Multigranulation Rough Set Theory Research Articles

Multi-granulation rough approximations under normal distribution

Multi-granulation decision-theoretic rough set effectively combines Bayesian decision approaches with multi-granulation rough set theory, and provides an important theoretical framework for studying rough set. In this paper, we explore several extensional models of multi-granulation decision-theoretic rough sets under the normal distribution of the decision loss function. Using the 3σ rule of normal distribution, we transform the decision loss of the multi-granulation decision-theoretic rough set into a set of interval values. We construct the upper and lower approximations from the optimistic, weakly optimistic, pessimistic, weakly pessimistic, optimistic-pessimistic, weakly optimistic-pessimistic, pessimistic-optimistic, and weakly pessimistic-optimistic viewpoints, and provide the decision rules of the proposed rough set models. The work in this paper brings the decision behavior based on a multi-granulation decision-theoretic rough set closer to the actual situation.

Matrix Approaches for Covering-Based Multigranulation Fuzzy Rough Set Models

Multigranulation rough set theory is one of the most effective tools for data analysis and mining in multicriteria information systems. Six types of covering-based multigranulation fuzzy rough set (CMFRS) models have been constructed through fuzzy β -neighborhoods or multigranulation fuzzy measures. However, it is often time-consuming to compute these CMFRS models with a large fuzzy covering using set representation approaches. Hence, presenting novel methods to compute them quickly is our motivation for this paper. In this article, we study the matrix representations of CMFRS models to save time in data processing. Firstly, some new matrices and matrix operations are proposed. Then, matrix representations of optimistic CMFRSs are presented. Moreover, matrix approaches for computing pessimistic CMFRSs are also proposed. Finally, some experiments are proposed to illustrate the effectiveness of our approaches.

Rough set-based entropy measure with weighted density outlier detection method

Abstract The rough set theory is a powerful numerical model used to handle the impreciseness and ambiguity of data. Many existing multigranulation rough set models were derived from the multigranulation decision-theoretic rough set framework. The multigranulation rough set theory is very desirable in many practical applications such as high-dimensional knowledge discovery, distributional information systems, and multisource data processing. So far research works were carried out only for multigranulation rough sets in extraction, selection of features, reduction of data, decision rules, and pattern extraction. The proposed approach mainly focuses on anomaly detection in qualitative data with multiple granules. The approximations of the dataset will be derived through multiequivalence relation, and then, the rough set-based entropy measure with weighted density method is applied on every object and attribute. For detecting outliers, threshold value fixation is performed based on the estimated weight. The performance of the algorithm is evaluated and compared with existing outlier detection algorithms. Datasets such as breast cancer, chess, and car evaluation have been taken from the UCI repository to prove its efficiency and performance.

Some New Covering-Based Multigranulation Fuzzy Rough Sets and Corresponding Application in Multicriteria Decision Making

Multigranulation rough set theory is an important tool to deal with the problem of multicriteria information system. The notion of fuzzy β -neighborhood has been used to construct some covering-based multigranulation fuzzy rough set (CMFRS) models through multigranulation fuzzy measure. But the β -neighborhood has not been used in these models, which can be seen as the bridge of fuzzy covering-based rough sets and covering-based rough sets. In this paper, the new concept of multigranulation fuzzy neighborhood measure and some types of covering-based multigranulation fuzzy rough set (CMFRS) models based on it are proposed. They can be seen as the further combination of fuzzy sets: covering-based rough sets and multigranulation rough sets. Moreover, they are used to solve the problem of multicriteria decision making. Firstly, the definition of multigranulation fuzzy neighborhood measure is given based on the concept of β -neighborhood. Moreover, four types of CMFRS models are constructed, as well as their characteristics and relationships. Then, novel matrix representations of them are investigated, which can satisfy the need of knowledge discovery from large-scale covering information systems. The matrix representations can be more easily implemented than set representations by computers. Finally, we apply them to manage the problem of multicriteria group decision making (MCGDM) and compare them with other methods.

