Topological approaches to covering rough sets

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper studies covering-based rough sets from the topological view. We explore the topological properties of this type of rough sets, study the interdependency between the lower and the upper approximation operations, and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation. Lastly, axiomatic systems for the lower approximation operation and the upper approximation operation are constructed.     

Generalized rough sets based on relations

Rough set theory has been proposed by Pawlak as a tool for dealing with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations. This paper studies arbitrary binary relation based generalized rough sets. In this setting, a binary relation can generate a lower approximation operation and an upper approximation operation, but some of common properties of classical lower and upper approximation operations are no longer satisfied. We investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations.This paper also explores the relationships between the lower or the upper approximation operation generated by the intersection of two binary relations and those generated by these two binary relations, respectively. Through these relationships, we prove that two different binary relations will certainly generate two different lower approximation operations and two different upper approximation operations.     

The further investigation of covering-based rough sets: Uncertainty characterization, similarity measure and generalized models

The notion of rough sets was originally proposed by Pawlak. In Pawlak’s rough set theory, the equivalence relation or partition plays an important role. However, the equivalence relation or partition is restrictive for many applications because it can only deal with complete information systems. This limits the theory’s application to a certain extent. Therefore covering-based rough sets are derived by replacing the partitions of a universe with its coverings. This paper focuses on the further investigation of covering-based rough sets. Firstly, we discuss the uncertainty of covering in the covering approximation space, and show that it can be characterized by rough entropy and the granulation of covering. Secondly, since it is necessary to measure the similarity between covering rough sets in practical applications such as pattern recognition, image processing and fuzzy reasoning, we present an approach which measures these similarities using a triangular norm. We show that in a covering approximation space, a triangular norm can induce an inclusion degree, and that the similarity measure between covering rough sets can be given according to this triangular norm and inclusion degree. Thirdly, two generalized covering-based rough set models are proposed, and we employ practical examples to illustrate their applications. Finally, relationships between the proposed covering-based rough set models and the existing rough set models are also made.     

Soft rough fuzzy sets and soft fuzzy rough sets

Fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the further discussion of the combinations of fuzzy set, rough set and soft set. A new soft rough set model is proposed and its properties are derived. Furthermore, fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established. The lower and upper approximation operators are presented and their related properties are surveyed.     

Invertible approximation operators of generalized rough sets and fuzzy rough sets

This paper studies the classes of rough sets and fuzzy rough sets. We discuss the invertible lower and upper approximations and present the necessary and sufficient conditions for the lower approximation to coincide with the upper approximation in both rough sets and fuzzy rough sets. We also study the mathematical properties of a fuzzy rough set induced by a cyclic fuzzy relation.     

Soft sets and soft rough sets

In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak’s rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak’s rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given.     

对模糊集、粗糙集和Vague集的比较研究

模糊集、粗糙集和Vague集三种理论都是对经典集合理论的扩展,使得集合论的应用扩展到了含糊的、不确定性的问题领域。介绍了三种集合的基本思想,重点分析三种理论的区别和内在联系,同时对三种理论的发展及应用作了一些探讨性研究。     

Tabu search for attribute reduction in rough set theory

In this paper, we consider a memory-based heuristic of tabu search to solve the attribute reduction problem in rough set theory. The proposed method, called tabu search attribute reduction (TSAR), is a high-level TS with long-term memory. Therefore, TSAR invokes diversification and intensification search schemes besides the TS neighborhood search methodology. TSAR shows promising and competitive performance compared with some other CI tools in terms of solution qualities. Moreover, TSAR shows a superior performance in saving the computational costs.     

粒度计算及其在数据挖掘中的应用

粒度计算涵盖了所有在处理问题过程中使用粒度的理论、方法、技术和工具。本文首先简要地介绍了粒度计算的基本思想、基本问题以及它的三个主要模型(模糊集、粗糙集和商空间),然后综述了粒度计算在数据挖掘中的应用。     

Adaptive multilevel rough entropy evolutionary thresholding

In this study, comprehensive research into rough set entropy-based thresholding image segmentation techniques has been performed producing new and robust algorithmic schemes. Segmentation is the low-level image transformation routine that partitions an input image into distinct disjoint and homogenous regions using thresholding algorithms most often applied in practical situations, especially when there is pressing need for algorithm implementation simplicity, high segmentation quality, and robustness. Combining entropy-based thresholding with rough set results in the rough entropy thresholding algorithm.The authors propose a new algorithm based on granular multilevel rough entropy evolutionary thresholding that operates on a multilevel domain. The MRET algorithm performance has been compared to the iterative RET algorithm and standard k-means clustering methods on the basis of β-index as a representative validation measure. Performance in experimental assessment suggests that granular multilevel rough entropy threshold based segmentations - MRET - present high quality, comparable with and often better than k-means clustering based segmentations. In this context, the rough entropy evolutionary thresholding MRET algorithm is suitable for specific segmentation tasks, when seeking solutions that incorporate spatial data features with particular characteristics.     

