Three-way Concept Analysis Research Articles

An efficient method of renewing object-induced three-way concept lattices involving decreasing attribute-granularity levels

In three-way concept analysis, changing (decreasing or increasing) attribute-granularity levels is needed to seek desirable information. Reconstructing three-way concept lattices often requires huge computation and long elapsed time when attribute-granularity levels are changed. To avoid this problem, a good strategy is indirectly renewing three-way concept lattices. Our paper studies how to renew object-induced three-way concept lattices involving decreasing attribute-granularity levels. Firstly, we analyze changes of object-induced three-way concept lattices when attribute-granularity levels are decreased. To classify changes of object-induced three-way concepts, we classify these concepts into six categories, derive sufficient and necessary conditions of identifying these categories, and investigate their properties. To explore changes of covering relations among object-induced three-way concepts, we classify covering relations into three categories, and identify them by finding which are the destructors of deleted object-induced three-way concepts before the decrease, and analyzing which are children concepts of object-induced three-way concepts as destructors after the decrease. Secondly, by using the above analysis results, we put forward a novel algorithm called OEL-Collapse to renew object-induced three-way concept lattices when attribute-granularity levels are decreased. Finally, experiments are conducted to illustrate the efficiency of the OEL-Collapse algorithm.

A novel method for generating a canonical basis for decision implications based on object-induced three-way operators

Three-way concept analysis can express not only the information that the objects of study have in common but also the information that they do not have in common, which is an extension of classical formal concept analysis. The decision implication is a special representation of the implication in the decision formal context, which reveals the dependency between conditional attributes and decision attributes. This paper is devoted to the study of a novel method for generating a canonical basis for decision implications based on object-induced operators (OE operators). First, the decision implication logic based on OE operators is described from the two aspects of semantics and syntax, which is applied to the decision formal context. Then, a method of obtaining the canonical basis for decision implications based on decision the premise of the OE operators is given, and the completeness, the nonredundancy and the optimality of the canonical basis are proven. Finally, a method of generating true premises based on OE operators is proposed by means of an incremental method, and the canonical basis for decision implications is then generated by the true premises based on OE operators.

Adjunctive three-way concepts from positive and negative concepts in lattice-valued formal contexts

Three-way concept analysis is an integration of formal concept analysis and three-way decision, which has been found wide applications in knowledge representation, cognitive computing and decision sciences. There have been already a huge number of literature on theory and applications on such topics. Most researches on formal concept analysis focus, independently, on specific scenarios such as Boolean formal contexts and fuzzy formal contexts. In this paper, a unified framework for three-way concept analysis is built in lattice-valued formal contexts. The notions of positive concept and negative concept are given in a lattice-valued formal context based on an overlapping function, one of significant operators for information fusion. Two pairs of object induced adjunctive three-way concept operators and attribute induced adjunctive three-way concept operators are introduced. Their properties have been fully investigated. With those operators, the algebraic structures of adjunctive three-way concepts, their extensions and intensions have been explored. Specifically, the local structure within pairs of sub-concept and sup-concept has been finely characterized. The derived results provide ones a way to construct the complete lattice of a lattice-valued formal context or its sublattice by determining all pairs of sub-concept and sup-concept provided that an object fuzzy set or an attribute fuzzy set is given. A case analysis on a real world dataset from UCI repository is presented to verify the rationality and significance of obtained results.