(α, β)-Multi-granulation bipolar fuzzified rough sets and their applications to multi criteria group decision making

Pawlak’s rough set theory based on single granulation has been extended to multi-granulation rough set structure in recent years. Multi-granulation rough set theory has become a flouring research direction in rough set theory. In this paper, we propose the notion of (α, β)-multi-granulation bipolar fuzzified rough set ((α, β)-MGBFRSs). For this purpose, a collection of bipolar fuzzy tolerance relations has been used. In the framework of multi-granulation, we proposed two types of (α, β)-multi-granulation bipolar fuzzified rough sets model. One is called the optimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) o-MGBFRSs) and the other is called the pessimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) p-MGBFRSs). Subsequently, a number of important structural properties and results of proposed models are investigated in detail. The relationships among the (α, β)-MGBFRSs, (α, β) o-MGBFRSs and (α, β) p-MGBFRSs are also established. In order to illustrate our proposed models, some examples are considered, which are helpful for applying this theory in practical issues. Moreover, several important measures associated with (α, β)-multi-granulation bipolar fuzzified rough set like the measure of accuracy, the measure of precision, and accuracy of approximation are presented. Finally, we construct a new approach to multi-criteria group decision-making method based on (α, β)-MGBFRSs, and the validity of this technique is illustrated by a practical application. Compared with the existing results, we also expound its advantages.

A dynamic approach for updating the lower approximation in adjustable multi-granulation rough sets

Granular computing ( $$ GrC $$ ) is one of the key issues in the field of information sciences. Research on the theory and algorithms of granular computing has very important practical significance in huge amounts of information. In multi-granulation rough set theory, two subsets are calculated to approximate the target concept, which are extremely time-consuming for large-scale data. In this paper, to address the issue above, we propose efficient algorithms for updating the lower approximation when a single object is added into or deleted from the target concept in an incomplete information system. Firstly, adjustable multi-granulation rough sets ( $$ AMGRSs $$ ) are introduced in an incomplete information system, and the related properties and theorems are explored. Secondly, it is proved that local- $$ AMGRSs $$ and $$ AMGRSs $$ are equivalent in an incomplete information system. Finally, dynamic algorithms for updating the lower approximation are proposed, and the efficiency of these algorithms is verified by an experiment.

Type-2 fuzzy multigranulation rough sets

Aiming at expanding the application range of the multigranulation rough set (MGRS) theory, a type-2 fuzzy multigranulation rough set (T2FMGRS) model is proposed by combining type-2 fuzzy sets with multigranulation rough sets (MGRSs) in this paper. At first, definitions and properties of optimistic and pessimistic type-2 fuzzy multigranulation rough sets (T2FMGRSs) are introduced. Then, the rough measure is proposed to measure the uncertainty of T2FMGRSs. Finally, T2FMGRSs over two universes are put forward and some examples of decision-making are given to illustrate the applicability of the newly proposed model. In summary, the establishment of T2FMGRSs is a meaningful generalization of the MGRS theory from both theoretical and practical aspects.

Incremental approaches to update multigranulation approximations for dynamic information systems

Multigranulation rough set (MGRS) theory provides an effective manner for the problem solving by making use of multiple equivalence relations. As the information systems always dynamically change over time due to the addition or deletion of multiple objects, how to efficiently update the approximations in multigranulation spaces by making fully utilize the previous results becomes a crucial challenge. Incremental learning provides an efficient manner because of the incorporation of both the current information and previously obtained knowledge. In spite of the success of incremental learning, well-studied findings performed to update approximations in multigranulation spaces have relatively been scarce. To address this issue, in this paper, we propose matrix-based incremental approaches for updating approximations from the perspective of multigranulation when multiple objects vary over time. Based on the matrix characterization of multigranulation approximations, the incremental mechanisms for relevant matrices are systematically investigated while adding or deleting multiple objects. Subsequently, in accordance with the incremental mechanisms, the corresponding incremental algorithms for maintaining multigranulation approximations are developed to reduce the redundant computations. Finally, extensive experiments on eight datasets available from the University of California at Irvine (UCI) are conducted to verify the effectiveness and efficiency of the proposed incremental algorithms in comparison with the existing non-incremental algorithm.

