Jian-Min Ma

Ranking interval sets based on inclusion measures and applications to three-way decisions

Three-way decisions provide an approach to obtain a ternary classification of the universe as acceptance region, rejection region and uncertainty region respectively. Interval set theory is a new tool for representing partially known concepts, especially it corresponds to a three-way decision. This paper proposes a framework for comparing two interval sets by inclusion measures. Firstly, we review the basic notations, interpretation and operation of interval sets and classify the orders on interval sets into partial order, preorder and quasi-order. Secondly, we define inclusion measure which indicates the degree to which one interval set is less than another one and construct different inclusion measures to present the quantitative ranking of interval sets. Furthermore, we present similarity measures and distances of interval sets and investigate their relationship with inclusion measures. In addition, we propose the fuzziness measure and ambiguity measure to show the uncertainty embedded in an interval set. Lastly, we study the application of inclusion measures, similarity measures and uncertainty measures of interval sets by a special case of three-way decisions: rough set model and the results show that these measures are efficient to three-way decision processing.

Attribute reductions in object-oriented concept lattices

Attribute reduction is one of the main issues in the study of concept lattice. This paper mainly deals with attribute reductions of an object-oriented concept lattice constructed on the basis of rough set. Attribute rank of object-oriented concept lattice is first defined, and relationships between attribute rank and object-oriented concepts are then discussed. Based on attribute rank, generating algorithm of object-oriented concepts is investigated. The object-oriented consistent set and object-oriented reduction of an object-oriented concept lattice are defined. Adjustment theorems of the object-oriented consistent set, and the necessary and sufficient conditions for a attribute subset to be an object-oriented consistent set of an object-oriented concept lattice are discussed. Then the object-oriented discernibility matrix of an object-oriented concept lattice is defined and its properties are also studied. Based on the object-oriented discernibility matrix, an approach to object-oriented reductions of an object-oriented concept lattice is proposed, and the attribute characteristics are also analyzed.

Axiomatic characterizations of dual concept lattices

Formal concept analysis is an algebraic model based on a Galois connection. It is used for symbolic knowledge exploration from an elementary form of a formal context. This paper mainly presents a general framework for concept lattice in which axiomatic approaches are used. The relationship between concept lattice and dual concept lattice is first studied. Based on set-theoretic operators, generalized concept systems are established. And properties of them are examined. By using axiomatic approaches, a pair of dual concept lattices is characterized by different sets of axioms. The connections between 0-1 binary relations and generalized concept systems are examined. And generalized dual concept systems can be constructed by a pair of dual set-theoretic operators. Axiomatic characterizations of the generalized concept systems guarantee the existence of a binary relation producing a formal context.

Dependence-space-based attribute reductions in inconsistent decision information systems

In rough set theory, attribute reduction is an important mechanism for knowledge discovery. This paper mainly deals with attribute reductions of an inconsistent decision information system based on a dependence space. Through the concept of inclusion degree, a generalized decision distribution function is first constructed. A decision distribution relation is then defined. On the basis of this decision distribution relation, a dependence space is proposed, and an equivalence congruence based on the indiscernibility attribute sets is also obtained. Applying the congruences on a dependence space, new approaches to find a distribution consistent set are formulated. The judgement theorems for judging distribution consistent sets are also established by using these congruences and the decision distribution relation.

Methods and Practices of Three-Way Decisions for Complex Problem Solving

A theory of three-way decisions is formulated based on the notions of three regions and associated actions for processing the three regions. Three-way decisions play a key role in everyday decision-making and have been widely used in many fields and disciplines. A group of Chinese researchers further investigated the theory of three-way decision and applied it in different domains. Their research results are highlighted in an edited Chinese book entitled “Three-way Decisions: Methods and Practices for Complex Problem Solving.” Based on the contributed chapters of the edited book, this paper introduces and reviews most recent studies on three-way decisions.

