Abstract-In fuzzy rule-based models acquired from numerical data, redundancy may be present in the form of similar fuzzy sets that represent compatible concepts. This results in an unnecessarily complex and less transparent linguistic description of the system. By using a measure of similarity, a rule base simplification method is proposed that reduces the number of fuzzy sets in the model. Similar fuzzy sets are merged to create a common fuzzy set to replace them in the rule base. If the redundancy in the model is high, merging similar fuzzy sets might result in equal rules that also can be merged, thereby reducing the number of rules as well. The simplified rule base is computationally more efficient and linguistically more tractable. The approach has been successfully applied to fuzzy models of real world systems.

Abstract-The use of genetic algorithms (GAs) and other evolutionary optimization methods to design fuzzy rules for systems modeling and data classification have received much attention in recent literature. Authors have focused on various aspects of these randomized techniques, and a whole scale of algorithms have been proposed. We comment on some recent work and describe a new and efficient two-step approach that leads to good results for function approximation, dynamic systems modeling and data classification problems. First fuzzy clustering is applied to obtain a compact initial rule-based model. Then this model is optimized by a real-coded GA subjected to constraints that maintain the semantic properties of the rules. We consider four examples from the literature: a synthetic nonlinear dynamic systems model, the iris data classification problem, the wine data classification problem, and the dynamic modeling of a diesel engine turbocharger. The obtained results are compared to other recently proposed methods.

Abstract-In our previous work we showed that genetic algorithms (GAs) provide a powerful tool to increase the accuracy of fuzzy models for both systems modeling and classification. In addition to these results, we explore the GA to find redundancy in the fuzzy model for the purpose of model reduction. An aggregated similarity measure is applied to search for redundancy in the rule base description. As a result, we propose an iterative fuzzy identification technique starting with data-based fuzzy clustering with an overestimated number of local models. The GA is then applied to find redundancy among the local models with a criterion based on maximal accuracy and maximal set similarity. After the reduction steps, the GA is applied with another criterion searching for minimal set similarity and maximal accuracy. This results in an automatic identification scheme with fuzzy clustering, rule base simplification and constrained genetic optimization with low-human intervention. The proposed modeling approach is then demonstrated for a system identification and a classification problem. Results are compared to other approaches in the literature. Attractive models with respect to compactness, transparency and accuracy, are the result of this symbiosis.Index Terms-Fuzzy classifier, genetic algorithm (GA), Iris data, rule base reduction, Takagi-Sugeno (T-S) fuzzy model, transparency and accuracy.

Abstract-This article is a reaction to recent publications on rulebased modeling using fuzzy set theory and fuzzy logic. The interest in fuzzy systems has recently shifted from the seminal ideas about complexity reduction toward data-driven construction of fuzzy systems. Many algorithms have been introduced that aim at numerical approximation of functions by rules, but pay little attention to the interpretability of the resulting rule base. We show that fuzzy rule-based models acquired from measurements can be both accurate and transparent by using a low number of rules. The rules are generated by product-space clustering and describe the system in terms of the characteristic local behavior of the system in regions identified by the clustering algorithm. The fuzzy transition between rules makes it possible to achieve precision along with a good qualitative description in linguistic terms. The latter is useful for expert evaluation, rule-base maintenance, operator training, control systems design, user interfacing, etc. We demonstrate the approach on a modeling problem from a recently published article.

Abstract-Two extensions to the objective function-based fuzzy clustering are proposed. First, the (point) prototypes are extended to hypervolumes, whose size can be fixed or can be determined automatically from the data being clustered. It is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function. Second, a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization. Starting with an overestimation of the number of clusters in the data, similar clusters are merged in order to obtain a suitable partitioning. An adaptive threshold for merging is proposed. The extensions proposed are applied to Gustafson-Kessel and fuzzy c-means algorithms, and the resulting extended algorithm is given. The properties of the new algorithm are illustrated by various examples.