Relative relation matrix-based approaches for updating approximations in multigranulation rough sets

Multigranulation rough set (MGRS) theory has attracted much attention. However, with the advent of big data era, the attribute values may often change dynamically, which leads to high computational complexity when handling large and complex data. How to effectively obtain useful knowledge from the dynamic information system becomes an important issue in MGRS. Motivated by this requirement, in this paper, we propose relative relation matrix approaches for computing approximations in MGRS and updating them dynamically. A simplified relative relation matrix is used to calculate approximations in MGRS, it is showed that the space and time complexities are no more than that of the original method. Furthermore, relative relation matrix-based approaches for updating approximations in MGRS while refining or coarsening attribute values are proposed. Several incremental algorithms for updating approximations in MGRS are designed. Finally, experiments are conducted to evaluate the efficiency and validity of the proposed methods.

Reduction foundation with multigranulation rough sets using discernibility

When multiple granulated knowledge in multigranulation spaces are involved in decision making, protocol principles are adopted to arrive at the final consensus. Multigranulation rough set theory utilizes a voting principle to combine the decision options derived from individual granulated knowledge. Note that those knowledge may provide different degrees of support to the final results, some are key, some are of less importance and some are even of no use. Selecting valuable knowledge and reducing worthless one are thus necessary for data processing, which can alleviate the storage occupancy and facilitate the logical and statistical analysis. However, the basic reduction foundation of multigranulation spaces has been rarely touched by researchers, which brings in many difficulties in algorithmic and real applications. This work aims to disclose the principles of multiple knowledge reduction in multigranulation spaces from the viewpoint of discernibility. First, the notions of knowledge reduction of multigranulation spaces are defined based on multigranulation rough set theory. Second, a decision function mapping each object into the decision options of its neighborhood granule is introduced. Third, several pairs of discernibility matrices and discernibility functions are successively developed using the decision function. We claim that the valuable and worthless knowledge in multigranulation spaces can be explicitly chose and eliminated respectively by using the proposed discernibility matrices and discernibility functions. That is to say, these discernibility tools provide a precise criterion for the knowledge reduction of multigranulation spaces. As a theoretical extension, a multigranulation information entropy is proposed and an approximate algorithm is constructed to compute a suboptimal reduct of a multigranulation space based on this entropy. In the end, numerical experiments are performed on public data sets to verify the effectiveness of the proposed reduction methods. This study can get us a grasp of the foundational principle of knowledge reduction and may bring a new insight for the designation of substantial reduction algorithms of multigranulation knowledge.

Variable precision multigranulation rough fuzzy set approach to multiple attribute group decision-making based on λ-similarity relation

Abstract This study proposes a fuzzy multigranulation rough set approach to the problem of multiple attribute group decision-making with uncertainty. Based on the classical Pawlak rough set theory, we define the λ - similarity ( 0 ⩽ λ ≤ 1 ) relation classes over the universe of discourse by introducing a distance measure to all alternatives with respect to attribute set. Subsequently, we present the α ( 0.5 α ≤ 1 ) rough approximation of a crisp decision-making object and a fuzzy decision-making object under the framework of multigranulation rough set theory, respectively. That is, we establish the variable precision multigranulation rough set model and variable precision multigranulation rough fuzzy set model based on λ - similarity relation, respectively. Meanwhile, we discuss the interrelationship between the proposed multigranulation rough fuzzy set model and the existing generalized rough set models. After that, we construct a new approach to multiple attribute group decision-making problems based on variable precision multigranulation rough fuzzy set theory. The decision-making procedure and the methodology as well as the algorithm of the proposed method are given and a detailed comparison of the traditional methods to multiple attribute group decision-making problems illustrates the advantages and limitations. Finally, an example of handling multiple criteria group decision-making problem of evaluation of emergency plans for unconventional emergency events illustrates this approach. The main contribution of this paper is twofold. One is to provide a new way to construct multigranulation rough set model with the fuzzy environment. Another is to try making a new way to handle multiple criteria group decision-making problems based on generalized rough set theory and methodologies.

The comparative study of covering rough sets and multi-granulation rough sets

The covering rough set (CRS) theory and the multi-granulation rough set (MGRS) theory are both the important generalizations of Pawlak rough set theory. Up to now, substantial contributions have been made to the development of CRS and MGRS. In this paper, in order to shed some light on the comparison and combination of CRS theory and MGRS theory, we investigate the relationship between CRS and MGRS based on different aspects. We firstly put forward an effective approach to describe the covering rough sets by means of the multi-granulation rough sets. Then, we, respectively, study the differences and relations of lower and upper operators, reduction, operation properties and algebraic properties between CRS and MGRS.

