Pessimistic Multigranulation Rough Set Research Articles

Pessimistic multigranulation rough bipolar fuzzy set and their application in medical diagnosis

Pessimistic multigranulation rough bipolar fuzzy set and their application in medical diagnosis

Pessimistic Multigranulation Rough Set of Intuitionistic Fuzzy Sets Based on Soft Relations

Qian presented multigranulation rough set (MGRS) models based on Pawlak’s rough set (RS) model. There are two types of MGRS models, named optimistic MGRS and pessimistic MGRS. Recently, Shabir et al. presented an optimistic multigranulation intuitionistic fuzzy rough set (OMGIFRS) based on soft binary relations. This paper explores the pessimistic multigranulation intuitionistic fuzzy rough set (PMGIFRS) based on soft relations combined with a soft set (SS) over two universes. The resulting two sets are lower approximations and upper approximations with respect to the aftersets and foresets. Some basic properties of this established model are studied. Similarly, the MGRS of an IFS based on multiple soft relations is presented and some algebraic properties are discussed. Finally, an example is presented that illustrates the importance of the proposed decision-making algorithm.

Knowledge reduction of pessimistic multigranulation rough sets in incomplete information systems

Knowledge reduction of pessimistic multigranulation rough sets in incomplete information systems

Relative Reduction of Neighborhood-Covering Pessimistic Multigranulation Rough Set Based on Evidence Theory

Relative reduction of multiple neighborhood-covering with multigranulation rough set has been one of the hot research topics in knowledge reduction theory. In this paper, we explore the relative reduction of covering information system by combining the neighborhood-covering pessimistic multigranulation rough set with evidence theory. First, the lower and upper approximations of multigranulation rough set in neighborhood-covering information systems are introduced based on the concept of neighborhood of objects. Second, the belief and plausibility functions from evidence theory are employed to characterize the approximations of neighborhood-covering multigranulation rough set. Then the relative reduction of neighborhood-covering information system is investigated by using the belief and plausibility functions. Finally, an algorithm for computing a relative reduction of neighborhood-covering pessimistic multigranulation rough set is proposed according to the significance of coverings defined by the belief function, and its validity is examined by a practical example.

Novel Three-Way Decisions Models with Multi-Granulation Rough Intuitionistic Fuzzy Sets

The existing construction methods of granularity importance degree only consider the direct influence of single granularity on decision-making; however, they ignore the joint impact from other granularities when carrying out granularity selection. In this regard, we have the following improvements. First of all, we define a more reasonable granularity importance degree calculating method among multiple granularities to deal with the above problem and give a granularity reduction algorithm based on this method. Besides, this paper combines the reduction sets of optimistic and pessimistic multi-granulation rough sets with intuitionistic fuzzy sets, respectively, and their related properties are shown synchronously. Based on this, to further reduce the redundant objects in each granularity of reduction sets, four novel kinds of three-way decisions models with multi-granulation rough intuitionistic fuzzy sets are developed. Moreover, a series of concrete examples can demonstrate that these joint models not only can remove the redundant objects inside each granularity of the reduction sets, but also can generate much suitable granularity selection results using the designed comprehensive score function and comprehensive accuracy function of granularities.

Multigranulation vague rough set over two universes and its application to group decision making

This article studies the rough approximation of a vague concept based on multiple granularity over two universes. With the description of the background of risk decision making problems in reality, we first present the optimistic multigranulation rough vague set, pessimistic multigranulation rough vague set and variable precision multigranulation rough vague set on the multigranulation approximation space over two universes. Then, under the multigranulation fuzzy approximate space over two universes, we establish the corresponding three types of multigranulation vague rough set over two universes. Subsequently, the interesting properties and results as well as the relationship between the multigranulation rough vague set models and multigranulation vague rough set model over two universes are investigated in detail. The results show that multigranulation vague rough set models are extensions of the existing generalized rough set models under the framework of two universes. At the same time, we construct a new approach to group decision making under uncertainty by using the theory of multigranulation vague rough set over two universes. The basic principal and the detailed steps of the decision making model given in this paper are presented in detail. Meanwhile, an example of handling a medical diagnosis group decision making problem illustrates this approach. The main contribution of this paper is twofold. One is to provide the theoretical model of multigranulation vague rough set over two universes. Another is to try making a new way to handle group decision making problems under uncertainty based on multigranulation vague rough set theory and methodologies over two universes.

