Lower Approximation Operators Research Articles

Data volume reduction in covering approximation spaces with respect to twenty-two types of covering based rough sets

In this paper, we investigate whether consistent mappings can be used as homomorphism mappings between a covering based approximation space and its image with respect to twenty-two pairs of covering upper and lower approximation operators. We also consider the problem of constructing such mappings and minimizing them. In addition, we investigate the problem of reducing the data volume using consistent mappings as well as the maximum amount of their compressibility. We also apply our algorithms against several datasets.

Algebraic and topological structures on factorizations of fuzzy sets1

In this paper, we introduce the notion of factorizations of fuzzy sets in complete residuated lattices. Moreover, we investigate fuzzy partially orders, the upper and lower approximation operators and Alexander (fuzzy) topologies on factorizations of fuzzy sets and give their examples.

Parametric Matroid of Rough Set

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

Rough Neutrosophic Multi-Attribute Decision-Making Based on Rough Accuracy Score Function

This paper presents multi-attribute decision making based on rough accuracy score function with rough neutrosophic attribute values. While the concept of neutrosophic sets is a powerful logicto handle indeterminate and inconsistent information, the theory of rough neutrosophic sets is also a powerful mathematical tool to deal with incompleteness. The rating of all alternatives is expressed with the upper and lower approximation operator and the pair of neutrosophic sets which are characterized by truth-membership degree, indeterminacy-membership degree, and falsity-membership degree. Weight of each attribute is partially known to decision-maker. We introduce a multi-attribute decision-making method in rough neutrosophic environment based on rough accuracy score function. Information entropy method is used to obtain the unknown attribute weights. Rough accuracy score function is defined to determine rough accuracy score values. Then weighted rough accuracy score value is defined to determine the ranking order of all alternatives. Finally, a numerical example is provided to illustrate the applicability and effectiveness of the proposed approach.

Rough approximations of multi-fuzzy sets

Rough sets have been invented to cope with uncertainty and to deal with intelligent systems characterised by insufficient and incomplete information. This paper proposes a general framework for the study of relation-based multi-fuzzy rough approximation operators. We define rough multi-fuzzy sets and study their properties. An arbitrary multi-fuzzy relation is used to approximate multi-fuzzy sets. A pair of lower and upper rough approximation operators is hence obtained and basic properties of these approximation operators will be examined. By introducing the cut sets of multi-fuzzy sets, we study the connections between multi-fuzzy rough approximation operators and crisp approximation operators. Finally, the multi-fuzzy approximation operators will be defined by axioms.

Parametric Rough Sets with Application to Granular Association Rule Mining

Granular association rules reveal patterns hidden in many-to-many relationships which are common in relational databases. In recommender systems, these rules are appropriate for cold-start recommendation, where a customer or a product has just entered the system. An example of such rules might be “40% men like at least 30% kinds of alcohol; 45% customers are men and 6% products are alcohol.” Mining such rules is a challenging problem due to pattern explosion. In this paper, we build a new type of parametric rough sets on two universes and propose an efficient rule mining algorithm based on the new model. Specifically, the model is deliberately defined such that the parameter corresponds to one threshold of rules. The algorithm benefits from the lower approximation operator in the new model. Experiments on two real-world data sets show that the new algorithm is significantly faster than an existing algorithm, and the performance of recommender systems is stable.

Properties of two types of covering-based rough sets

Rough set theory is a tool to deal with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations, or partitions. Recently, some sufficient and necessary conditions were given in Zhu (Proceedings of the first international workshop granular computing and brain informatics (Gr BI’06), IEEE international conference on web intelligence (WI 06), pp. 494–497, 2006), Zhu and Wang (Proceedings of the IEEE international conference on data mining (ICDM’06) workshop foundation of data mining and novel techniques in high dimensional structural and unstructured data, pp. 407–411, 2006), (IEEE Trans Knowl Data Eng, 19, pp 1131–1144, 2007, Proceedings of the sixth international conference on machine learning and cybernetics, 33), under which a lower approximation operator and an upper approximation operator satisfy certain classical properties. In this paper, we give a counterexample to show that the condition given in Theorem 13 in (Proceedings of the first international workshop granular computing and brain informatics (Gr BI’06), IEEE international conference on web intelligence (WI 06), pp. 494–497, 2006) is not necessary. The correct formulation is stated. We also give other conditions for a covering, under which certain classical properties hold for the second and third types of covering-based lower and upper approximation operators.

