A Note on Computing the Crisp Order Context of a Fuzzy Formal Context for Knowledge Reduction

- Journal title : Journal of Information Processing Systems
- Volume 11, Issue 2, 2015, pp.184-204
- Publisher : Korea Information Processing Society
- DOI : 10.3745/JIPS.04.0009

Title & Authors

A Note on Computing the Crisp Order Context of a Fuzzy Formal Context for Knowledge Reduction

Singh, Prem Kumar; Kumar, Ch. Aswani;

Singh, Prem Kumar; Kumar, Ch. Aswani;

Abstract

Fuzzy Formal Concept Analysis (FCA) is a mathematical tool for the effective representation of imprecise and vague knowledge. However, with a large number of formal concepts from a fuzzy context, the task of knowledge representation becomes complex. Hence, knowledge reduction is an important issue in FCA with a fuzzy setting. The purpose of this current study is to address this issue by proposing a method that computes the corresponding crisp order for the fuzzy relation in a given fuzzy formal context. The obtained formal context using the proposed method provides a fewer number of concepts when compared to original fuzzy context. The resultant lattice structure is a reduced form of its corresponding fuzzy concept lattice and preserves the specialized and generalized concepts, as well as stability. This study also shows a step-by-step demonstration of the proposed method and its application.

Keywords

Crisp Context;Concept Lattice;Formal Concept Analysis;Fuzzy Formal Concept;Fuzzy Relation;Knowledge Reduction;

Language

English

Cited by

References

1.

R. Wille, Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts. Darmstadt: Technische Hochschule, Fachbereich Mathematik, 1981.

2.

B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations. Berlin: Springer, 1999.

3.

C. Carpineto and G. Romano, Concept Data Analysis: Theory and Applications. Chichester: John Wiley & Sons, 2004.

4.

A. Burusco and R. Fuentes-Gonzalez, "The study of the L-fuzzy concept lattice," Matheware and Soft Computing, vol. 1, no. 3, pp. 209-218, 1994.

5.

R. Belohlavek and V. Vychodil, "What is fuzzy concept lattice," in Proceedings of Concept Lattices and Their Applications, Olomuc, Czech Republic, 2005, pp. 34-45.

6.

S. Ayouni, S. Ben Yahia, and A. Laurent, "Extracting compact and information lossless sets of fuzzy association rules," Fuzzy Sets and Systems, vol. 183, no. 1, pp. 1-25, 2011.

7.

P. Ghosh, K. Kundu, and D. Sarkar, "Fuzzy graph representation of a fuzzy concept lattice," Fuzzy Sets and Systems, vol. 161, no. 12, pp. 1669-1675, 2010.

8.

C. De Maio, G. Fenza, V. Loia, and S. Senatore, "Hierarchal web resources retrieval by exploiting fuzzy formal concept analysis," Information Processing and Management, vol. 48, no. 2, pp. 399-418, 2012.

9.

T. T. Nguyen, S. C. Hui, and K. Chang, "A lattice-based approach for mathematical search using formal concept analysis," Experts Systems with Applications, vol. 39, no. 5, pp. 5820-5828, 2012.

10.

C. Aswani Kumar and S. Srinivas, "Concept lattice reduction using fuzzy K-means clustering," Expert Systems with Applications, vol. 37, no. 3, pp. 2696-2704, 2010.

11.

C. Aswani Kumar, "Knowledge discovery in data using formal concept analysis and random projections," International Journal of Applied Mathematics and Computer Science, vol. 21, no. 4, pp. 745-756, 2011.

12.

A. Gely, "Links between modular decomposition of concept lattices and bimodular decomposition of a context," in Proceedings of the Concept Lattices and Their Applications, Nancy, France, 2011, pp.393-403.

13.

L. Guo, F. Huang, Q. Li, and G. Q. Zhang, G. Q. "Power contexts and their concept lattices," Discrete Mathematics, vol. 311, no. 18, pp. 2049-2063, 2011.

14.

Q. Hu, J. Liu, and D. Yu, "Mixed feature selection based on granulation and approximation," Knowledge-Based Systems, vol. 21, no. 4, pp. 294-304, 2008.

15.

J. Konecny and M. Krupka, "Block relations in fuzzy Settings", in Proceedings of the Concept Lattices and Their Applications, Nancy, France, 2011, pp.115-130.

16.

L. Li and J. Zhang, "Attribute reduction in fuzzy concept lattices on the T-implication," Knowledge-Based Systems, vol. 23, no. 6, pp. 497-503, 2010.

17.

