Granular fuzzy modeling with evolving hyperboxes in multi-dimensional space of numerical data

Clustering has been applied to numerous areas, including signal and image processing. Many approaches have been developed over the years to efficiently construct granular models on a basis of numerical experimental data. In this study, we propose a novel approach to construct a granular model that is fundamentally designed around information granules regarded as hyperboxes. Several studies have been focused on building a set of hyperboxes around data; one of them being a Min-Max Neural Network (NN) algorithm. Here we develop two different methods to construct these information granules, nevertheless some essential similarities to previous studies can be found. In particular, hyperboxes are constructed by using some reference data, and they are endowed with some parametric flexibility to facilitate controlling their size, whereas the construction of the hyperboxes involve elimination or reduction of possible overlaps between them. In the proposed approach, given a set of input and output data pairs, we construct interval-based information granules to partition the output space (viz. the space of the output variable). On a basis of these intervals, we carry out a so-called context-based Fuzzy C-Means algorithm to construct cluster centers (prototypes) in the multivariable input space. These prototypes serve as hyperbox cores. To construct the information granules, two methods are studied: one develops a family of hyperboxes by realizing some constrictions, while the other one engages Differential Evolution (DE) to realize further optimization. To reduce overlap, two methods are tested: one being previously proposed for the min-max NN and a new one, which engages DE to optimize the overlap reduction. Experimental studies involve synthetic data and publicly available real-world data. The results are compared with the outcomes produced by the algorithm proposed by Simpson. The performance of the method is quantified and it is demonstrated that the obtained results are substantially better when dealing with multi-dimensional data.