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.

Granular computing (GrC) is an efficient way to reveal descriptions of data in line with human cognition and plays a critical role in knowledge discovery. Information granules (IGs), the basic computing unit of GrC, is the key component of knowledge representation and processing. Rough sets are one of the classical GrC models and generate IGs based on indiscernibility relations. The relations can effectively achieve the granulation of nominal attributes and generate desirable IGs, but they may cause information loss when achieving the granulation of numerical attributes. To overcome this issue, fuzzy rough sets (FRS) and neighborhood rough sets (NRS) were developed based on the rough sets. However, to generate high-performance IGs in practice, the FRS model requires prior knowledge to determine a fuzzy operation in advance, and NRS needs to calculate an optimal neighborhood radius. In addition, regardless of FRS or NRS, each object is taken as a computing unit to generate IGs that constitute a covering rather than a partition for the universe. This process is not only time-consuming but also prone to generate redundant IGs. Therefore, in this study, a data-driven interval granulation approach based on the uncertainty principle is proposed to generate justifiable interval neighborhood IGs with flexibility and tolerance. First, the interval granulation of attribute values and interval equivalence relation are defined. Next, with the interval equivalence relation, a novel interval rough sets model is developed to unify numerical and nominal attributes into one framework, and a membership function is developed without requiring prior knowledge in advance. Then, a highly effective classifier named CAR-ING integrating attribute reduction technique is developed from the perspective of interval neighborhood IGs. Finally, experiments and comparisons on 17 widely used UCI benchmark datasets and 3 real Biobank medical datasets from the U.K. demonstrated that CAR-ING performs significantly better than four state-of-the-art classifiers based on GrC and five classical classifiers in machine learning. Additionally, the efficiency of CAR-ING is demonstrated on 20 datasets.

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.