A new extension of fuzzy sets using rough sets: R-fuzzy sets

This paper presents a new extension of fuzzy sets: R-fuzzy sets. The membership of an element of a R-fuzzy set is represented as a rough set. This new extension facilitates the representation of an uncertain fuzzy membership with a rough approximation. Based on our definition of R-fuzzy sets and their operations, the relationships between R-fuzzy sets and other fuzzy sets are discussed and some examples are provided.     

基于T模剩余蕴涵算子的直觉模糊粗糙集

构造并系统研究了直觉模糊T模的剩余蕴涵。在此基础上,推导出了直觉模糊粗糙集的一种构造模型,证明了Pawlak粗糙集、直觉模糊集、模糊粗糙集、粗糙模糊集及模糊T粗糙集都是直觉模糊粗糙集的特殊情形。最后给出并证明了直觉模糊粗糙集的一些性质。     

Approximations and reducts with covering generalized rough sets

The covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle. It is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed. The first one is to present reasonable definitions of set approximations, and the second one is to develop reasonable algorithms for attributes reduct. The existing covering generalized rough sets, however, mainly pay attention to constructing approximation operators. The ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable. Furthermore, less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets. In this paper we concentrate our discussion on the above two issues. We first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones. Then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets. With these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications.     

A Note on Z-numbers

Decisions are based on information. To be useful, information must be reliable. Basically, the concept of a Z-number relates to the issue of reliability of information. A Z-number, Z, has two components, Z = (A, B). The first component, A, is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, B, is a measure of reliability (certainty) of the first component. Typically, A and B are described in a natural language. Example: (about 45 min, very sure). An important issue relates to computation with Z-numbers. Examples: What is the sum of (about 45 min, very sure) and (about 30 min, sure)? What is the square root of (approximately 100, likely)? Computation with Z-numbers falls within the province of Computing with Words (CW or CWW). In this note, the concept of a Z-number is introduced and methods of computation with Z-numbers are outlined. The concept of a Z-number has a potential for many applications, especially in the realms of economics, decision analysis, risk assessment, prediction, anticipation and rule-based characterization of imprecise functions and relations.     

基于直觉模糊粗糙集的属性约简

针对Jensen下近似定义的局限性,提出一种新的等价类形式的近似算子表示,并将其推广到直觉模糊环境.在此基础上,将相对正域、相对约简、相对核等粗糙集的知识约简概念推广到直觉模糊环境,提出一种直觉模糊信息系统的启发式属性约筒算法.实例计算表明.该方法比Jensen的属性约简方法更为合理有效.     

An efficient accelerator for attribute reduction from incomplete data in rough set framework

Feature selection (attribute reduction) from large-scale incomplete data is a challenging problem in areas such as pattern recognition, machine learning and data mining. In rough set theory, feature selection from incomplete data aims to retain the discriminatory power of original features. To address this issue, many feature selection algorithms have been proposed, however, these algorithms are often computationally time-consuming. To overcome this shortcoming, we introduce in this paper a theoretic framework based on rough set theory, which is called positive approximation and can be used to accelerate a heuristic process for feature selection from incomplete data. As an application of the proposed accelerator, a general feature selection algorithm is designed. By integrating the accelerator into a heuristic algorithm, we obtain several modified representative heuristic feature selection algorithms in rough set theory. Experiments show that these modified algorithms outperform their original counterparts. It is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets.     

A generalized model of fuzzy rough sets

The consideration of approximation problem of fuzzy sets in fuzzy information systems results in theory of fuzzy rough sets. This paper focuses on models of generalized fuzzy rough sets, a generalized model of fuzzy rough sets based on general fuzzy relations are studied, properties and algebraic characterization of the model are revealed, and relationships between this model and related models are also discussed.     

Fuzzy preference based rough sets

Preference analysis is an important task in multi-criteria decision making. The rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations. The existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria. In this paper, we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria. Fuzzy preference relations are incorporated into a fuzzy rough set model, which leads to a fuzzy preference based rough set model. The measure of attribute dependency of the Pawlak’s rough set model is generalized to compute the relevance between criteria and decisions. The definitions of upward dependency, downward dependency and global dependency are introduced. Algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets.     

On fuzzy rough sets based on tolerance relations

Dubois and Prade (1990) [1] introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets, which was originally proposed by Pawlak (1982) [8]. Later, Radzikowska and Kerre introduced the so-called (I,T)-fuzzy rough sets, where I is an implication and T is a triangular norm. In the present paper, by using a pair of implications (I,J), we define the so-called (I,J)-fuzzy rough sets, which generalize the concept of fuzzy rough sets in the sense of Radzikowska and Kerre, and that of Mi and Zhang. Basic properties of (I,J)-fuzzy rough sets are investigated in detail.