Concept approximations based on rough sets and similarity measures

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Concept approximations based on rough sets and similarity measures

Journal Article•

Concept approximations based on rough sets and similarity measures

01 Jan 2001-International Journal of Applied Mathematics and Computer Science (University of Zielona Gora Press)-Vol. 11, Iss: 3, pp 655-674

TL;DR: This paper presents two different approaches to the concept approximation, one based on rough set theory while the other based on a similarity measure, and presents algorithms for the two approaches.

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Abstract: The formal concept analysis gives a mathematical definition of a formal concept. However, in many real-life applications, the problem under investigation cannot be described by formal concepts. Such concepts are called the non-definable concepts (Saquer and Deogun, 2000a). The process of finding formal concepts that best describe non-definable concepts is called the concept approximation. In this paper, we present two different approaches to the concept approximation. The first approach is based on rough set theory while the other is based on a similarity measure. We present algorithms for the two approaches.

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Journal Article•DOI•

Three-way cognitive concept learning via multi-granularity

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Jinhai Li¹, Chenchen Huang¹, Jianjun Qi², Yuhua Qian³, Wenqi Liu¹ - Show less +1 more•Institutions (3)

Kunming University of Science and Technology¹, Xidian University², Shanxi University³

01 Feb 2017-Information Sciences

TL;DR: An axiomatic approach to describe three-way concepts by means of multi-granularity is put forward and a three- way cognitive computing system is designed to find composite three-Way cognitive concepts.

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319 citations

Journal Article•DOI•

Concept learning via granular computing

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Jinhai Li¹, Changlin Mei², Weihua Xu³, Yuhua Qian⁴•Institutions (4)

Kunming University of Science and Technology¹, Xi'an Jiaotong University², Chongqing University of Technology³, Shanxi University⁴

20 Mar 2015-Information Sciences

TL;DR: Cognitive mechanism of forming concepts is analyzed based on the principles from philosophy and cognitive psychology, including how to model concept-forming cognitive operators, define cognitive concepts and establish cognitive concept structure to improve efficiency of concept learning.

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240 citations

Cites background or methods from "Concept approximations based on rou..."

...Then one can derive lower and upper approximations of an arbitrary subset X of U which are respectively defined as: AðXÞ ¼ [ Y2U=INDðAÞ;Y # X Y and AðXÞ ¼ [ Y2U=INDðAÞ;Y\X–; Y: In rough set theory [23], the pair ½AðXÞ;AðXÞ is referred to as the rough set of X with respect to the attribute set A. Note that the idea of lower and upper approximations in rough set theory was further extended by Saquer and Deogun [32], Yao and Chen [51], and Zhang and Qiu [55] to Wille’s concept lattice, and by Shao et al. [34] to fuzzy concept lattice....
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...Note that the idea of lower and upper approximations in rough set theory was further extended by Saquer and Deogun [32], Yao and Chen [51], and Zhang and Qiu [55] to Wille’s concept lattice, and by Shao et al....
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...Motivated by the existing set approximation ideas [23,32,34,51,55], we respectively define the lower and upper approximations of an object set X0 in the cognitive concept lattice BðUn;An;Ln;HnÞ as follows:...
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Journal Article•

Rough set approximations in formal concept analysis

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Yiyu Yao¹, Yaohua Chen¹•Institutions (1)

University of Regina¹

01 Jan 2006-Lecture Notes in Computer Science

TL;DR: The notion of rough set approximations is introduced into formal concept analysis and approximation operators are defined based on both lattice- theoretic and set-theoretic operators to provide a better understanding of data analysis.

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Abstract: A basic notion shared by rough set analysis and formal concept analysis is the definability of a set of objects based on a set of properties. The two theories can be compared, combined and applied to each other based on definability. In this paper, the notion of rough set approximations is introduced into formal concept analysis. Rough set approximations are defined by using a system of definable sets. The similar idea can be used in formal concept analysis. The families of the sets of objects and the sets of properties established in formal concept analysis are viewed as two systems of definable sets. The approximation operators are then formulated with respect to the systems. Two types of approximation operators, with respect to lattice-theoretic and set-theoretic interpretations, are studied. The results provide a better understanding of data analysis using rough set analysis and formal concept analysis.

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128 citations

Cites background from "Concept approximations based on rou..."

...Many studies considered rough set approximations in formal concept lattice [4, 8, 11, 16, 17, 20, 23]....
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...Reversely, the third group focuses on applying concepts and methods of rough set analysis into formal concept analysis [4, 8, 11, 16, 17, 20, 23, 34, 39]....
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...Saquer and Deogun proposed to approximate a set of objects, a set of properties, and a pair of a set of objects and a set of properties, based on a formal concept lattice [16, 17]....
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...Saquer and Deogun suggested that all concepts in a concept lattice can be considered as definable, and a set of objects can be approximated by concepts whose extensions approximate the set of objects [16, 17]....
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Journal Article•DOI•

Relations of attribute reduction between object and property oriented concept lattices

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Xia Wang¹, Wenxiu Zhang¹•Institutions (1)

Xi'an Jiaotong University¹

01 Jul 2008-Knowledge Based Systems

TL;DR: Relations of attribute reduction between object and property oriented formal concept lattices are discussed and beautiful results are obtained that attribute reducts and attribute characteristics in the two concept lattice are the same based on new approaches to attribute reduction by means of irreducible elements.

