Granular Fuzzy Modeling for Multidimensional Numeric Data: A Layered Approach Based on Hyperbox

At present, the development of most of the granular fuzzy models depends upon some well-established numeric ones. In this study, a layered approach used to directly construct granular fuzzy models based on multidimensional numeric data is presented by engaging design methodology of granular computing. The crux of the approach involves a construction of interval information granules in the output space and the corresponding hyperbox information granules in the input space. A method of constructing these information granules and the hyperbox-based granular fuzzy model formed around them is studied in detail. Two different schemes to decode the formed hyperbox-based granular fuzzy model are also presented. Furthermore, a measure of a composite quality of the formed hyperbox-based granular fuzzy model is proposed along with the concept of coverage and specificity of resulting information granules. A number of experimental studies are reported, which offer a useful insight into the effectiveness of the presented approach, as well as reveal the impact of critical parameters on the performance of the established models.

I. Introductory Notes

As a fundamental and algorithmic construct in granular computing (GrC) [1]–[3], information granules (IGs) play a pivotal role in compressed representation, structuralization, and characterization of numeric data [4]–[10]. IGs [11], which are regarded as abstract entities delivering essential characteristics of numeric data in concise manner, emerge through forming pieces of knowledge from data at a higher level of abstraction (generality) and representing them in the form of a certain existing formalism such as sets (intervals) [12], fuzzy sets [13], rough sets [14], [15], shadowed sets [16], probabilistic sets [17], and alike. The emergence of IGs comes through the process of abstraction of data in which information granularity (abstraction level) plays a pivotal role. Namely, the size (specificity) of IGs can be dominated by information granularity. In the framework of GrC, by admitting some level of information granularity, IGs becomes more reflective of the main features of raw numeric data.