A neighborhood rough set model with nominal metric embedding

The variable precision rough set (VPRS) model generalizes the Pawlak rough set model with variable parameters. The Bayesian rough set (BRS) model improves the VPRS model with non-parametric modification by using the prior probability as a reference. This paper presents two research results related to rough set model and rough vague sets. One is a modification to the Bayesian rough set model and the other is a modification to rough vague sets. First, the Bayesian rough set model is analyzed and discussed. Second, a modification to this model is proposed and verified. Finally, a modification to rough vague sets is presented and its related properties are discussed..

Distance metric learning-based multi-granularity neighborhood rough sets for attribute reduction

It is an important subject to mine valuable knowledge from complex and massive data in the era of big data. Rough set theory is a new mathematical tool for dealing with uncertain and inaccurate data, decision‐theoretic rough set model (DTRS), as an extension of classical rough set model, is used to analyze decision information systems and multi‐granulation decision‐theoretic rough set model (MG‐DTRS) can analyze and process target concepts from different angles and levels. However, the classical DTRS model exists some limitations in dealing numerical or hybrid‐valued data. Considering the different influence of numerical features and symbolic features on decision‐making, the paper proposes an incomplete neighborhood multi‐granulation decision‐theoretic rough set model in hybrid‐valued decision system through integrating MG‐DTRS with neighbourhood rough sets, and two types of neighborhood multi‐granulation decision‐theoretic set models are emphatically analysed. Furthermore, taking pessimistic and optimistic neighborhood multi‐granulation decision‐theoretic rough sets as examples, the implementation algorithms and related properties of the two type of models are studied. Finally, the relationship between the proposed model and other models is analyzed through formula derivation. The model proposed in this paper can effectively solve the decision‐making problem of hybrid‐valued incomplete information system through multi‐angle and multi‐level analysis.

Mixed feature selection based on granulation and approximation

Six-Set Approximation Theorem of neighborhood related rough sets

Covering based intuitionistic fuzzy (IF) rough set is a generalization of granular computing and covering based rough sets. By combining covering based rough sets, IF sets and fuzzy rough sets, we introduce three classes of coverings based IF rough set models via IF$$\beta $$-neighborhoods and IF complementary $$\beta $$-neighborhood (IFC$$\beta $$-neighborhood). The corresponding axiomatic systems are investigated, respectively. In particular, the rough and precision degrees of covering based IF rough set models are discussed. The relationships among these types of coverings based IF rough set models and covering based IF rough set models proposed by Huang et al. (Knowl Based Syst 107:155–178, 2016). Based on the theoretical analysis for coverings based IF rough set models, we put forward intuitionistic fuzzy TOPSIS (IF-TOPSIS) methodology to multi-attribute decision-making (MADM) problem with the evaluation of IF information problem. An effective example is to illustrate the proposed methodology. Finally, we deal with MADM problem with the evaluation of fuzzy information based on CFRS models. By comparative analysis, we find that it is more effective to deal with MADM problem with the evaluation of IF information based on CIFRS models than the one with the evaluation of fuzzy information based on CFRS models.

The basic notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation. This paper proposes an intuitionistic fuzzy rough set (IFRS) model which is a combination of intuitionistic fuzzy set (IFS) and rough set. We approximate an IFS by using soft binary relations instead of fixed binary relations. By using this technique, we get two pairs of intuitionistic fuzzy (IF) soft sets, called the upper approximation and lower approximation with respect to foresets and aftersets. Properties of newly defined rough set model (IFRS) are studied. Similarity relations between IFS with respect to this rough set model (IFRS) are also studied. Finally, an algorithm is constructed depending on these approximations of IFSs and score function for decision-making problems, although a method of decision-making algorithm has been introduced for fuzzy sets already. But, this new IFRS model is more accurate to solve the problem because IFS has degree of nonmembership and degree of hesitant.

Based on the analysis of the rough set model on a tolerance relation and the fuzzy rough set, two types of fuzzy rough sets models on tolerance relations are constructed and researched. Then we propose the optimistic and pessimistic multi-granulation fuzzy rough sets models in a fuzzy tolerance approximation space with the point view of granular computing. In these models, the fuzzy lower and upper approximations of a fuzzy set are defined on multiple fuzzy tolerance relations. It follows the research on the properties of the fuzzy lower and upper approximations of the new models. The fuzzy rough set model, rough set model on a tolerance relation and multi-granulation rough sets models are special cases of the new ones from the perspective of the considered concepts, related relations and granular computing. The new models are meaningful generalizations of the classical rough sets.

A generalization of the original idea of rough sets and variable precision rough sets is introduced. This generalization is based on the concept of absolute and relative rough membership. Similarly to variable precision rough set model, the generalization called parameterized rough set model, is aimed at modeling data relationships expressed in terms of frequency distribution rather than in terms of a full inclusion relation used in the classical rough set approach. However, differently from variable precision rough set model, one or more parameters modeling the degree to which the condition attribute values confirm the decision attribute value, are considered. The properties of this extended model are investigated and compared to the classical rough set model and the variable precision rough set model.

Neighborhood margin rough set: Self-tuning neighborhood threshold

The uncertainty of probabilistic rough sets in multi-granulation spaces

Neighborhood rough set theory is an important method for dealing with uncertainty, fuzziness and undefined objects, and neighborhood rough set theory based on binary relation is studied by scholars a lot. However, most of the current research is used to deal with equivalent binary relations or other binary relations with stringent condition, which has been unable to meet the rapid development of data. This paper proposes a new multi-granulation rough set that can deal with general binary relations based on the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> neighborhoods, and verifies its related properties. At the same −time, it has richer meanings through weighted methods. Theoretical analysis and practical examples show that this rough set has better ability to process data. Finally, a weighted attribute reduction algorithm is designed based on the significant function under the general binary relation. The experimental results show that this rough set can handle complex data well. The research in this paper develops the theory of classical rough sets, and provides a theoretical basis for the knowledge acquisition of information systems under the general binary relation.

Rough set theory for the interval-valued fuzzy information systems

A novel approach to attribute reduction based on weighted neighborhood rough sets

Most previous studies on rough sets focused on attribute reduction and decision rule mining on a single concept level. Data with attribute value taxonomies (AVTs) are, however, commonly seen in real-world applications. In this paper, we extend Pawlak's rough set model, and propose a novel multi-level rough set model (MLRS) based on AVTs and a full-subtree generalization scheme. Paralleling with Pawlak's rough set model, some conclusions related to the MLRS are given. Meanwhile, a novel concept of cut reduction based on MLRS is presented. A cut reduction can induce the most abstract multi-level decision table with the same classification ability on the raw decision table, and no other multi-level decision table exists that is more abstract. Furthermore, the relationships between attribute reduction in Pawlak's rough set model and cut reduction in MLRS are discussed. We also prove that the problem of cut reduction generation is NP-hard, and develop a heuristic algorithm named CRTDR for computing the cut reduction. Finally, an approach named RMTDR for mining multi-level decision rule is provided. It can mine decision rules from different concept levels. Example analysis and comparative experiments show that the proposed methods are efficient and effective in handling the problems where data is associated with AVTs.