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Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy

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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.

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Acknowledgments

Authors sincerely acknowledge the financial support from National Board of Higher Mathematics, Dept. of Atomic Energy, Govt. of India under the grant number 2/48(11)/2010-R&D II/10806.

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Authors and Affiliations

  1. School of Information Technology and Engineering, VIT University, Vellore, 632014, Tamilnadu, India

    Prem Kumar Singh & Aswani Kumar Cherukuri

  2. Faculty of Science, Kunming University of Science and Technology, Kunming, 650500, China

    Jinhai Li

Authors
  1. Prem Kumar Singh
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  2. Aswani Kumar Cherukuri
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  3. Jinhai Li
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Corresponding author

Correspondence to Aswani Kumar Cherukuri.

Appendix

Appendix

Nomenclature

Meaning

L

Scale of truth degree

L

Residuated lattice

F

Fuzzy formal context

O

Set of objects

o

An object

P

Set of attributes

p

An attribute

P

Probability

\(\tilde{R}\)

\(L\)–relation between \(O\) and \(P\)

\(\otimes \)

Multiplication

\(\rightarrow \)

Residuum

\(a, b, c\)

Elements in L

(\(\uparrow , \downarrow \))

Galois connection

\(A\)

Extent

\(B\)

Intent

\(L^{{\textit{O}}}\)

\(L\)–set of objects

\(L^{{\textit{P}}}\)

\(L\)–set of attributes

\(\bigcup \)

Union

\(\bigcap \)

Intersection

\(\wedge \)

Infimum

\(\vee \)

Supremum

\(\theta , \theta _{1}, \theta _{2} \)

Granulation

\(E\)

Average information weight

\(\sum \)

Summation

\(m\)

Total number of attributes

\(w_{j}\)

Weight of attribute

\(Weight(k)\)

Weight of \(k\)–th formal concept

\(D\)

Deviation

\(||\)

Absolute difference

\(C\)

Single fuzzy formal concept

FC \(_\mathbf{{F} }\)

Set of fuzzy formal concepts

\(o_{i}\)

\(i\)–th objects

\(p_{j}\)

\(j\)–th attibute

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Singh, P.K., Cherukuri, A.K. & Li, J. Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy. Int. J. Mach. Learn. & Cyber. 8, 179–189 (2017). https://doi.org/10.1007/s13042-014-0313-6

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  • DOI: https://doi.org/10.1007/s13042-014-0313-6

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