Multi-Granulation Rough Set for Incomplete Interval-Valued Decision Information Systems Based on Multi-Threshold Tolerance Relation

A relation is viewed as a granularity from a granular computing perspective. A classic rough set contains only one granularity. A multi-granulation rough set contains multiple granularities, which promotes the applications of classical rough set. Firstly, this paper uses the incomplete interval-valued decision information system (IIVDIS) as research object and constructs two rough set models in the light of single granularity rough set model for applying the rough set theory to real life more widely, which are optimistic multi-granulation rough set (OMGRS) model and pessimistic multi-granulation rough set (PMGRS) model in the IIVDIS. Secondly, we design two algorithms to compute the roughness and the degree of dependence that are two tools for measuring uncertainty of rough set. Finally, several experiments are performed on six UCI data sets to verify the validity of the proposed theorems.

SDMGRS: Soft Dominance Based Multi Granulation Rough Sets and Their Applications in Conflict Analysis Problems

The classical rough set theory was presented by Pawlak, which is mainly concerned with the approximation of sets described by a single binary relation on the universe. In this paper, we initiate a multi attribute group decision making problems in the presence of multi attribute and multi decision in decision making with preferences. Then resolving the problem, using two different approximation strategies, i.e., seeking common reserving difference and seeking common rejecting difference, four kinds of soft dominance-based multi-granulation rough sets are presented, namely, soft dominance based optimistic multi-granulation rough sets and soft dominance based pessimistic multi-granulation rough sets and their applications in solving a multi agent conflict analysis decision problem. The proposed method addresses the limitations of the Pawlak model and Sun's conflict analysis model and thus improve these models. Finally, the results on labor management negotiation problems show that the proposed algorithms are more effective and efficient for feasible consensus strategy when compared with Sun's technique.

Matrix-based approaches for dynamic updating approximations in multigranulation rough sets

Multigranulation rough set, which is constructed by a family of equivalence relations has attracted much attention, because it offers a theoretical framework for the problem solving in the view of multigranulation. However, the granular structure in the information systems often dynamically evolves over time. How to dynamically obtain the potential useful knowledge for decision making is of great significance in the context of multigranulation. Motivated by this requirement, in this paper, we present the definitions of equivalence relation matrix, diagonal matrix and cut matrix for a single granular structure in multigranulation rough set, and propose a matrix representation of multigranulation approximations in optimistic and pessimistic multigranulation rough set. Then, corresponding matrix-based dynamic approaches for updating approximations are proposed in multigranulation rough set when a single granular structure evolves over time. The experimental evaluations show the effectiveness of the proposed matrix-based dynamic updating algorithms compared with the matrix-based static algorithm.

Cardiac arrhythmia classification using multi-granulation rough set approaches

Cardiovascular disease is a most important reason for human death in modern society. Electrocardiogram (ECG) signal deals with valuable information about functioning of the heart. For that reason, ECG investigation signifies an efficient way to identification and treat different types of cardiac arrhythmia diseases. Nowadays various pattern classification methods has been developed for the classification of ECG signals. These classification methods helps to physician for diseases diagnosis. A multi-granulation rough set (MGRS) has become a new direction of rough set theory, which is based on multiple binary relations on the universe of discourse. In this present study, Multi-granulation rough set based classification approaches (Pessimistic Multi-Granulation Rough Set (PMGRS) and Optimistic Multi-Granulation Rough Set (OMGRS)) are applied to mine appropriate rules to explore better decision making process. The experiments were conducted on the ECG data from the Physionet arrhythmia database to classify five kinds of normal and abnormal signals. In the classification process, Feature extraction played an important role. And we have used two kinds feature extraction methods (1) Pan Tomkins (PT) feature extraction method. This method used to extract the morphological features are P, Q, R, S, T peak intervals, which is also used to determine heart rate. (2) Wavelet transform (WT) feature extraction method. This method used to extract the wavelet coefficients. Both two methods are successfully applied to (ECG signal) classification. The proposed multi-granulation rough set rule based classification methods is validated using the first 24 channel of the ECG signal records of the MIT-BITH arrhythmia database, and achieves finding high accuracies. Experimental results show that the proposed classification techniques significantly outperforms other well-known techniques.