Tri-granularity attribute reduction of three-way concept lattices

A real-world dataset often generates a huge concept lattice that may be difficult to understand and impractical to use. The principle of granular computing is that we can understand and use a complex whole at multiple levels of granularity. In light of this, we propose a unified tri-granularity model of various types of concept lattices, which allows us to examine a huge concept lattice at three levels of granularity (i.e., the elementary granularity, local granularity and global granularity). Attribute reduction plays a fundamental role in three-way concept analysis, it simplifies the expression of three-way concepts and thus contributes to a better perception of the knowledge in three-way concept lattices. The tri-granularity model suggests an opportunity to investigate the tri-granularity attribute reduction of three-way concept lattices. The existing research on attribute reduction works at the global granularity but pays little attention to the local granularity, or, even less, to the elementary granularity. Driven by these issues, we supply definitions and methods of local granularity and elementary granularity attribute reduction of three-way concept lattices. These newly proposed two levels of attribute reduction with the existing global granularity attribute reduction together provide a framework for the tri-granularity attribute reduction of three-way concept lattices. We further analyze the relationships among the three levels of attribute reduction. Moreover, the efficacy of our suggested approach is illustrated by an example via the trisections induced by three-way concepts. Finally, we design two tri-granularity attribute reduction algorithms whose effectiveness is further examined by numerical experiments.

Multiview granular data analytics based on three-way concept analysis

Multiview granular data analytics based on three-way concept analysis

Distributed three-way formal concept analysis for large formal contexts

Three-way concept analysis (3WCA) is a framework based on Formal concept analysis and three-way decisions is used in the field of knowledge discovery to solve uncertainties in many domains like machine learning, data mining and software engineering. The 3WCA requires both the formal context and its complement context for generating concepts and constructing the concept lattice. In three-way concept analysis, data are analyzed using two types of concept lattices constructed using classical formal concept analysis (FCA) algorithms: object-induced (OE) concept lattices and attribute-induced (AE) concept lattices. The existing formal concept analysis algorithms focus on the sequential generation of OE and AE concepts rather than finding them in parallel and cannot process large datasets efficiently. The main contribution of this paper is that we propose a novel parallel algorithm for concept generation and construction of the three way concept lattice for knowledge discovery and representation in large datasets. Aiming to construct an efficient algorithm for 3WCA, this paper primarily discusses the existing algorithms for concept generation. Further we develop an efficient algorithm for OE and AE concept generation and lattice construction. Extensive experiments are conducted on various datasets to evaluate the efficiency of the proposed algorithm. Both the experimental and statistical results demonstrate the efficacy of the algorithm on larger datasets. Also the proposed algorithm can significantly decrease the time required for OE/AE concept generation and lattice construction compared to the existing classical FCA algorithms.

A constructing approach to multi-granularity object-induced three-way concept lattices

Three-way concept analysis is a research hotspot. Inspired by multi-scale information system, some scholars put forward and researched multi-granularity formal context. Most of the existing algorithms for constructing three-way concept lattice take care of the static formal contexts and can not deal with the multi-granularity formal context. To address this problem, this paper primarily focuses on a constructing approach to object-induced three-way concept (OE-concept) lattices for multi-granularity formal contexts. Firstly, transformation relationships between OE-concepts based on different granularities are researched. Secondly, the mutual transformation between OE-concept lattices under different granularities is proposed via these relationships, and then corresponding algorithms are proposed. Finally, several groups of datasets are selected from UCI for comparative experiments. Experimental results exhibit that our algorithms are more effective and advantageous than the latest construction algorithms.

Network rule extraction under the network formal context based on three-way decision.

Knowledge discovery combined with network structure is an emerging field of network data analysis and mining. Three-way concept analysis is a method that can fit the human mind in uncertain decisions and analysis. In reality, when three-way concept analysis is placed in the background of a network, not only the three-way rules need to be obtained, but also the network characteristic values of these rules should be obtained, which is of great significance for concept cognition in the network. This paper mainly combines complex network analysis with the formal context of three-way decision. Firstly, the network formal context of three-way decision (NFC3WD) is proposed to unify the two studies mentioned above into one data framework. Then, the network weaken-concepts of three-way decision (NWC3WD) and their corresponding sub-networks are studied. Therefore, we can not only find out the network weaken-concepts but also know the average influence of the sub-network, as well as the influence difference within the sub-network. Furthermore, the concept logic of network and the properties of its operators are put forward, which lays a foundation for designing the algorithm of rule extraction. Subsequently, the bidirectional rule extraction algorithm and reduction algorithm based on confidence degree are also explored. Meanwhile, these algorithms are applied to the diagnosis examples of COVID-19 from which we can not only get diagnostic rules, but also know the importance of the population corresponding to these diagnostic rules in the network through network eigenvalues. Finally, experimental analysis is made to show the superiority of the proposed method.

