similarity between two
neighbouring fuzzy subsets.
s1 < S(A,B) < s2,
where 0 < s1 < s2 < 1, S(A,B) is a fuzzy similarity
measure. Notice that S(A,B) =0 means that the two fuzzy subsets do not overlap
and S(A,B) = 1 that the two fuzzy subsets are the same. Both situation
should be avoided. In fact, S(A,B) should be siginificantly
less than 1 so that A is distinguishable.
Consistency of Fuzzy Rules
Fuzzy rules should be consisteny with each other and consistent with human
heuristics (a priori knowledge). Intuitively, if the condition part of
fuzzy rules are similar, then the consequent part of the two rules should
also be similar. A consistency definition based on this heuristics has been
suggested in .
Compactness of fuzzy systems includes two aspects: a small number of
conditions in the rule premise and a small number of fuzzy rules in the rule
A method considering these interpretability conditions in addition to
approximation performance has been proposed
in  using evolution strategies to generate interpretable
fuzzy rules from data.
 Y. Jin, W. von Seelen and B. Sendhoff. On generating flexible, complete,
consistens and compact (FC3) fuzzy rules from data using evolution strategies.
IEEE Transactions on Systems, Man, and Cybernetics, 29(4):829-845, 1999
For discussions, please contact Yaochu Jin.