As a novel fuzzy covering, fuzzy $\beta$ covering has attracted considerable attention. However, the traditional fuzzy- $\beta$ -covering-based rough set and most of its extended models cannot well fit the distribution of samples in real data, which limits their application in classification learning and decision making. First, the upper and lower approximations of these models have no inclusion relation, so they cannot characterize a given objective concept accurately. Moreover, most of these models are hard to resist the influence of noise data, resulting in poor robustness in feature learning. For these reasons, a robust rough set model is set forth by combining fuzzy rough sets, covering-based rough sets, and multigranulation rough sets. To this end, the optimistic and pessimistic lower and upper approximations of a target concept are reconstructed by means of the fuzzy $\beta$ neighborhood related to a family of fuzzy coverings, and a new multigranulation fuzzy rough set model is presented. Furthermore, a fuzzy dependence function is introduced to evaluate the classification ability of a family of fuzzy $\beta$ coverings at different granularity levels. The dimensionality reduction of a given fuzzy covering decision table is carried out from the perspective of maintaining the discrimination power, and a forward algorithm for feature selection is developed by using the optimistic significance of candidate features as heuristic information. Three groups of numerical experiments on 16 different types of datasets demonstrate that the proposed model exhibits good robustness on datasets contaminated with noise and outperforms some state-of-the-art feature learning algorithms in terms of classification accuracy and the size of the selected feature subset.

As a combination of fuzzy sets and covering rough sets, fuzzy $\beta $ covering has attracted much attention in recent years. The fuzzy $\beta $ neighborhood serves as the basic granulation unit of fuzzy $\beta $ covering. In this article, a new discernibility measure with respect to the fuzzy $\beta $ neighborhood is proposed to characterize the distinguishing ability of a fuzzy covering family. To this end, the parameterized fuzzy $\beta $ neighborhood is introduced to describe the similarity between samples, where the distinguishing ability of a given fuzzy covering family can be evaluated. Some variants of the discernibility measure, such as the joint discernibility measure, conditional discernibility measure, and mutual discernibility measure, are then presented to reflect the change of distinguishing ability caused by different fuzzy covering families. These measures have similar properties as the Shannon entropy. Finally, to deal with knowledge reduction with fuzzy $\beta $ covering, we formalize a new type of decision table, that is, fuzzy $\beta $ covering decision tables. The data reduction of fuzzy covering decision tables is addressed from the viewpoint of maintaining the distinguishing ability of a fuzzy covering family, and a forward attribute reduction algorithm is designed to reduce redundant fuzzy coverings. Extensive experiments show that the proposed method can effectively evaluate the uncertainty of different types of datasets and exhibit better performance in attribute reduction compared with some existing algorithms.

As a typical multigranularity data analysis model, multi-scale rough sets have attracted considerable attention in recent years. However, classical multi-scale rough sets and most of its extended models can only deal with discrete data, which limits its popularization and application. To overcome this problem, we investigate the fuzzy generalization of multi-scale rough sets as well as their application in feature selection for continuous data. To this end, a new type of decision systems, i.e., multi-scale fuzzy decision systems, is formalized to represent knowledge at different scales. Scaled fuzzy granules in terms of a family of scaled fuzzy relations are introduced, using which the granular structures of fuzzy lower and upper approximations are presented. A heuristic lattice-based optimal scale selection algorithm is then put forward from the viewpoint of maintaining the consistency of decision systems. Decision rules with strong generalization ability can be obtained by selecting appropriate scales. Finally, a forward feature selection algorithm was developed by means of the optimal scale to reduce redundant fuzzy relations. Extensive numerical experiments are further conducted to compare the proposed algorithm with some state-of-the-art algorithms. The experimental results show that our model can improve the generalization ability of fuzzy rough set, so as to be more feasible and effective.

Information entropy and its extended forms have been widely used in various rough sets to evaluate the uncertainty of a fuzzy relation or a fuzzy granularity. However, they are difficult to estimate the distinguishing information of feature subsets derived from multisource data. On the one hand, most of them only have a single granulation and cannot characterize and fuse the distinguishing information from multiple granularity levels. On the other hand, they generally lack the ability to resist noise. When the data is mingled with noise, the classification performance will be greatly affected. To overcome these problems, this article put forth some novel distinguishing indicators for fuzzy $\beta $ covering (FBC) decision systems. First, we discussed the granulation structure of FBC and introduced a new multigranulation distinguishing indicator. Second, some variants of the distinguishing indicator, i.e., multigranulation variable-precision conditional distinguishing indicator, multigranulation variable-precision joint distinguishing indicator, and multigranulation variable-precision mutual distinguishing indicator, are proposed to dynamically reflect the change of distinguishing information with respect to a given granulation structure. Finally, knowledge reduction is processed by the view of keeping the classification capacity, and a forward algorithm for feature subset selection with an optimistic variable-precision conditional distinguishing indicator is developed. Two kinds of numerical experiments are conducted on 12 benchmark datasets to examine the robustness and classification performance of the presented algorithm. Experimental results indicate that our method attains competitive classification performance compared with four popular dimensionality reduction algorithms and exhibits strong robustness when facing data noise.