Knowledge discovery and updating under the evolution of network formal contexts based on three-way decision

Three-way concept analysis was proposed by the combination of three-way decision and formal concept analysis. Since it can provide more information than formal concept analysis, the new model has been widely used in the field of knowledge discovery. However, three-way concept analysis mainly deals with object-attribute data. If there is structure information between objects, how to remould traditional three-way concepts to describe both object-object information and object-attribute information is a topic worthy of discussion. To analyze object-object information and object-attribute information at the same time, we first propose the notion of a network formal context, and then define global network OE-concept and local network OE-concept. After that we investigate knowledge discovery based on the global network OE-concept and local network OE-concept. We further discuss the dynamic updating mechanism of global network OE-concept and local network OE-concept under the evolution of a network formal context. The corresponding algorithms are designed to update old network OE-concepts, and the time complexity is analyzed. Note that the classical formal context itself can also represent a network formal context, so in order to show the necessity of the proposed network formal context and network OE-concepts, we clarify the advantages of the network formal context and network OE-concepts from the aspects of knowledge representation, efficiency of computing network OE-concepts, dynamic updating of concepts and stability analysis of concepts. In addition, we conduct some experiments to argue that knowledge discovery of local network OE-concepts is better than that of global network OE-concepts as well as the classical OE-concepts. Meanwhile, the experimental results also show the feasibility and effectiveness of the proposed network OE-concept updating methods.

The Feature Description of Formal Context Based on the Relationships among Concept Lattices.

Three-way concept analysis (3WCA) is extended research of formal concept analysis (FCA) by combining three-way decision. The three-way object oriented concept lattice (OEOL) is one of the important data structures which integrates rough set, concept lattice and three-way decision in 3WCA. In the paper, we investigate the characteristics of formal context based on the isomorphic relationship among the kinds of concept lattices with OEOL. Firstly, II-dual intersectable attributes and II-dual intersectable context are proposed and the relationship between the type I-dual intersectable context(dual intersectable context) and the type II-dual intersectable context are studied. In addition, the relationship among the kinds of concept lattices with OEOL are studied when the formal context is both I-dual intersectable context and II-dual intersectable context. Finally, the inverse problems of the above conclusions are discussed and the following two conclusions are obtained: (1) the formal context is the type II-dual intersectable context, when the object oriented concept lattice and OEOL are isomorphic. (2) In addition, the formal context is the type I-dual intersectable context, when the concept lattice and OEOL are anti-isomorphic.

Neutrosophic three-way concept lattice and its application in conflict analysis

The theory of three-way concept analysis has been developed into an effective tool for data analysis and knowledge discovery. In this paper, we propose neutrosophic three-way concept lattice by combining neutrosophic set with three-way concept analysis and present an approach for conflict analysis by using neutrosophic three-way concept lattice. Firstly, we propose the notion of neutrosophic formal context, in which the relationships between objects and attributes are expressed by neutrosophic numbers. Three pairs of concept derivation operators are proposed. The basic properties of object-induced and attribute-induced neutrosophic three-way concept lattices are examined. Secondly, we divide the neutrosophic formal context into three classical formal contexts and propose the notions of the candidate neutrosophic three-way concepts and the redundant neutrosophic three-way concepts. Two approaches of constructing object-induced (attribute-induced) neutrosophic three-way concept lattices are presented by using candidate, redundant and original neutrosophic three-way concepts respectively. Finally, we apply the neutrosophic formal concept analysis to the conflict analysis and put forward the corresponding optimal strategy and the calculation method of the alliance.

