Rough set classification based on quantum logic

Intuitionistic fuzzy (IF) rough sets are the generalization of traditional rough sets obtained by combining the IF set theory and the rough set theory. The existing research on IF rough sets mainly concentrates on the establishment of lower and upper approximation operators using constructive and axiomatic approaches. Less effort has been put on the attribute reduction of databases based on IF rough sets. This paper systematically studies attribute reduction based on IF rough sets. Firstly, attribute reduction with traditional rough sets and some concepts of IF rough sets are reviewed. Then, we introduce some concepts and theorems of attribute reduction with IF rough sets, and completely investigate the structure of attribute reduction. Employing the discernibility matrix approach, an algorithm to find all attribute reductions is also presented. Finally, an example is proposed to illustrate our idea and method. Altogether, these findings lay a solid theoretical foundation for attribute reduction based on IF rough sets.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

Rough set theory for the interval-valued fuzzy information systems

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

In this paper we present a new approach to rough set approximations that permits to distinguish between two kinds of “imperfect” knowledge in a joint framework: on one hand, vagueness, due to imprecise knowledge and uncertainty typical of fuzzy sets, and on the other hand, ambiguity, due to granularity of knowledge originating from the coarseness typical of rough sets. The basic idea of our approach is that each concept is represented by an orthopair, that is, a pair of disjoint sets in the universe of knowledge. The first set in the pair contains all the objects that are considered as surely belonging to the concept, while the second set contains all the objects that surely do not belong to the concept. In this context, following some previous research conducted by us on the algebra of rough sets, we propose to define as rough approximation of the orthopair representing the considered concept another orthopair composed of lower approximations of the two sets in the first orthopair. We shall apply this idea to the classical rough set approach based on indiscernibility, as well as to the dominance-based rough set approach. We discuss also a variable precision rough approximation, and a fuzzy rough approximation of the orthopairs. Some didactic examples illustrate the proposed methodology.

Rough set can be established on mutually-inversistic set theory constructed by the author, forming mutually-inversistic rough set. The main functionality of traditional rough set is to mine decision rules from lower approximations. From the point of view of mutually-inversistic rough set, it is to mine mutually inverse being contained connections from lower approximations. Mutually-inversistic rough set can also mine mutually inverse intersection connections from upper approximations. It can mine not only the strong mutually inverse intersection connections, but also weak mutually inverse intersection connections. It can also make chain reasoning on the mutually inverse intersection connections mined. Therefore, mutually-inversistic rough set excels traditional rough set in functionality.

A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets

In order to effectively handle the real-valued data sets in practice, it is valuable from theoretical and practical aspects to combine fuzzy rough set and variable precision rough set so that a powerful tool can be developed. That is, the model of fuzzy variable precision rough set, which not only can handle numerical data but also is less sensitive to misclassification and perturbation,In this paper, we propose a new variable precision rough fuzzy set by introducing the variable precision parameter to generalized rough fuzzy set, i.e., the variable precision rough fuzzy set based on general relation. We, respectively, define the variable precision rough lower and upper approximations of any fuzzy set and it level set with variable precision parameter by constructive approach. Also, we present the properties of the proposed model in detail. Meanwhile, we establish the relationship between the variable precision rough approximation of a fuzzy set and the rough approximation of the level set for a fuzzy set. Furthermore, we give a new approach to uncertainty measure for variable precision rough fuzzy set established in this paper in order to overcome the limitations of the traditional methods. Finally, some numerical example are used to illuminate the validity of the conclusions given in this paper.

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfy many interesting properties. Many types of generalized rough set models have been developed in the literature. This paper presents a general framework for the study of rough intuitionistic fuzzy sets in which intuitionistic fuzzy sets are approximated with respect to a crisp approximation space. A pair of lower and upper rough intuitionistic fuzzy approximation operators are first defined. Properties of rough intuitionistic fuzzy approximations are then examined. The relationships between special types of crisp binary relations and properties of rough intuitionistic fuzzy approximation operators are further established.

Product service system (PSS) is an important strategy for manufacturing companies to improve competitiveness. The extension of supply range forces the manufacturing companies to select service suppliers to support PSS development due to the constraint of resources and capability. VIKOR approach is applied to evaluate the service suppliers based on compromise programming. To deal with the uncertain evaluation information, evaluation data are expressed as vague values by means of vague set theory. In the vague VIKOR, the maximised group utility and minimised individual regret are obtained based on similarity measure between two vague values. A set of criteria weights is a critical input of VIKOR. The traditional weights-determining approaches are subjective and have higher uncertainty such as analytic hierarchy process and analytic network process. Rough set mining approach can be used to obtain the criteria weights according to the surveyed performance information on each criterion and on the whole. The traditional rough set has poor noise-tolerance capability, the variable precision rough set approach is used to mine the criteria weights, and a mining algorithm is proposed in this paper. A case study of rescue service supplier evaluation in excavator PSS development is presented to illustrate the effectiveness of the proposed approach.

Pawlak’s classical model of rough set approximations provides an efficient tool for extracting information exactly by employing available knowledge (i.e., known knowledge) in an information system, since many problems in rough set theory are NP-hard and their solution process is therefore greedy and approximate. Many extensions of Pawlak’s classical model have been proposed in recent years. Most of them are considered over one or two sets, that is, one- or two-dimensional space or one- or two-dimensional data. Aided by relation-based rough set models, a few of these extensions are considered over three sets. However, the real world is in three-dimensional space. Therefore, it is necessary to solve these problems with other models, such as covering rough set models. For this purpose, we propose the TP-matroid—a matroidal structure over three sets. Employing the family of feasible sets of a TP-matroid as the available knowledge, a pair of rough set approximations—lower and upper approximations—is provided. In addition, for an information system defined over three sets, assisted by formal concept analysis, we establish a pair of rough set approximations. Furthermore, two TP-matroids are established based on the above pair of rough set approximations. The integration between the two pairs of rough set approximations presented here is discussed. The results show that for an information system in three-dimensional space, the rough set approximations provided here can effectively explore unknown knowledge by using available knowledge based on the family of feasible sets of a TP-matroid.

Matrix approach to decision-theoretic rough sets for evolving data

The integration of mathematical and statistical data analysis research can engender a novel and better approach, especially for survival analysis. This paper is devoted to Professor Pawlak and his ideas about rough sets and its applications. We propose MULTIHYRIS, an alternative hybrid intelligent system with a rough sets and population based approach for survival analysis. MULTIHYRIS is designed to increase the versatility and efficiency of survival analysis techniques. The MULTIHYRIS architecture incorporates mathematics - rough sets (with discernibility relations and individual patient consideration) - with statistics - Kaplan-Meier and Cox methods (with population estimates). The central idea behind MULTIHYRIS is to perform univariate analysis by using rough sets, database management and the Kaplan-Meier method with soft computing. All results from the univariate analysis are subsequently used in further mulitvariate analysis. In this stage, we provide two optional approaches to serve different requirements; rough sets integrated with database management and the Cox method. The former approach is able to produce decision rules while the latter generates a Cox model. Furthermore, set operations are used to unite these two outcomes and generate new reducts - hybrid reducts based on our rough sets-population based system. The informativeness of the rules and models can be verified within this analysis by validation processes and statistical tests. To demonstrate MULTIHYRIS, we have implemented it on a real-world geriatric data set, collected from the Dalhousie Medical School.

Incremental fuzzy cluster ensemble learning based on rough set theory

Information fusion in rough set theory : An overview