Multigranularity-Layer Shadowed Set: A Three-Way Approximation Framework for Fuzzy Information

The shadowed set, as a three-way approximation model for a fuzzy set, has been extensively studied in model construction, theoretical analysis, data analysis, and applications. However, the current research on the construction of a shadowed set is all based on a single attribute to complete the approximate partition of a simple target concept, i.e., single-granularity-layer space. Multiple attributes are not considered comprehensively to achieve an approximate partition of the complex target concept, i.e., multigranularity-layer space. In addition, the current evaluation criteria of the shadowed set are all reasonable explanations for model construction and threshold determination, lacking an effective evaluation of approximate partition results. Therefore, the multigranularity-layer shadowed set (MGLSS) is proposed in this article, which aims to extend the construction of the shadowed set from single-granularity-layer space to multigranularity-layer space. First, MGLSS is analyzed based on information systems and divided into two models: optimistic-MGLSS (OPT-MGLSS) and pessimistic-MGLSS (PES-MGLSS). Second, the expression form, threshold selection, semantic interpretation, partition rules, basic mathematical theorems, and the definition of fusion operators of MGLSS are analyzed and discussed. Third, two evaluation criteria of coverage and accuracy of approximate partition results are proposed. Finally, six cases are analyzed to illustrate the application scenarios of MGLSS, and one instance and algorithm analysis are given to demonstrate the construction steps, and through the real datasets experiment and statistical hypothesis testing analysis, to provide an objective and quantitative scientific basis for the research conclusion. The experimental results demonstrate the validity and rationality of the MGLSS construction mechanism.