Reduction Of Concept Lattices Research Articles

Concept lattice reduction using different subset of attributes as information granules

In recent years, the output of formal concept analysis has been widely spread in various research fields for knowledge processing tasks. In this process, a major issues arises when large number of formal concepts are generated from the given context. Available approaches lacks in user required dynamic reduction of concept lattice based on shape and size of the given problem. To overcome this problem, the current paper proposes a method to control the size of concept lattice based on user defined subset of attributes (or objects). Further the proposed method provides a way to select some of the important concepts generated from chosen subset of attributes. For this purpose properties of Shannon entropy is utilized by the proposed method to select some of the important concepts at different granulation of their computed weight. The analysis derived from the proposed method is also compared with recently published granulation tree method with an empirical analysis.

Representing attribute reduction and concepts in concept lattice using graphs

Concept lattice is an area of research which is based on a set-theoretical model for concepts and conceptual hierarchies. It is better for the studying of concept lattice to minimize the input data before revealing the construction of a concept lattice. Actually, this duty can be done by attribute reduction for a context. Graph is useful in data analysis since it gives us a visual trend on the behavior of our data points and allows us to test some laws in data analysis. This paper is a preliminary attempt to study how directed graph can be used on attribute reduction and conceptual construction in concept lattices. We investigate all the reducible attributes and concepts in a context with the aid of graph theory. For a context, we define a relevant graph on the set of attributes and, further, define a pre-weighted relevant graph. Afterward, using relevant graphs and pre-weighted relevant graphs with the method of deleting vertices in a directed graph, we find all the reducible attributes in a context. After that, we seek out all of concepts and the concept lattice for a given context. All these results may be not only used to improve the visibility and readability of attribute reduction in a context and the construction of concept lattice, but also broaden the applied range of directed graph such as in the field of attribute reduction.

Attribute and size reduction mechanisms in multi-adjoint concept lattices

Attribute reduction and size reduction in concept lattices are key research topics in Formal Concept Analysis (FCA). This paper combines both strategies in the multi-adjoint concept lattice framework in order to simplify the information provided by the original context. Specifically, we present three procedures which merge the attribute reduction and the size reduction by means of an irreducible α-cut concept lattice, analyzing the obtained properties.

Residuated lattices of block relations: size reduction of concept lattices

We deal with size reduction of concept lattices by means of factorization by block relations defined on corresponding formal context. We show that all block relations with a multiplication defined by means of relational composition form a (non-commutative) residuated lattice. Such residuated lattice can be then thought of as a scale of truth degrees using which we evaluate formulas of predicate logic specifying the desired parameters of the factorization. We also introduce efficient algorithms computing operations on a residuated lattice of block relations. The naive way how to design such algorithms is to compute all the formal concepts of a given context in advance, and then apply some well-known properties of residuated lattices. Our algorithms get rid of the time-consuming precomputation of all concepts.

Multi-Value Attribute Concept Lattice Reduction Based on Granular Computing

Concept lattice essentially describes the relationship between objects and attributes. The reduction of multi-value attribute concept lattice is a hot topic in the fields of information retrieval, knowledge discovery and data mining etc., while the granular computing emphasizes observing and analyzing the same problem from different granular worlds. It makes the complex problems around us be mapped to an easy to handle and more simple theory of calculation. The paper gives definitions of the concept granule and the compatible concept granular set by application of information granular and the granular of layered theory, and provides an algorithm to compute concept granular set through calculation of the compatible relationship. The paper further constructs the concept granule lattice, and then deletes the attribute of smaller contribution to concept granule. Through the comparison of the concept granule lattices, the multi-value attribute reduction could be achieved and the core attribute set in the formal context could be obtained. Instances could demonstrate the high efficiency and accuracy of this algorithm that is easier to realize through programming. Through the resolution of attribute, the calculation complexity could be reduced and the efficiency of calculation could be improved.

