Learning consistent, complete and compact sets of fuzzy rules in conjunctive normal form for regression problems

Abstract

When a flexible fuzzy rule structure such as those with antecedent in conjunctive normal form is used, the interpretability of the obtained fuzzy model is significantly improved. However, some important problems appear related to the interaction among this set of rules. Indeed, it is relatively easy to get inconsistencies, lack of completeness, redundancies, etc. Generally, these properties are ignored or mildly faced. This paper, however, focuses on the design of a multiobjective genetic algorithm that properly considers all these properties thus ensuring an effective search space exploration and generation of highly legible and accurate fuzzy models.

Appendix: Wang–Mendel algorithm

The ad hoc data-driven Mamdani-type fuzzy rule set generation process proposed by Wang and Mendel(1992) is widely known and used because of its simplicity. In our algorithm, Pitts-DNF, it is used in the initialization process and completeness operator. Therefore, for the sake of readability we briefly introduce the algorithm in this appendix.

It is based on working with an input-output data pair set representing the behavior of the problem being solved:

$$ E = \{ e_1, \ldots, e_N \},\ e_l=(x^l_1,\ldots,x^l_n,y^l_1,\ldots,y^l_m), $$

with N being the data set size, n the number of input variables, and m the number of output variables. The algorithm consists of the following steps:

• 1. Consider a fuzzy partition (definitions of the membership functions parameters) for each input/output variable.

• 2. Generate a candidate fuzzy rule set: This set is formed by the rule best covering each example contained in E. Thus, N candidate fuzzy rules, CR ^l, are obtained. The structure of each rule is generated by taking a specific example, i.e., an (n + m)-dimensional real vector, and setting each one of the variables to the linguistic term (associated fuzzy set) best covering every vector component:

  $$ \begin{aligned} CR^l: &\hbox{IF }X_1\hbox{ is }A_1^l\hbox{ and }\ldots\hbox{ and }X_n\hbox{ is }A_n^l\\ &\hbox{THEN }Y_1\hbox{ is }B^l_1\hbox{ and }\ldots\hbox{ and }Y_m\hbox{ is }B^l_m \end{aligned} $$
  $$ A_i^l = \hbox{arg }\mathop {\max }\limits_{A^{\prime} \in {\mathbf{A}}_i } \mu _{A^{\prime}} (x_i^l ),\quad B^l_j = \hbox{arg } \mathop {\max }\limits_{B^{\prime}\in {\mathbf{B}}_{\mathbf j}} \mu _{B^{\prime}} (y^l_j ) $$

• 3. Give an importance degree to each candidate rule:

  $$ D(CR^l) = \prod_{i=1}^n{\mu_{A^l_i}(x^l_i)} \cdot \prod_{j=1}^{m}{\mu_{B^l_j}(y^l_j)} $$

• 4. Obtain a final fuzzy rule set from the candidate fuzzy rule set: To do so, the N candidate rules are first grouped in g different groups, each one of them composed of all the candidate rules containing the same antecedent combination. To build the final fuzzy rule set, the rule with the highest importance degree is chosen in each group. Hence, g will be both the number of different antecedent combinations in the candidate rule set and the number of rules in the Mamdani-type fuzzy rule set finally generated.