Information fusion and numerical characterization of a multi-source information system

The existing research of multi-source information system, pessimistic or optimistic multi-granulation fusion functions, provided by multi-granulation rough set (MGRS) theory, which apply disjunctive and conjunctive operators of multiple binary relations to aggregate multiple granular structures induced by different binary relations, are too relax or too restrictive to solve the practical problems. In this paper, we employ evidence theory, probability theory and information entropy to address the information fusion and numerical characterization of uncertain data in a multi-source information system (MSIS). First, we propose novel definitions of multi-source rough approximations and corresponding multi-granulation rough approximations, probability distribution and basic probability assignment, which can be used to construct the connection between rough approximations and evidence theory. Second, the above ideas are extended to multi-source covering information system (MCIS). Finally, Shannon’s fusion algorithm based on equivalence relations or coverings, involved in the significance degree of condition attributes set with respect to a sample, conditional probability and information entropy, is presented to calculate the uncertainty degree of a decision, respectively. Then, based on the defined conditional probability in this paper, we design a multi-granulation variable precision rough set and consider the relationship with MGRS. And, the illustrative examples are given to elaborate the operation mechanism of the above conclusions. This study will be helpful for integrating the uncertain information come from multiple sources and eventful for creating a route of granular computing.

New and improved: grey multi-granulation rough sets

ABSTRACTRough set has drawn great attention in recent decades, among which multi-granulation rough set (MGRS) is an arresting direction. It constructs a formal theoretical framework to solve complex problems under the circumstance of multiple binary relations. However, the fusion of multi-granulation rough set and grey system for acquiring knowledge is still a gap. Toward this end, we devise a grey multi-granulation rough set (GMGRS) by taking multiple grey relational relations into consideration under the framework of MGRS. In grey information system, the constructed grey relational relation that measures the relationship among objects can be used to further establish multiple binary relations. Based on two different approximate strategies (seeking common reserving difference and seeking common eliminating difference), two types of GMGRS are presented, respectively. After discussing several important properties of GMGRS, we discover that the properties of the proposed GMGRS are synchronous with the classical MGRS. Meanwhile, to obtain the attribute reduction under GMGRS, we reconstruct significance measure and termination criterion based on the θ-precision pessimistic GMGRS. Last but not least, theoretical studies and practical examples demonstrate that our proposed GMGRS largely enrich the MGRS theory and provide a new technique for knowledge discovery, which is practical in real-world scenarios.

Generalized multigranulation rough sets and optimal granularity selection

Multigranulation rough set theory is a desirable direction in the field of rough set, in which upper and lower approximations are approximated by multiple granular structures. However, classic multigranulation rough set is studied from two kinds of qualitative combination rules which were generated by pessimistic and optimistic viewpoints, respectively. The two combination rules seem to lack of practicability since one is too restrictive and the other too relaxed. To overcome this disadvantage, we propose a generalized multigranulation rough set model in this paper. First, we discuss upper and lower approximation sets of a generalized multigranulation rough set by introducing a support characteristic function and an information level. Then, as one of the most important problems in granular computing, we carefully study how to select optimal granularity in generalized multigranulation rough sets. Furthermore, algorithms of optimal granularity selection are constructed, by which we can provide an efficient approach to compute the optimal granularity based on generalized multigranulation rough sets. Finally, an illustrative example is given to show the effectiveness of the proposed approach. The main contribution of this paper is to construct the model of the optimal particle size selection on account of the generalized multi granularity, and overcome the limitation of the classical multi granularity.

A multiple-valued logic approach for multigranulation rough set model

Rough sets have often been studied under a three-valued logic framework. In this paper, we attempt to extend the previous study in two ways. Firstly, we extend the previous study from single-granulation to multigranulation. Secondly, we study multigranulation rough set theory from the viewpoint of three-way decision. More precisely, we embody the idea of three-way decision theory in the definition of multigranulation rough set theory. This leads to an axiomatic definition of decision-oriented aggregation operators on 3={0,12,1}, which are quite different from those conjunctions proposed so far. Moreover, considering that a multigranulation rough set also divides the universe into five disjoint subsets, we present a five-valued semantics for multigranulation rough set model, and a kind of non-deterministic matrices is thus given.