Concept lattices reduction: Definition, analysis and classification

Formal concept analysis (FCA) is currently considered an important formalism for knowledge representation, extraction and analysis with applications in different areas. A problem identified in several applications is the computational cost due to the large number of formal concepts generated. Even when that number is not very large, the essential aspects, those effectively needed, can be immersed in a maze of irrelevant details. In fact, the problem of obtaining a concept lattice of appropriate complexity and size is one of the most important problems of FCA. In literature, several different approaches to control the complexity and size of a concept lattice have been described, but so far they have not been properly analyzed, compared and classified. We propose the classification of techniques for concept lattice reduction in three groups: redundant information removal, simplification, and selection. The main techniques to reduce concept lattice are analyzed and classified based on seven dimensions, each one composed of a set of characteristics. Considerations are made about the applicability and computational complexity of approaches of different classes.

Fuzzy concept lattice reduction using Shannon entropy and Huffman coding

In the last decade, formal concept analysis (FCA) in a fuzzy setting has received more attention for knowledge processing tasks in various fields. The hierarchical order visualisation of generated formal concepts is a major concern for the practical application of FCA. In this process, a major issue is the huge number of formal concepts generated from ‘a large context’, and another problem is their ‘storage’ complexity. To deal with these issues a method is proposed in this paper based on Shannon entropy and Huffman coding. The proposed method is illustrated using crisply generated concepts such that the changes between obtained concepts can be measured using Levenshtein distance. The analysis derived from the proposed method is illustrated with an example for FCA in a fuzzy setting.

Note on the characterization and reduction of concept lattices through matroid theory

Mao[Characterization and reduction of concept lattices through matroid theory, Information Sciences 281, 10 (2014), 338–354] claims to make contributions to the study of concept lattices by using matroids. This note shows that his results are either trivial or wrong.

Relations of reduction between covering generalized rough sets and concept lattices

The reduction theory plays an important role in data analysis. This paper studies the relation between the reduction of a covering and the attribute reduction of a concept lattice. The reduction of a covering from the perspective of concept lattices is investigated. Conversely, the attribute reduction of a formal context is studied in the framework of covering generalized rough sets. The results in this paper show that the reduction of a covering can be viewed as the attribute reduction of a derivative formal context. Moreover, every reduct of a given formal context can be seen as the reduct of an induced covering. As an application of the theoretical results, an approach to the attribute reduction of concept lattices based on covering generalized rough sets is proposed. Furthermore, experiments are given to show the effectiveness of the proposed method.

Attribute Reduction of Intuitionistic Fuzzy Concept Lattices

In this paper, we turn intuitionistic fuzzy information systems into 0-1 formal contexts by using dominance relation. A pair of operators is defined to get the formal concept lattices in the intuitionistic fuzzy information systems. Furthermore, some properties and attribute reduction based on discernibility matrices is investigated.

Formal concept analysis and concept lattice: perspectives and challenges

Formal concept analysis FCA is a powerful tool for data mining, ontology research, web semantic retrieval, software engineering, and knowledge discovery. Concept lattice is the core data structure of FCA. Association rules mining methods based on concept lattices are discussed. The algorithms of constructing concept lattices are introduced, and the merits and drawbacks of these algorithms are compared. The research situation about attribute reduction of concept lattice is given. Furthermore, the extended models of concept lattice and the challenges to development of concept lattice are introduced. At last, many problems on FCA and concept lattice needed to study deeply are given.

Attribute reduction in multi-adjoint concept lattices

Knowledge reduction is one of the key issues in formal concept analysis and there have been many studies on this topic. The irreducible elements in a lattice are also very important, since they form the basic information of a relational system. Moreover, they are also important from the viewpoint of attribute reduction.Both topics are notably more complicated in a fuzzy setting since not only the size of the sets of attributes and objects influence the size of the fuzzy concept lattice, but the truth-value sets, where the sets of objects, attributes and the relation are evaluated, are important.This paper presents, in the general fuzzy framework of multi-adjoint concept lattices, a characterization of the meet-irreducible elements, from which a classification of attributes and its application to attribute reduction is introduced.

Characterization and reduction of concept lattices through matroid theory

Concept lattice theory is an efficient tool for data analysis. Reduction of attributes and objects for a context is an important issue in concept lattice theory. We explore the relationships between matroidal spaces and concepts for a given context. With the assistance of matroid theory, we conduct the attribute reduction and object reduction for a context and formulate the concepts for a given context. As a generalization of the linear independence in vector spaces, matroids provide well established platforms for some algorithms such as greedy algorithm.In this paper, we mainly propose a notion of matroidal space by the family of circuits of a matroid. These relationships characterize the constructions of not only concepts, but also concept lattices. Meanwhile, we demonstrate that reduction of attributes and objects proposed in this paper preserve the original hierarchy order in the concept lattice for a given context.

