Multi-label classification represented by hierarchical classification (HC) plays an important role in current large-scale problems, which can acquire a more accurate expression of data that conforms to the human multi-granularity cognitive process. To compress the original dataset and simultaneously enhance the expressive force of models, selecting an appropriate granularity for approximately describing the classification is the main task in the rough set theory. Nevertheless, the current rough set theory merely concerns flat classification and encounters new problems when approximately describing HC. 1) There lacks a measure to correctly reflect misclassification in accordance with the hierarchical accuracy of HC on the training set. 2) There lacks a measure relying on the distribution of the dataset to reflect the difference between two distinct feature sets describing HC in generalization ability. To address the mentioned issues, this paper utilizes the knowledge distance to characterize HC and proposes a cost-sensitive granularity selection for HC. First, HC and features are respectively granulated according to hierarchical quotient space and neighborhood granular structures. Then, knowledge distance and its extended form are employed to formulate misclassification and test costs. On this basis, a cost-sensitive neighborhood granularity selection is presented for HC. Finally, we experimentally demonstrate the excellent performance of the proposed method in terms of efficiency and HC accuracy both in synthetic and real datasets.

Three-way decision with neighborhood rough sets (3WDNRS) is adept at addressing uncertain problems involving continuous data by configuring the neighborhood radius. However, on one hand, the inputs of 3WDNRS are individual neighborhood granules, which reduce the decision efficiency and generality; on other hand, the thresholds of 3WDNRS require prior knowledge to be approximately set in advance, making it difficult to apply in cases where such knowledge is unavailable. To address these issues, we introduce granular-ball computing (GBC) into 3WDNRS from the perspective of uncertainty. Firstly, we propose an enhanced granular-ball generation method based on DBSCAN called DBGBC. Subsequently, we present an improved granular-ball neighborhood rough sets model (GBNRS++) by combining DBGBC with a quality index. Furthermore, we construct a three-way classifier with granular-ball neighborhood rough sets (3WC-GBNRS++) based on the principle of minimum fuzziness loss. This approach provides an objective and efficient way to determine the thresholds. To further enhance classification accuracy, we design an adaptive granular-ball neighborhood within the subsequent classification process of 3WC-GBNRS++. Finally, experimental results demonstrate that, 3WC-GBNRS++ almost outperformed other comparison methods in terms of effectiveness and robustness, including 4 state-of-the-art granular-balls-based classifiers and 5 classical machine learning classifiers on 12 public benchmark datasets. Moreover, we discuss the limitations of our work and the outlook for future research.

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.

Feature selection is an important preprocessing step for high-dimensional data mining and machine learning; it is viewed as the selection of the optimal granularity to describe the target concept in rough set theory. Currently, research on rough sets mainly focuses on granularity selection in flat classification scenarios, while organizing hundreds of labels for hierarchical classification (HC) can provide additional external information and achieve better performance in terms of both accuracy and efficiency. However, HC also faces the following problems: 1) the current measures’ failure to characterize the uncertainty in HC; 2) the inability to select the optimal granularity of the target concept in HC; and 3) no valid approach to select features in a decision system with hierarchical classification (HieDS). To address these problems, this article introduces HC into rough set theory and proposes an approach to granularity selection for HC. First, we introduce the knowledge distance to reflect the uncertainty of HC and define related important characteristic functions to describe a HieDS. Then, from the perspective of uncertainty, granularity selection for the target concept and feature selection are presented based on these characteristic functions. Finally, we experimentally realize granularity selection and demonstrate excellent performance of feature selection in a HieDS in terms of both feature selection and classification accuracy.

Depending on whether α is an idempotent element of the t-norm T0 or not, we study and characterize the structure of a continuous crossmigrative t-norm T with respect to a fixed and continuous t-norm T0, which is indeed the conjecture presented by Fodor et al. (Int J Intell Syst 2012;27:411–428) Results of this paper show that the -cross-migrativity is completely determined by the restriction on a portion of domain of T and have nothing to do with the remaining of [0, 1]2. Moreover, we also clarify the relations between our present results and the previous results.