A Note on Constructing Fuzzy Homomorphism Map for a Given Fuzzy Formal Context
01 Jan 2014-pp 845-855
TL;DR: This study focuses on constructing a fuzzy homomorphism map h that shows that reduced fuzzy concept lattice preserves the generalization and specialization with an illustrative example.
Abstract: Formal Concept Analysis is a well established mathematical model for data analysis and processing tasks. Computing all the fuzzy formal concepts and their visualization is an important concern for its practical applications. In this process a major problem is how to control the size of concept lattice. For this purpose current study focus on constructing a fuzzy homomorphism map h:F = \( (O_{i} ,P_{j} ,\tilde{R}) \to {\mathbf{D}} = (X_{m} ,Y_{n} ,\tilde{\varphi }) \) for the given fuzzy formal context F where, m ≤ i and n ≤ j. We show that reduced fuzzy concept lattice preserves the generalization and specialization with an illustrative example.
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TL;DR: This work proposes an algorithm for generating the bipolar fuzzy formal concepts, a method for ( α, β ) -cut ofipolar fuzzy formal context and its implications with illustrative examples.
112 citations
TL;DR: The results obtained from the proposed method are in good agreement with Levenshtein distance method and interval–valued fuzzy formal concepts method but with less computational complexity.
Abstract: In this paper we propose a method for reducing the number of formal concepts in formal concept analysis of data with fuzzy attributes. We compute the weight of fuzzy formal concepts based on Shannon entropy. Further, the number of fuzzy formal concepts is reduced at chosen granulation of their computed weight. We show that the results obtained from the proposed method are in good agreement with Levenshtein distance method and interval–valued fuzzy formal concepts method but with less computational complexity.
77 citations
01 Apr 2016
TL;DR: The current paper solves the problem of reducing the number of fuzzy formal concepts in FCA with fuzzy setting thereby simplifying the corresponding fuzzy concept lattice structure by linking an interval-valued fuzzy graph to the fuzziness in a given many-valued context which is transformed into a fuzzy formal context.
Abstract: Formal concept analysis (FCA) is a mathematical framework for data analysis and processing tasks. Based on the lattice and order theory, FCA derives the conceptual hierarchies from the relational information systems. From the crisp setting, FCA has been extended to fuzzy environment. This extension is aimed at handling the uncertain and vague information represented in the form of a formal context whose entries are the degrees from the scale [0, 1]. The present study analyzes the fuzziness in a given many-valued context which is transformed into a fuzzy formal context, to provide an insight into generating the fuzzy formal concepts from the fuzzy formal context. Furthermore, considering that a major problem in FCA with fuzzy setting is to reduce the number of fuzzy formal concepts thereby simplifying the corresponding fuzzy concept lattice structure, the current paper solves the problem by linking an interval-valued fuzzy graph to the fuzzy concept lattice. For this purpose, we propose an algorithm for generating the interval-valued fuzzy formal concepts. To measure the weight of fuzzy formal concepts, an algorithm is proposed using Shannon entropy. The knowledge represented by formal concepts using interval-valued fuzzy graph is compared with entropy-based-weighted fuzzy concepts at chosen threshold.
64 citations
TL;DR: A method is proposed based on Shannon entropy and Huffman coding to deal with the huge number of formal concepts generated from ‘a large context’ and another problem is their ‘storage’ complexity.
Abstract: 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.
50 citations
Posted Content•
01 Mar 2018TL;DR: A method for adequate analysis of vagueness and uncertainty in data with fuzzy attributes using the amplitude and phase term of a defined complex vague set based concept lattice is proposed.
Abstract: Recently, the calculus of concept lattice is extended from unipolar to bipolar fuzzy space for precise measurement of vagueness in the attributes based on their acceptation and rejection part. These extensions still unable to highlight the uncertainty in vague attributes and measurement of fluctuation at given phase of time. To conquer this problem, current paper proposed a method for adequate analysis of vagueness and uncertainty in data with fuzzy attributes using the amplitude and phase term of a defined complex vague set based concept lattice. In addition, the analysis derived from the proposed method is compared with CVSS method through an illustrative example.
43 citations
References
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Book•
01 Jan 1988TL;DR: The fuzzy sets uncertainty and information is one book that the authors really recommend you to read, to get more solutions in solving this problem.
Abstract: (1990). Fuzzy Sets, Uncertainty, and Information. Journal of the Operational Research Society: Vol. 41, No. 9, pp. 884-886.
3,120 citations
"A Note on Constructing Fuzzy Homomo..." refers background in this paper
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...A fuzzy homomorphism is a mapping between two defined informational systems [16–18]....
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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.
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.
2,407 citations
Proceedings Article•
01 Jan 1994
TL;DR: This paper begins with the L-Fuzzy concept definition that generalizes the definitions of the formal concept theory, and studies the lattice structure of the L/Omega concept set, giving a constructive method for calculating this lattice.
Abstract: The L-Fuzzy concept theory that we have developed sets up classifications from the objects and attributes of a context through L-Fuzzy relations. This theory generalizes the formal concept theory of R. Wille. In this paper we begin with the L-Fuzzy concept definition that generalizes the definitions of the formal concept theory, and we study the lattice structure of the L-Fuzzy concept set, giving a constructive method for calculating this lattice. At the end, we apply this constructive method to an example that has been studied by other methods.
272 citations
TL;DR: A novel method for building the approximate concept lattice of an incomplete context, the notion of an approximate decision rule and an approach for extracting non-redundant approximate decision rules from an incomplete decision context are presented.
239 citations
TL;DR: This second part of a large survey paper analyzes recent literature on Formal Concept Analysis (FCA) and some closely related disciplines using FCA and uses the visualization capabilities of FCA to explore the literature, to discover and conceptually represent the main research topics in the FCA community.
Abstract: This is the second part of a large survey paper in which we analyze recent literature on Formal Concept Analysis (FCA) and some closely related disciplines using FCA. We collected 1072 papers published between 2003 and 2011 mentioning terms related to Formal Concept Analysis in the title, abstract and keywords. We developed a knowledge browsing environment to support our literature analysis process. We use the visualization capabilities of FCA to explore the literature, to discover and conceptually represent the main research topics in the FCA community. In this second part, we zoom in on and give an extensive overview of the papers published between 2003 and 2011 which applied FCA-based methods for knowledge discovery and ontology engineering in various application domains. These domains include software mining, web analytics, medicine, biology and chemistry data.
223 citations