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.

Hierarchical quotient space structure (HQSS), as a typical description of granular computing (GrC), focuses on hierarchically granulating fuzzy data and mining hidden knowledge. The key step of constructing HQSS is to transform the fuzzy similarity relation into fuzzy equivalence relation. However, on one hand, the transformation process has high time complexity. On the other hand, it is difficult to mine knowledge directly from fuzzy similarity relation due to its information redundancy, i.e., sparsity of effective information. Therefore, this article mainly focuses on proposing an efficient granulation approach for constructing HQSS by quickly extracting the effective value of fuzzy similarity relation. First, the effective value and effective position of fuzzy similarity relation are defined according to whether they could be retained in fuzzy equivalence relation. Second, the number and composition of effective values are presented to confirm that which elements are effective values. Based on these above theories, redundant information and sparse effective information in fuzzy similarity relation could be completely distinguished. Next, both isomorphism and similarity between two fuzzy similarity relations are researched based on the effective value. The isomorphism between two fuzzy equivalence relations is discussed based on the effective value. Then, the algorithm with low time complexity for extracting effective values of fuzzy similarity relation is introduced. On the basis, the algorithm for constructing HQSS is presented to realize efficient granulation of fuzzy data. The proposed algorithms could accurately extract effective information from the fuzzy similarity relation and construct the same HQSS with the fuzzy equivalence relation while greatly reducing the time complexity. Finally, relevant experiments on 15 UCI datasets, 3 UKB datasets, and 5 image datasets are shown and analyzed to verify the effectiveness and efficiency of the proposed algorithm.

Feature selection method with rough sets based on incremental learning has the major advantage of the higher efficiency in a dynamic information system, which has attracted extensive research. However, the incremental approximation feature selection with an accelerator (IAFSA) remains ambiguous for a dynamic information system with fuzzy decisions (ISFD). Driven by this concern, the nonincremental approximation feature selection is first presented by fuzzy knowledge distance (FKD). Second, the incremental theory of FKD is constructed with a batch of objects appended to or removed from the dynamic ISFD. Subsequently, an acceleration mechanism to eliminate redundant information granules is developed to reduce the sample space. Eventually, two categories of IAFSA based on FKD are presented. The experiments reflect the efficiency and effectiveness of the developed IAFSA algorithms.

Granular computing, a new paradigm for solving large-scale and complex problems, has made significant progresses in knowledge discovery. Granular ball computing (GBC) is a novel granular computing method, which can rapidly generate scalable and robust information granules, that is, granular balls. However, a comprehensive index for measuring the performance of a granular ball does not exist. Furthermore, GBC lacks a mechanism to deal with dynamic decision systems. Therefore, in this study, the quality index of a granular ball is first formulated. Next, with this index, a novel granular ball rough sets model (GBRS) based on GBC is proposed. GBRS is more conducive to learning knowledge from uncertain datasets and more suited to incremental learning than the latest granular ball neighborhood rough sets model based on GBC. Subsequently, an incremental mechanism is introduced into GBRS, and two incremental learning models are developed for objects increasing in stream patterns and batch patterns, respectively. In the incremental learning process, three patterns of granular balls, that is, update, fusion, and split, were well studied when a set of objects was added to the decision system. Finally, to verify the effectiveness and efficiency, we apply GBRS and these two incremental learning models into classification tasks. Compared with four current state-of-the-art classification methods based on granular computing and four classical classifiers in machine learning, the proposed classifiers in this paper achieve higher classification accuracy as well as better efficiency on benchmark datasets.

As a three-way approximation of fuzzy sets, shadowed sets have attracted extensive attention in recent years. A fundamental issue in the process of constructing shadowed sets is the interpretation and determination of the threshold pair$({\alpha, \beta } )$, and the uncertainty consistency, that is, the consistency of fuzzy entropy. However, there may be a large fuzzy entropy loss between a fuzzy set and its corresponding game theoretic shadowed sets (GTSS), and the GTSS model is also accompanied by a large time cost when the precision of$({\alpha, \beta } )$is improved. Therefore, the fuzzy-entropy-based GTSS (FeGTSS) is proposed in this article from the perspective of fuzzy entropy loss. First, based on the compromise principle of game theory, the fuzzy entropy loss of shadowed sets is analyzed in this article. Second, in the process of calculating$({\alpha, \beta } )$, the optimal game strategy is searched based on the dichotomy algorithm. Third, the FeGTSS model is extended and discussed based on the analysis of different data distribution types. Finally, the rationality and validity of the FeGTSS model are illustrated through instances and experimental analysis.