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Abstract: As one of the basic problems of knowledge discovery and data analysis, knowledge reduction can make the discovery of implicit knowledge in data easier and the representation simpler. In this paper, relations of attribute reduction between object and property oriented formal concept lattices are discussed. And beautiful results are obtained that attribute reducts and attribute characteristics in the two concept lattices are the same based on new approaches to attribute reduction by means of irreducible elements. It turns out to be meaningful and effective in dealing with knowledge reduction, as attribute reducts and attribute characteristics in the object and property oriented formal concept lattices can be acquainted by only investigating one of the two concept lattices.

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113 citations

Journal Article•DOI•

Concept analysis via rough set and AFS algebra

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Lidong Wang¹, Xiaodong Liu¹•Institutions (1)

Dalian University of Technology¹

01 Nov 2008-Information Sciences

TL;DR: AFS formal concept is proposed, which can be viewed as the generalization and development of monotone concept proposed by Deogun and Saquer (2003), and it is shown that the set of all AFS formal concepts forms a complete lattice.

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107 citations

Cites background from "Concept approximations based on rou..."

...Combination of FCA and rough set theory provides some new approaches for data analysis and knowledge discovery [20,21,26,34]....
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References

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Open Access

More filters

Book•

Fuzzy sets

[...]

Lotfi A. Zadeh

01 Aug 1996

TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

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Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

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52,705 citations

Journal Article•DOI•

Rough sets

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Zdzisław Pawlak¹, Jerzy W. Grzymala-Busse², Roman Słowiński³, Wojciech Ziarko⁴•Institutions (4)

Warsaw University of Technology¹, University of Kansas², Poznań University of Technology³, University of Regina⁴

01 Nov 1995-Communications of The ACM

TL;DR: This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

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Abstract: Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

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7,185 citations

Book•

Formal Concept Analysis: Mathematical Foundations

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Bernhard Ganter, Rudolf Wille, C. Franzke

04 Dec 1998

TL;DR: This is the first textbook on formal concept analysis that gives a systematic presentation of the mathematical foundations and their relation to applications in computer science, especially in data analysis and knowledge processing.

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Abstract: From the Publisher: This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relation to applications in computer science, especially in data analysis and knowledge processing. Above all, it presents graphical methods for representing conceptual systems that have proved themselves in communicating knowledge. Theory and graphical representation are thus closely coupled together. The mathematical foundations are treated thoroughly and illuminated by means of numerous examples.

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4,757 citations

"Concept approximations based on rou..." refers background or methods in this paper

...where the infimum and supremum were found using the Fundamental Theorem of formal concept analysis (Ganter and Wille, 1999; Wille, 1982)....
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...This context has 23 formal concepts which were generated using Algorithm Next described in (Ganter and Wille, 1999)....
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...The Fundamental Theorem of FCA states that the set of all the formal concepts on a given context with the ordering (A1, B1) ≤ (A2, B2) iff A1 ⊆ A2 is a complete lattice called the concept lattice, in which the infima and suprema are given by (Ganter and Wille, 1999; Wille, 1982):...
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...This algorithm is also described in (Ganter and Wille, 1999)....
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...1, where the concepts are labeled using reduced labeling (Ganter and Wille, 1999)....
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Book Chapter•DOI•

Restructuring lattice theory: an approach based on hierarchies of concepts

[...]

Rudolf Wille¹•Institutions (1)

Darmstadt University of Applied Sciences¹

12 May 2009

TL;DR: Restructuring lattice theory is an attempt to reinvigorate connections with the authors' general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattices theory.

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Abstract: Lattice theory today reflects the general Status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

...read moreread less

2,407 citations

Journal Article•DOI•

Concept lattices and conceptual knowledge systems

[...]

Rudolf Wille¹•Institutions (1)

Darmstadt University of Applied Sciences¹

01 Mar 1992-Computers & Mathematics With Applications

TL;DR: A mathematical model for conceptual knowledge systems allows us to study mathematically the representation, inference, acquisition, and communication of conceptual knowledge.

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Abstract: “Concept Lattice” is the central notion of “Formal Concept Analysis”, a new area of research which is based on a set-theoretical model for concepts and conceptual hierarchies. This model yields not only a new approach to data analysis but also methods for formal representation of conceptual knowledge. These methods are outlined on three levels. First, basics on concept lattices are explained starting from simple data contexts which consist of a binary relation between objects and attributes indicating which object has which attribute. On the second level, conceptual relationships are discussed for data matrices which assign attribute values to each of the given objects. Finally, a mathematical model for conceptual knowledge systems is described. This model allows us to study mathematically the representation, inference, acquisition, and communication of conceptual knowledge.

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590 citations

"Concept approximations based on rou..." refers methods in this paper

...The formal concept analysis (FCA) is a mathematical framework, developed by Wille and his colleagues at Darmstadt University, which is useful for representation and an analysis of data (Wille, 1992; 1982; 1989)....
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...Introduction The formal concept analysis (FCA) is a mathematical framework, developed by Wille and his colleagues at Darmstadt University, which is useful for representation and an analysis of data (Wille, 1992; 1982; 1989)....
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