L -fuzzy multigranulation rough set based on residuated lattices

This paper studies L-fuzzy multigranulation rough set based on residuted lattices, which is considered as an extension of the classical L-fuzzy rough set. Two models of L-fuzzy optimistic multigrannulation rough set and L-fuzzy pessimistic multigrannulation rough set based on residuted latteices are suggested. Then, we mainly investigate some funda- mental properties of the L-fuzzy multigranulation rough set model. With regards to residuated lattices' topology, we study the topological and the lattice structure in the L-fuzzy optimistic multigranulation rough set model and L-fuzzy pessimistic multigranulation rough set model, respectively.

Pessimistic multi-granulation rough set-based classification for heart valve disease diagnosis

The primary contribution of this study relies on proposing a new method, which can detect heart diseases in respective heart valve data. In this work, supervised quick reduct feature selection algorithm is applied for selecting important features from heart valve data. The classification method is applied only for relevant features selected using supervised quick reduct from heart valve data. In this paper, a new classification approach based on pessimistic multi-granulation rough sets (PMGRS) is applied for heart valve disease diagnosis. In multi-granulation rough sets, set approximations are well-defined by multiple equivalence relations on the universe, leading to an effective model for classification. This is confirmed by experimental evaluation, which shows excellent classification performance and also demonstrates that the proposed approach is superior to other benchmark classification algorithms including naive Bayes, multi-layer perceptron (MLP), and J48 and decision table classifiers.

Rough Set Approximations in Multi-granulation Fuzzy Approximation Spaces

Pawlak’s rough set model considers the rough approximations based on an equivalence relation. Multi-granulation rough set models concern rough approximations based on multiple equivalence relations. In this paper, we examine six types of rough set approximations in multi-granulation fuzzy approxima tion spaces (MGFASs). We construct a partition of the given universe based on a fuzzy binary relation in a fuzzy approximation space. Based on the partition, we introduce a pair of rough set approximations. In a multi-granulation fuzzy approximation space, by a family of fuzzy binary relations, we introduce two kinds of rough set approximations in terms of the union and intersection of fuzzy relations, respectively. A pair of rough set approximations based on the family of fuzzy binary relations is also discussed. Furthermore, the optimistic and pessimistic multi-granulation rough set approximations are investigated due to the fuzzy binary relations in aMGFAS. Properties of these rough set approximations are demonstrated. Finally, we examine relationships of them. It is proved that the lower and upper approximations generated by a family of fuzzy binary relations are the pair nearest to the undefinable set, and the pessimistic multi-granulation lower and upper approximations are the pair farthest to the undefinable set.

Multigranulation rough set theory over two universes

Recently, a multigranulation rough set MGRS has become a new direction in rough set theory, which is based on multiple binary relations on the universe of discourse. The existing literature about multigranulation rough set is based on the assumption of the same universe. In reality, however, a good deal of practical decision making may relate to the possibility of two or more different universes. In this paper, we consider the rough approximation of a given concept over two different universes with respect to the multigranulation space formed by different mappings of the two universes, i.e., the multigranulation rough set model. We respectively define the optimistic multigranulation rough set, pessimistic multigranulation rough set and variable precision multigranulation rough set over two universes, each of which can be appropriate to a different real-world decision-making problem in management science. Then several important properties of these models are discussed in detail. Also, the relationship between the multigranulation rough set over two universes and the existing models in the literature is investigated. At last, the entropy and granularity of the binary mapping between two different universes are defined, and then we give an approach to uncertainty measurement based on the granularity of binary mapping for multigranulation rough set over two universes. The multigranulation rough set model over two universes provides a new, effective approach for practical decision problems in management science.

Incomplete Variable Multigranulation Rough Sets Decision

Recently, a multigranulation rough set (MGRS) has become a new direction in rough set theory, which is different from the Pawlake's rough set since the former takes multiple granulations on the un iverse into account. In this paper, by analyzing the limitations of optimistic multigranulation rough set (OMGRS) and pessimistic multigranulation rough set (PMGRS) in incomplete information system, the incomplete variable multigranulation rough set (VMGRS) is proposed, and the relationships among VMGRS, OMGRS and PMGRS are deeply explored, From the properties, it can be found that OMGRS and PMGRS are the special cases compared to our VMGRS. Furthermore, several important measurements are introduced into the VMGRS; it is shown that the measurements of the VMGRS are between the measurements of OMGRS and PMGRS. Finally, som e numerical examples are employed to substantiate the conceptual arguments.

