Density peak clustering is an effective and interpretable method for uncovering potential knowledge in unlabeled datasets with arbitrary shapes. It has been extensively studied by researchers, and a series of extended models have been proposed. The performances of these algorithms largely depend on the positions and number of cluster centers. However, accurately selecting these centers remains a challenging problem. Therefore, to address this issue, an adaptive density distribution clustering (ADDC) method based on graph theory and $k$ -nearest neighbors is developed in this study. ADDC is a decentralized and robust clustering approach, which consists of three main components. First, an undirected neighborhood graph is constructed based on the neighbor degree defined in this article to implement a decentralized allocation strategy. Second, componentwise local density is introduced, and a new criterion for selecting density peaks is established to serve as one of the guidelines for determining the number of clusters. Third, with the neighborhood graph and density peaks, criterion-based decomposition and fusion strategies are formulated to identify clusters with multiple peaks or to detect low-density clusters without peaks. Finally, experiments and comparisons on widely used real datasets and synthetic datasets demonstrated that ADDC significantly outperforms five classical clustering methods and seven state-of-the-art density-based cluster approaches.

Clustering is a significant technique in data mining, which can uncover the hidden correlation information and obtain deeper understanding of the inherent structure of data. However, when dealing with the data with extremely uneven density and increasingly complex structure, most current clustering algorithms only obtain results at a single granular level, resulting in a unilateral understanding of the data. Therefore, a novel multi-granularity clustering algorithm based on grid partition and fuzzy quotient space (MGCGF) is proposed in this paper. Firstly, by introducing the Gaussian kernel density function to characterize the distribution characteristics of data, a grid partition method is designed to select representative points for clustering. Secondly, based on the results of grid partition, the representative points composed of the maximum density values in each dimension of the grids are used for multi-granularity clustering to improve the clustering efficiency. Finally, a multi-granularity clustering algorithm is proposed by introducing fuzzy quotient space theory with representative points as input. MGCGF can be used to uncover the hierarchical structure of the data itself and form a multi-granularity space. And the clustering results with multi-granularity can be directly provided from multi-granularity spaces without re-clustering. By comparing with the other six clustering algorithms, the feasibility is verified in terms of both efficiency and accuracy.

In recent years, with the rapid development of the Internet, multisource information fusion has become a forefront issue due to its ability to merge different information. Granular computing (GrC), as a methodology simulating human hierarchical cognition, provides a new approach for multisource information fusion. However, on one hand, the existing information fusion studies in GrC all focus on feature-level fusion and decision-level fusion based on multisource data, neglecting the basic characteristics and advantages of GrC: granulation. On the other hand, the existing methods for fusing the knowledge spaces in GrC suffer from losing the necessary information or artificially adding information. In order to address these issues, a novel information fusion mode from the perspective of GrC is proposed in this article, named knowledge-level fusion. First, by introducing a new step, that is, granulate data to construct the knowledge space, into the multisource information fusion process, the knowledge-level fusion mode is proposed. Second, the optimistic core quotient space is proposed to characterize the information consensus and information gap of multisource knowledge spaces in the static data environment. The pessimistic core quotient space is proposed to characterize the information consensus in the dynamic data environment. Related theorems are given to describe the characteristics of the core quotient spaces. Then, the knowledge-level fusion method driven jointly by the data space and the knowledge space is introduced based on the principle of extracting the core quotient space first and then allocating other objects in the candidate set. On the basis, the superiority of the proposed method over the existing methods is demonstrated through theoretical analysis. Finally, experiments on 12 UCI datasets and three UKB datasets are carried out to verify the promoting effect on classification and clustering algorithms, the effectiveness compared to feature-level and decision-level fusion modes, efficiency and statistical significance of the proposed knowledge-level fusion method and mode.

