Generating Complete, Distinguishable, Consistent and Compact
Fuzzy Systems Using Evolutionary Algorithms

Completeness

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.

Fig. 1

Distinguishability

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.

Fig. 2

The completeness and distinguishability of a fuzzy partition can be
guaranteed by imposing a constraint on the similarity between two
neighbouring fuzzy subsets.

s_{1} < S(A,B) < s_{2},

where 0 < s_{1} < s_{2} < 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
two
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 [1].
#### Compactness

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
base.

A method considering these interpretability conditions in addition to
approximation performance has been proposed
in [1] using evolution strategies to generate interpretable
fuzzy rules from data.
#### References

[1] Y. Jin, W. von Seelen and B. Sendhoff. On generating flexible, complete,
consistens and compact (FC^{3}) 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.