Rough set model based on formal concept analysis

This paper proposes a rough set model based on formal concept analysis. In this model, a solution to an algebraic structure problem is first provided in an information system: a lattice structure is inferred from the information system and corresponding nodes are called rough concepts. How to deal with common problems in rough set theory based on rough concepts is then explored, such as upper and lower approximation operators, reducts and cores. Decision dependency has become a common form of knowledge representation owing to its properties of expressiveness and ease of understanding, so it has been widely used in practice. Finally, application of rough concepts to the extraction of decision dependencies from a decision table is studied; a complete and non-redundant set of decision dependencies can be obtained from a decision table. Examples demonstrate that application of the method presented in this paper is valid and practicable. The results not only provide a better understanding of rough set theory from the perspective of formal concept analysis, but also demonstrate a new way of combining rough set theory and formal concept analysis.

Quantitative analysis for covering-based rough sets through the upper approximation number

Coverings are a useful form of data, while covering-based rough sets provide an effective tool for dealing with this data. Covering-based rough sets have been widely used in attribute reduction and rule extraction. However, few quantitative analyses for covering-based rough sets have been conducted, while many advances for classical rough sets have been obtained through quantitative tools. In this paper, the upper approximation number is defined as a measurement to quantify covering-based rough sets, and a pair of upper and lower approximation operators are constructed using the approximation number. The operators not only inherit some important properties of existing approximation operators, but also exhibit some new quantitative characteristics. It is interesting to note that the upper approximation number of a covering approximation space is similar to the dimension of a vector space or the rank of a matrix.

Algorithms and Algorithm Analysis of Logical Difference Operation of Variable Precision Lower Approximation Operator and Grade Upper Approximation Operator

This paper aims to construct new operation of approximation operators, and explore its calculation. First it proposes logical difference operation of variable precision lower approximation operator and grade upper approximation operator. Then regular algorithm and structural algorithm are proposed and analyzed, and furthermore, a conclusion is drawn that structural algorithm has advantages in time complexity and space complexity. Finally a practical example is given to illustrate the new operation and its algorithms.

Textural approach to generalized rough sets based on relations

This paper presents a discussion on rough set theory from the textural point of view. A texturing is a family of subsets of a given universal set U satisfying certain conditions which are generally basic properties of the power set. The suitable morphisms between texture spaces are given by direlations defined as pairs ( r , R ) where r is a relation and R is a corelation. It is observed that the presections are natural generalizations for rough sets; more precisely, if ( r , R ) is a complemented direlation, then the inverse of the relation r (the corelation R) is actually a lower approximation operator (an upper approximation operator).

FRSVMs: Fuzzy rough set based support vector machines

This paper aims to improve hard margin support vector machines (SVMs) by considering the membership of every training sample in constraints. The membership is computed by employing the technique of fuzzy rough sets so that hard margin SVMs can be combined with fuzzy rough sets and the inconsistence between conditional features and decision labels can be taken into account at the same time. In this paper, we first propose fuzzy transitive kernel based fuzzy rough sets. For binary classification, we use a lower approximation operator in fuzzy transitive kernel based fuzzy rough sets to compute the membership for every training input. And then we reformulate hard margin support vector machines into fuzzy rough set based SVMs (FRSVMs) with new constraints in which the membership is taken into account. Finally, comparisons with soft margin SVMs and fuzzy SVMs are made. The experimental results show that the proposed approach is feasible and valid. It significantly improved the performance of the hard margin SVMs.

A rough logic based on incomplete information and its application

This paper discusses a new rough logic based on incomplete information. Originally rough logic was constructed upon knowledge systems of complete information. In this paper, we consider incomplete knowledge representation systems in which information functions are nondeterministic. After modifying the usual definitions of upper and lower approximation operations in rough logic, we examine some properties of these operations. Also, we consider a relationship of the proposed logic to KTB modal logic. Then, we remark about an application of incomplete information systems to decision logic.