Fuzzy rule base simplification using multidimensional scaling and constrained optimization

This paper proposes a novel approach for the development of highly accurate and interpretable fuzzy models with Gaussian fuzzy sets, under the framework of multidimensional scaling and non-linear constrained optimization. Upon the assumption that an accurate initial fuzzy model has already been designed, we introduce an effective methodology to approximate the similarity measure between fuzzy sets. The resulting similarity degrees guide the quantification of the dissimilarity degrees between rule antecedents. We, then, put the multidimensional scaling in place in order to transform the rule antecedents into points in a low dimensional Euclidean space. The elaboration on the distribution of these points is carried out by means of an objective-function based fuzzy clustering using a cluster validity index. Rules that correspond to points belonging to the same cluster are similar and therefore, are unified through a specialized merging process. With respect to each dimension, the aforementioned merging process acts to create a topology of fuzzy sets that enables us to elicit interpretability constraints, which are used to minimize the model's performance index in terms of non-linear constrained optimization. The established fuzzy model appears to possess a simple and transparent (i.e. interpretable) structure while maintaining a highly accurate behavior. The overall method is rigorously tested and evaluated through a number of simulation experiments that involve low and high dimensional data sets.