Attribute Reduction in Intuitionistic Fuzzy Concept Lattices

As an effective tool for knowledge discovery, concept lattice has been successfully applied to various fields. And one of the key problems of knowledge discovery is attribute reduction. In order to understand the problems better, the attribute reduction is necessary to perfect the theory as well as expand application of concept lattice. This paper introduces the intuitionistic fuzzy theory into the concept lattice theory and proposes a kind of intuitionistic fuzzy concept lattice. Then, an approach to attribute reduction based on the discernibility matrix is proposed and investigated, which makes the discovery of implicit knowledge easier and the representation simpler in data; furthermore, the theory of concept lattice is perfected. The theory of intuitionistic fuzzy concept lattice is useful and meaningful in view of the complexity and fuzziness of information in real world, and the potential value of dealing with information is expected in the future.

Relating attribute reduction in formal, object-oriented and property-oriented concept lattices

Attribute reduction is an important step in reducing computational complexity in order to extract information from relational systems.Three of these systems are the formal, object-oriented and property oriented concept lattices. Attribute reduction in the last two concept lattices has recently been studied. The relation with the first concept lattice is very important since two important, independent tools to extract information from databases–the formal concept analysis and rough set theory–will be related. This paper studies attribute reduction in these three frameworks. The main results are that the classification of each attribute into absolutely necessary, relatively necessary and absolutely unnecessary attributes is independent of the framework considered and that an attribute reduct in one of these relational systems is also an attribute reduct in the others.

On complexity reduction of concept lattices: three counterexamples

We discuss the paper by Cheung and Vogel, Inf Retr 8(2), pp 285---299 (2005). We introduce some counterexamples showing that the theoretical results, the paper is based upon, are incorrect.

Attribute reduction in decision formal context based on homomorphism

Attribute reduction in concept lattices is one of the key problems in the study of concept lattice theory. This paper defines a homomorphism consistent set and provides an approach to attribute reduction of a consistent decision formal context based on concept lattices. We also examine the relationship between homomorphism consistent set and attribute consistent set. The method for judging a homomorphism consistent set is simpler than the existing ones.

The Research of Constructing Rough Concept Lattices Model

Formal concept lattices and rough set theory are two kinds of complementary mathematical tools for data analysis and data processing. The algorithm of concept lattice reduction based on variable precision rough set is proposed by combining the algorithms of β-upper and lower distribution reduction in variable precision rough set. The traditional algorithms aboutβvalue select algorithm, attribute reduction based on discernibility matrix and extraction rule in VPRS are discussed, there are defects in these traditional algorithms which are improved. Finally, the generation system of concept lattice based on variable precision rough set is designed to verify the validity of the improved algorithm and a case demonstrates the whole process of concept lattice construction.

Relation between concept lattice reduction and rough set reduction

One of the key problems of knowledge discovery is knowledge reduction. Rough set theory and the theory of concept lattices are two efficient tools for knowledge discovery. Attribute reduction based on rough set theory and the theory of concept lattices both have been researched. Since an information system, the data description of rough set theory, and a formal context, the data description of concept lattice theory, can be taken as the other one, the attribute reduction based on the same data base can be studied from these two perspectives, and researching their relation is significant. This paper mainly discusses the relation between concept lattice reduction and rough set reduction based on classical formal context, which will be meaningful for the relation research between these two theories, and for their knowledge discovery.

Approaches to attribute reduction in concept lattices induced by axialities

This paper investigates approaches to attribute reduction in concept lattices induced by axialities. Based on an axiality, a type of covariant Galois connection between power sets, or equivalently a binary relation between the ground sets, the lattice of all concepts associated with a formal context is studied. Some judgment theorems for attribute reduction in such a lattice are proposed and proved. Extended from the idea of knowledge reduction in rough set theory, a Boolean approach to calculating all reducts of a context is formulated via the use of discernibility function. Finally, all attributes are classified into three types by their significance in constructing the concept lattice. The characteristics of these types of attributes are also analyzed.