As an innovative methodology in data processing and knowledge representation, granular-ball computing (GBC) adaptively generates distinct neighborhoods for individual objects, thereby improving both generality and flexibility. By replacing point inputs with granular-balls (GBs), GBC achieves substantial efficiency gains. However, traditional GB-based classifiers may produce unreliable classifications under uncertain conditions. To address this limitation, we propose a novel approach that integrates three-way decision (3WD) theory with GBC, enabling robust handling of uncertain classification problems. This study first introduces a sequential three-way decision with fuzzy granular-ball rough sets (S3WD-FGBRS). We systematically analyze the changing rules of the multilevel decision cost in S3WD-FGBRS and its three regions. Building upon the principle of justifiable granularity, we develop a cost-sensitive three-way granular-ball generation method (CS3W-GBG) based on S3WD-FGBRS that incorporates a granularity optimization mechanism. To validate our approach, we conduct comprehensive experiments using three state-of-the-art GB classifiers and two benchmark classifiers on 12 publicly available datasets. Experimental results demonstrate that CS3W-GBG exhibits strong resilience in processing uncertain data through its 3WD strategy. Furthermore, our method achieves competitive performance compared to existing approaches in terms of classification accuracy and robustness.

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.

As one of the rapidly developing methodologies for dealing with complex problems in line with human cognition, granular computing has made significant achievements in knowledge discovery. Neighborhood classifier, as a typical description of granular computing, is an effective method for the classification of continuous data. However, in the phase of constructing neighborhood rules, the existing neighborhood classifiers are divided into the following two modes and both have defects: 1)the strategy driven by center:there are a lot of overlap and inclusion among neighborhood rules, that is, it needs to spend much time to reduce redundancy rules; and 2)the strategy driven by rule:there are many rules containing only one object, which cannot form an effective covering and would affect knowledge acquisition in the incremental environment. Therefore, in this article, fuzzy quotient space theory is introduced to construct neighborhood rules. Based on the optimal granularity of fuzzy quotient space, first, a neighborhood covering classifier, which has no redundant rules and could form the effective covering, is proposed without any artificial parameter. Second, comprehensively considering the knowledge purity and complexity, the quality measure of granularity is proposed, which guides the optimal granularity selection of hierarchical quotient space structure. Third, neighborhood rules are constructed in optimal granularity. Then, the offset of center is introduced to describe the membership degree of the test object to different neighborhood rules, and the neighborhood allocation strategy is proposed accordingly. Next, an algorithm for the neighborhood covering classifier based on these theories is proposed. Finally, experiments on 13 UCI datasets and three real datasets are carried out to verify the performance of the proposed classifier from four common classification indexes.

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.

Multi-scale decision system (MDS) is an effective tool to describe hierarchical data in machine learning. Optimal scale combination (OSC) selection and attribute reduction are two key issues related to knowledge discovery in MDSs. However, searching for all OSCs may result in a combinatorial explosion, and the existing approaches typically incur excessive time consumption. In this study, searching for all OSCs is considered as an optimization problem with the scale space as the search space. Accordingly, a sequential three-way decision model of the scale space is established to reduce the search space by integrating three-way decision with the Hasse diagram. First, a novel scale combination is proposed to perform scale selection and attribute reduction simultaneously, and then an extended stepwise optimal scale selection (ESOSS) method is introduced to quickly search for a single local OSC on a subset of the scale space. Second, based on the obtained local OSCs, a sequential three-way decision model of the scale space is established to divide the search space into three pair-wise disjoint regions, namely the positive, negative, and boundary regions. The boundary region is regarded as a new search space, and it can be proved that a local OSC on the boundary region is also a global OSC. Therefore, all OSCs of a given MDS can be obtained by searching for the local OSCs on the boundary regions in a step-by-step manner. Finally, according to the properties of the Hasse diagram, a formula for calculating the maximal elements of a given boundary region is provided to alleviate space complexity. Accordingly, an efficient OSC selection algorithm is proposed to improve the efficiency of searching for all OSCs by reducing the search space. The experimental results demonstrate that the proposed method can significantly reduce computational time.