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In this article, we advocate that a knowledge tidbit residing in the output space could be helpful in improving the performance (accuracy) of the fuzzy rule-based model. It states that if two outputs are far apart from each other , it is advisable to place their corresponding inputs in different clusters when forming subspaces of the input space . Considering this knowledge guidance mechanism, we propose two different methods to partition the input space. In the first method, input data are first partitioned with the use of the standard clustering algorithm, say fuzzy C-means; here, a constructed partition matrix is reflective of the structure present in the input space. Then, the knowledge tidbit is used to adjust the entries of the original partition matrix in such a way that those input data whose corresponding output data are far apart from each other are assigned with low values of proximity. In the second method, we propose two strategies to modify the distance between input data and a prototype (cluster center) identified in the input space. The crux of this method is that if there are many input data (which, in virtue of the knowledge tidbit, are regarded as being far-apart from the input data of interest) around a certain prototype, the distance between the input data of interest and this prototype should be penalized. Thus, the membership of these input data to the prototype is reduced. The comprehensive experimental studies carried out on both synthetic and publicly available data are used to examine the usefulness of the proposed methods.

In this article, we are concerned with a problem of aggregation of order-2 information granules, and fuzzy sets, in particular. When processing order-1 fuzzy sets, the structural information about the space over which fuzzy sets are defined is not taken into account at all. In contrast, the aggregation of order-2 fuzzy sets requires a careful attention that needs to be paid both to the closeness determined in the space of membership degrees and the collection of information granules over which such fuzzy sets are defined. We formulate an original optimization problem that simultaneously involves considerations of distances in the membership space (space of membership grades) and some measure of resemblance formed in the space of relationships of reference information granules. The gradient-based learning scheme is constructed. Some illustrative examples are included.

Big data with a large number of observations (samples) have posed genuine challenges for fuzzy clustering algorithms and fuzzy C-means (FCM), in particular. In this article, we propose an original algorithm referred to as a hyperplane division method to split the entire data set into disjoint subsets. By disjoint subsets, we mean that the data subspaces (parts of the entire data space), each of which is supported or spanned by the data points in the corresponding subset, do not overlap each other. The disjoint subsets turned out to be beneficial to the improvement of the quality of the clusters formed by the clustering algorithms. Moreover, considering that either a large number (say, thousands) or a small number (say, a few) of clusters may be pursued in the clustering task, we propose corresponding strategies (based on the hyperplane division method) to make clustering processes feasible, efficient, and effective. By validating the proposed strategies on both synthetic and publicly available data, we show their superiority (in terms of both efficiency and effectiveness) manifested in a visible way over the method of clustering the entire data and over some representative big data clustering methods.

Fuzzy models are regarded as numeric constructs and as such are optimized and evaluated at the numeric level. In this study, we depart from this commonly accepted position and propose a granular evaluation of fuzzy models and present an augmentation of fuzzy models by forming information granules around numeric values of the parameters and constructions of the models. The concepts and algorithms of granular fuzzy models are discussed in the setting of Takagi–Sugeno rule-based architectures. We show how different protocols of forming and allocating information granules lead to the improvement of the granular performance of the models. Different from the standard numeric performance measure of fuzzy models coming in the form of the root mean squared error index, two performance measures are introduced that are pertinent to granular constructs, namely coverage and specificity. Furthermore, we propose a global indicator implied by these two measures, called an area under the curve, being computed for the characteristics of the granular model expressed in the coverage-specificity coordinates. A series of experimental studies is reported, which offers a comprehensive overview of the introduced performance measure criteria as well as the underlying realization of the granular fuzzy models.

This study is concerned with a design of membership functions of fuzzy sets. The membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics. These two requirements are articulated through the principle of justifiable granularity. The parametric version of the principle is discussed in detail. We show linkages with type-2 fuzzy sets, which are constructed on a basis of type-1 fuzzy sets. Several experimental studies are reported, which illustrate a behavior of the introduced method.