Stability of three-way concepts and its application to natural language generation

• Three-way concept stability and its properties. • Algorithm for the computation of three-way concept stability. • A case study on natural language generation scenario. Three-way concept analysis (3WCA) has been an emerging and important methodology for knowledge discovery and data analysis. Particularly, 3WCA can efficiently characterize the information of “jointly possessed” and “jointly not possessed” compared to the classical formal concept only can describe common attributes owned by objects. This property, typical of 3WCA has a huge potential in the field of Natural Language Generation (NLG). However, the construction of a three-way concept lattice is proved as an NP-complete problem and even harder than the construction of conventional concept lattice. This could negatively affect the use of 3WCA for NLG in real contexts. Hence, it is necessary to prune the three-way concept lattice and extract more interesting three-way concepts for knowledge acquisition. To this end, this paper defines the stability of a three-way concept and analyzes the relevant properties. An efficient computational algorithm for calculating the stability of three-way concepts is developed and evaluated by an experiment. In addition, a case study on NLG is conducted for demonstrating the applicability of the proposed technique.

Incremental construction of three-way concept lattice for knowledge discovery in social networks

Three-way concept analysis (3WCA), a combination of three-way decision and formal concept analysis, is widely used in the field of knowledge discovery. Generally, constructing three-way concept lattices requires the original formal context and its complement context simultaneously. Additionally, the existing three-way concept lattice construction algorithms focus on the static formal context, and cannot cope with the dynamic formal context that is an essential representation in social networks. Toward this end, this paper pioneers a novel problem and method for the incremental construction of three-way concept lattice for knowledge discovery in social networks. Aiming to facilitate the construction efficiency, this paper firstly investigates the three-way concept lattice construction for attribute-incremental/object-incremental formal contexts, respectively. Then, the dynamic formal context of a social network can be viewed as a special formal context with both attribute-increment and object-increment. Further, we develop the AE/OE concept lattice incremental construction algorithms, called SNS-AE and SNS-OE. Extensive experiments are conducted on various formal contexts to evaluate the effectiveness of our incremental algorithms. The experimental results demonstrate that our proposed incremental algorithms can significantly decrease the construction time of three-way concept lattice compared to the non-incremental algorithm.

Matrix-based reduction approach for one-sided fuzzy three-way concept lattices

Three-way concept analysis is a mathematical model of the combination of formal concept analysis and three-way decision, and knowledge discovery plays a significant impact on formal fuzzy contexts since such datasets are frequently encountered in real life. In this paper, a novel type of one-sided fuzzy three-way concept lattices is presented in a given formal fuzzy context with its complement, in which a ternary classification is available. In such case, we comprehensively explore the connections between the proposed models and classical fuzzy concept lattices among elements, sets, and orders. Furthermore, approaches to granular matrix-based reductions are investigated, by which granular consistent sets, and granular reducts via discernibility Boolean matrices are tectonically put forward. At last, the demonstrated results are performed by several experiments which enrich the research of three-way concept analysis.

Attribute reduction of SE-ISI concept lattices for incomplete contexts.

Three-way concept analysis in incomplete contexts lays the theory dealing with the data in incomplete contexts, especially three kinds of partially known formal concepts including SE-ISI formal concept, ISE-SI formal concept and ISE-ISI formal concept. Generally speaking, not every attribute is essential in an incomplete context since the purpose of research is different. Thus, we propose four kinds of attribute reduction of SE-ISI concept lattices based on different criteria. Then, we discuss the relationships among the four kinds of attribute reduction, including the relationships among the consistent sets and relationships among the reducts. Finally, based on discernibility matrices and discernibility functions, the approaches to obtaining these attribute reduction are presented.

Constructing three-way concept lattice based on the composite of classical lattices

Three-way concept analysis provides a novel model for three-way decisions, and also extends the classical formal concept analysis, which has attracted many attentions in recent years. In order to construct three-way concept lattices, original formal context and its complement context need to be considered simultaneously. In this paper, a new method of constructing attribute-induced three-way (AE) and object-induced three-way (OE) concept lattices is proposed. Firstly, we introduce an AE-oriented composite operator and an OE-oriented composite operator that combine a pair of formal concepts coming from the concept lattices of original formal context and its complement context. Secondly, we define the candidate AE concepts and redundant AE concepts based on the AE-oriented composite operator, and define the candidate OE concepts and redundant OE concepts based on the OE-oriented composite operator, respectively. Finally, we propose an AE lattice construction algorithm and an OE lattice construction algorithm. Theoretical analysis and experimental results demonstrate that the proposed method is simple and effective, and it is a useful supplement to current three-way concept lattice construction methods.