Generalized multigranulation double-quantitative decision-theoretic rough set

The principle of the minority subordinate to the majority is the most feasible and credible when people make decisions in real world. So generalized multigranulation rough set theory is a desirable fusion method, in which upper and lower approximations are approximated by granular structures satisfying a certain level of information. However, the relationship between a equivalence class and a concept under each granular structure is very strict. Therefore, more attention are paid to fault tolerance capabilities of double-quantitative rough set theory and the feasibility of majority principle. By considering relative and absolute quantitative information between the class and concept, we propose two kinds of generalized multigranulation double-quantitative decision-theoretic rough sets(GMDq-DTRS). Firstly, we define upper and lower approximations of generalized multigranulation double-quantitative rough sets by introducing upper and lower support characteristic functions. Then, important properties of two kinds of GMDq-DTRS models are explored and corresponding decision rules are given. Moreover, internal relations between the two models under certain constraints and GMDq-DTRS and other models are explored. The definition of the approximation accuracy in GMDq-DTRS is proposed to show the advantage of GMDq-DTRS. Finally, an illustrative case is shown to elaborate the theories advantage of GMDq-DTRS which are valuable to deal with practical problems. Generalized multigranulation double-quantitative decision-theoretic rough set theory will be more feasible when making decisions in real life.

Evidence-theory-based numerical characterization of multigranulation rough sets in incomplete information systems

Multigranulation rough sets are desirable features in the field of rough set, where this concept is approximated by multiple granular structures. In this study, we employ belief and plausibility functions from evidence theory to characterize the set approximations and attribute reductions in multigranulation rough set theory. First, we show that in an incomplete information system, the pessimistic multigranulation approximations can be measured by belief and plausibility functions, whereas the optimistic multigranulation approximations do not possess this characteristic in general. We also give a sufficient and necessary condition for the numerical measurement of optimistic multigranulation approximations by belief and plausibility functions. Second, in an incomplete decision system, the pessimistic multigranulation approximations are also measured by belief and plausibility functions. In the end, an attribute reduction algorithm for multigranulation rough sets is proposed based on evidence theory, and its efficiency is examined by an example. Thus, belief and plausibility functions can be employed to numerically characterize the attribute reductions and to construct an attribute reduction algorithm for multigranulation rough sets.

Uncertainty Measures for Multigranulation Approximation Space

Multigranulation rough set theory is a relatively new mathematical tool for solving complex problems in the multigranulation or distributed circumstances which are characterized by vagueness and uncertainty. In this paper, we first introduce the multigranulation approximation space. According to the idea of fusing uncertain, imprecise information, we then present three uncertainty measures: fusing information entropy, fusing rough entropy, and fusing knowledge granulation in the multigranulation approximation space. Furthermore, several essential properties (equivalence, maximum, minimum) are examined and the relationship between the fusion information entropy and the fusion rough entropy is also established. Finally, we prove these three measures are monotonously increasing as the partitions become finer. These results will be helpful for understanding the essence of uncertainty measures in multigranulation rough space and enriching multigranulation rough set theory.

An information fusion approach by combining multigranulation rough sets and evidence theory

Multigranulation rough set (MGRS) theory provides two kinds of qualitative combination rules that are generated by optimistic and pessimistic multigranulation fusion functions. They are used to aggregate multiple granular structures from a set theoretic standpoint. However, the two combination rules seem to lack robustness because one is too relaxed and the other too restrictive to solve some practical problems. Dempster’s combination rule in the evidence theory has been employed to aggregate information coming from multiple sources. However, it fails to deal with conflict evidence. To overcome these limitations, we focus on the combination of granular structures with both reliability and conflict from multiple sources, which has been a challenging task in the field of granular computing. We first address the connection between multigranulation rough set theory and the evidence theory. Then, a two-grade fusion approach involved in the evidence theory and multigranulation rough set theory is proposed, which is based on a well-defined distance function among granulation structures. Finally, an illustrative example is given to show the effectiveness of the proposed fusion method. The results of this study will be useful for pooling the uncertain data from different sources and significant for establishing a new direction of granular computing.