Local rough set: A solution to rough data analysis in big data

Abstract As a supervised learning method, classical rough set theory often requires a large amount of labeled data, in which concept approximation and attribute reduction are two key issues. With the advent of the age of big data however, labeling data is an expensive and laborious task and sometimes even infeasible, while unlabeled data are cheap and easy to collect. Hence, techniques for rough data analysis in big data using a semi-supervised approach, with limited labeled data, are desirable. Although many concept approximation and attribute reduction algorithms have been proposed in the classical rough set theory, quite often, these methods are unable to work well in the context of limited labeled big data. The challenges to classical rough set theory can be summarized with three issues: limited labeled property of big data, computational inefficiency and over-fitting in attribute reduction. To address these three challenges, we introduce a theoretic framework called local rough set, and develop a series of corresponding concept approximation and attribute reduction algorithms with linear time complexity, which can efficiently and effectively work in limited labeled big data. Theoretical analysis and experimental results show that each of the algorithms in the local rough set significantly outperforms its original counterpart in classical rough set theory. It is worth noting that the performances of the algorithms in the local rough set become more significant when dealing with larger data sets.

Knowledge reduction of dynamic covering decision information systems caused by variations of attribute values

In practical situations, it is time-consuming to conduct knowledge reduction of dynamic covering decision information systems caused by variations of attribute values with the non-incremental approaches. In this paper, motivated by the need for knowledge reduction of dynamic covering decision information systems, we introduce incremental approaches to computing the type-1 and type-2 characteristic matrices for constructing the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces caused by revising attribute attributes. We also employ several examples to explain how to compute the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Then we propose the incremental algorithms for computing the second and sixth lower and upper approximations of sets and employ experimental results to illustrate the incremental algorithms are effective to calculate the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Finally, we give two examples to show how to conduct knowledge reduction of dynamic covering decision information systems caused by altering attribute values.

Concept acquisition approach of object-oriented concept lattices

Formal concept analysis is an effective tool for data analysis and knowledge discovery. Corresponding to concept lattice in a formal context, object-oriented concept lattice is introduced based on rough set. Obtaining object-oriented concepts is important but difficult because of the higher time complexity. In order to solve this question, we first divide the power set of the attribute set into the layered sets in this paper. Since for any object-oriented concept, the object-oriented concept extension and object-oriented concept intension determine each other uniquely, we introduce the layered extension sets. By discussing the properties of layered extension sets, the approach to acquire object-oriented concepts is investigated, and related concept acquirement algorithm is also depicted. Examples prove that the concept acquirement approach is valid.

Structured probabilistic rough set approximations

Abstract Probabilistic rough set approximations are proposed based on a conditional probability to describe the desired levels of precision between the equivalence classes and an approximated set. This definition shows the detailed information on individuals satisfying some conditions but ignores the structural information. In this paper, applying the structured granules in a coarsened-grained universe, we introduce structured probabilistic rough set approximations between the power sets of the original universe and the coarsened-grained universe. By using the zooming-in and structured probabilistic rough approximation operators, two pairs of probabilistic rough lower and upper approximations on the same universe are investigated. Related properties and relationships of them are investigated. Furthermore, applying the Bayesian decision procedure, conditional probability and loss functions, three-way classifications in structured probabilistic rough set approximations are then proposed to classify the structured granules of the coarsened-grained universe into three disjoint structured probabilistic regions. This method gives the values of the pair of thresholds. Meanwhile, by using the minimum-risk decision rules, we also can construct the structured probabilistic rough lower and upper approximations. Finally, we discuss the monotonicity of structured probabilistic positive and negative regions.

Dependence Space-Based Model for Constructing Dual Concept Lattice in Sub-formal Context

This paper mainly studies a dependence space-based model for constructing dual concept lattice in sub-formal context. Based on the operators of a dual concept lattice , a ∩ −congruence on the power set of objects is defined, and ∩ −dependence space is then obtained. By the ∩ −congruence, dual concept granules and an inner operator decided by any subset of attributes are introduced. It is proved that each open element of the inner operator is just the minimal element in an dual concept granule. Furthermore, the open element is also the extension of a dual concept lattice.