Conflict analysis under one-vote veto based on approximate three-way concept lattice

Conflict analysis, which plays an important role in our society, has attracted more and more attentions. However, the existing conflict analysis models are not effective enough to evaluate the inconsistency degree of cliques that have more than two individuals. Besides, for each clique, its allied, conflict and neutral attributes have not been explicitly defined and thoroughly explored. Therefore, due to the lack of enough explicit clues, we cannot further formulate some strategies manipulating the conflict degrees of related cliques to realize the specific objectives. Last but not least, one-vote veto is seldom considered, although it plays a vital role in many fields, such as venture capital and United Nations Security Council resolutions. In order to solve these problems, we resort to three-way concept analysis and describe a clique by using the intent of a specific concept. On the basis of the obtained specific concepts, we derive the allied, conflict and neutral attributes of cliques, and further make decisions and explore the maximal coalitions and minimum conflict sets. Finally, in order to cater dynamic environment, we also describe an incremental algorithm for conflict analysis.

Three-way dual concept analysis

Three-way concept analysis is a mathematical theory, which combines formal concept analysis and three-way decision. The existing models, i.e., three-way concept lattices and three-way object oriented concept lattices, have been used successfully in many fields. However, these two models are established to cater some specific applications and cannot fulfill some special requirements such as the needs to seek potential collaborators in international import and export transactions. In this paper, a novel type of three-way concept lattices is presented based on dual concept analysis, which enables ones to characterize specific set by pointing out the attributes that are not possessed by at least one object in the complement of this specific set. And then, the connections between three-way dual concept lattices and classical dual concept lattices are explored. Besides, the relationships among four types of three-way concept analysis models, i.e., three-way concept lattices, three-way dual concept lattices, three-way object oriented concept lattices and three-way property oriented concept lattices, are investigated.

Three-Way Concept Analysis for Incomplete Formal Contexts

Recently, incomplete formal contexts have received more and more attention from the communities of formal concept analysis. Different from a complete context where the binary relations between all the objects and attribute are known, an incomplete formal context has at least a pair of object and attribute with a completely unknown binary relation. Partially known formal concepts use interval sets to indicate the incompleteness. Three-way formal concept analysis is capable of characterizing a target set by combining positive and negative attributes. However, how to describe target set, by pointing out what attributes it has with certainty and what attributes it has with possibility and what attributes it does not has with certainty and what attributes it does not has with possibility, is still an open problem. This paper combines the ideas of three-way formal concept analysis and partially known formal concepts and presents a framework of approximate three-way concept analysis. At first, approximate object-induced and attribute-induced three-way concept lattices are introduced, respectively. And then, the relationship between approximate three-way concept lattice and classical three-way concept lattice are investigated. Finally, examples are presented to demonstrate and verify the obtained results.

Characteristics of three-way concept lattices and three-way rough concept lattices

In classical formal concept (Wille’s concept) analysis, a basic theorem about concept lattices is that every concept lattice is a complete lattice and conversely, every complete lattice is isomorphic to a concept lattice. Three-way concept analysis is an extended theory of formal concept analysis. Similarly, three-way concept lattices and three-way rough lattices are also complete lattices. However, unlike the classical case, not every complete lattice arises as a three-way concept lattice (or as a three-way rough concept lattice). In this paper, we focus on characterising those complete lattices which can be represented by three-way concept lattices. In order to achieve this, we first discuss some properties of special elements such as atoms and irreducible elements, and complements of three-way concept lattices. Then we give our main theorem by displaying conditions under which any complete lattice can be realised as a three-way concept lattice. Similar results are discussed and obtained for three-way rough concept lattices.