Pessimistic rough set based decisions: A multigranulation fusion strategy

Multigranulation rough sets (MGRS) is one of desirable directions in rough set theory, in which lower/upper approximations are approximated by granular structures induced by multiple binary relations. It provides a new perspective for decision making analysis based on rough set theory. In decision making analysis, people often adopt the decision strategy “Seeking common ground while eliminating differences” (SCED). This strategy implies that one reserves common decisions while deleting inconsistent decisions. From this point of view, the objective of this study is to develop a new multigranulation rough set based decision model based on SCED strategy, called pessimistic multigranulation rough sets. We study this model from three aspects, which are lower/upper approximation and their properties, decision rules and attribute reduction, in this paper.

Multigranulation rough sets: From partition to covering

The classical multigranulation rough set (MGRS) theory offers a formal theoretical framework for solving the complex problem under multigranulation environment. However, it is noticeable that MGRS theory cannot be applied in multi-source information systems with a covering environment in the real world. To address this issue, we firstly present in this paper three types of covering based multigranulation rough sets, in which set approximations are defined by different covering approximation operators. Then, by using two different approximation strategies, i.e., seeking common reserving difference and seeking common rejecting difference, two kinds of covering based multigranulation rough set are presented, namely, a covering based optimistic multigranulation rough set and a covering based pessimistic multigranulation rough sets. Finally, we develop some properties and several uncertainty measures of the covering based multigranulation rough sets. These results will enrich the MGRS theory and enlarge its application scope.

Hierarchical Structures on Multigranulation Spaces

Though many hierarchical structures have been proposed to analyze the flner or coarser relationships between two granulation spaces, these structures can only be used to compare the single granulation spaces. However, it should be noticed that the concept of multigranulation plays a fundamental role in the development of granular computing. Therefore, the comparison between two multigranulation spaces has become a necessity. To solve such problem, two types of the multigranulation spaces are considered: one is the partition-based multigranulation space, the other is the covering-based multigranulation space. Three difierent hierarchical structures are then proposed on such two multigranulation spaces, respectively. Not only the properties about these hierarchical structures are discussed, but also the relationships between these hierarchical structures and the multigranulation rough sets are deeply investigated. It is shown that the flrst hierarchical structure is consistent with the monotonic varieties of optimistic multigranulation rough set, and the second hierarchical structure is consistent to the monotonic varieties of pessimistic multigranulation rough set, the third hierarchical structure is consistent to the monotonic varieties of both optimistic and pessimistic multigranulation rough sets.

On the structure of the multigranulation rough set model

The original rough set model, i.e., Pawlak’s single-granulation rough set model has been extended to a multigranulation rough set model, where two kinds of multigranulation approximations, i.e., the optimistic and pessimistic approximations were introduced. In this paper, we consider some fundamental properties of the multigranulation rough set model, and show that(i)Both the collection of lower definable sets and that of upper definable sets in the optimistic multigranulation rough set model can form a lattice, such lattices are not distributive, not complemented and pseudo-complemented in the general case. The collection of definable sets in the optimistic multigranulation rough set model does not even form a lattice in general conditions.(ii)The collection of (lower, upper) definable sets in the optimistic multigranulation rough set model forms a topology on the universe if and only the optimistic multigranulation rough set model is equivalent to Pawlak’s single-granulation rough set model.(iii)In the context of the pessimistic multigranulation rough set model, the collections of three different kinds of definable sets coincide with each other, and they determine a clopen topology on the universe, furthermore, they form a Boolean algebra under the usual set-theoretic operations.

On multigranulation rough sets in incomplete information system

Multigranulation rough set is a new and interesting topic in the theory of rough set. In this paper, the multigranulation rough sets approach is introduced into the incomplete information system. The tolerance relation, the similarity relation and the limited tolerance relations are employed to construct the optimistic and the pessimistic multigranulation rough sets, respectively. Not only the properties about these multigranulation rough sets are discussed, but also the relationships among these multigranulation rough sets models are explored. It is shown that by the multigranulation rough sets theory, the limited tolerance relations based multigranulation lower approximations fall between the tolerance and the similarity relations based multigranulation lower approximations, the limited tolerance relations based multigranulation upper approximations fall between the similarity and the tolerance relations based multigranulation upper approximations. Such results are consistent to those in single-granulation based rough sets models.