J. Li, C. Mei, and Y. Lv, "A heuristic knowledge reduction method for decision formal contexts," Computers and Mathematics with Applications, vol. 61, no. 4, pp. 1096-1106, 2011.

18.

P. K. Singh and C. Aswani Kumar, "A method for reduction of fuzzy relation in given fuzzy formal context," in Proceedings of International Conference on Mathematical Modeling and Scientific Computation (ICMMSC2012), Gandhigram, India, 2012, pp. 343-350.

19.

P. K. Singh and C. Aswani Kumar, "A method for decomposition of fuzzy formal context," Procedia Engineering, vol. 38, pp. pp. 1852-1857, 2012.

20.

I. Beg and S. Ashraf, "Numerical representation of product transitivity complete fuzzy ordering," Mathematical & Computer Modelling, vol. 53, no. 5, pp. 617-623, 2011.

21.

K. H. Lee, First Course on Fuzzy Theory and Applications. Heidelberg: Springer, 2005.

22.

L. A. Zadeh, "Similarity relations and fuzzy orderings," Information Sciences, vol. 3, no. 2, pp. 177-200, 1971.

23.

W. B. V. Kandasamy and F. Smarandache, Fuzzy Interval Matrices, Neutrosophic Interval Matrices and Their Applications. Phoenix, AZ: Hexis, 2006.

24.

Y. Djouadi, "Extended Galois connection derivational operators for information retrieval based on fuzzy formal concept lattice," in Proceedings of 5th International Conference on Scalable Uncertainty Management (SUM2011), Dayton, OH, 2011, pp. 346-358.

25.

D. Dubois and H. Prade, "Possibility theory and formal concept analysis: characterizing independent subcontexts," Fuzzy Sets and Systems, vol. 196, pp. 4-16, 2012.

26.

M. Krupka and J. Lastovica, "Concept lattices of incomplete data," in Proceedings of 10th International Conference on Formal Concept Analysis (ICFCA2012), Leuven, Belgium, 2012, pp. 180-194.

27.

J. Li, C. Mei, and Y. Lv, "Incomplete decision contexts: approximate concept construction rule acquisition and knowledge reduction," International Journal of Approximate Reasoning, vol. 54, no. 1, pp. 191-207, 2013.

28.

J. J. Buckley and E. Eslami, An Introduction to Fuzzy Logic and Fuzzy Sets. Heidlberg: Physica-Verlag, 2002.

29.

R. Belohlavek, "A note on variable threshold concept lattices: threshold-based operators are reducible to classical concept-forming operators," Information Sciences, vol. 177, no. 15, pp. 3186-3191, 2007.

30.

R. Belohlavek, "Fuzzy galois connections," Mathematical Logic Quarterly, vol. 45, no. 4, pp. 497-504, 1999.

31.

J. Pocs, "Note on generating fuzzy concept lattices via Galois connection," Information Sciences, vol. 185, no. 1, pp. 128-136, 2012.

32.

W. X. Zhang, J. M. Ma, and S. Q. Fan, "Variable threshold concept lattices," Information Sciences, vol. 177, no. 22, pp. 4883-4892, 2007.

33.

R. Belohlavek and J. Konecny, "Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited," Information Sciences, vol. 199, pp. 133-137, 2012.

34.

J. Medina, "Relating attribute reduction in formal, object-oriented and property-oriented concept lattices," Computers and Mathematics with Applications, vol. 64, no. 6, pp. 1992-2002, 2012.

35.

M. W. Shao, M. Liu, and W. X. Zhan, "Set approximations in fuzzy formal concept analysis," Fuzzy Sets and Systems, vol. 158, no. 23, pp. 2627-2640, 2007.

36.

V. Cross and M. Kandasamy, "Creating fuzzy concepts: the one sided threshold, fuzzy closure and factor analysis method," in Proceedings of the 13th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing (RSFDGrC2011), Moscow, Russia, 2011, pp.127-134.

37.

S. O. Kuznetsov and S. A. Obiedkov, "Comparing performance of algorithms for generating concept lattices," Journal of Experimental & Theoretical Artificial Intelligence, vol. 14, no. 2-3, pp. 189-216, 2002.

38.

X. Kang, D. Li, S. Wang, and K. Qu, "Formal concept analysis based on fuzzy granularity base for different granulations," Fuzzy Sets and Systems, vol. 203, pp. 33-48, 2012.

39.

W. Z. Wu, Y. Leung, and J. S. Mi, "Granular computing and knowledge reduction in formal context," IEEE Transaction on Knowledge and Data Engineering, vol. 21, no. 10, pp. 1461-1474, 2009.

40.

S. O. Kuznetsov, "On stability of a formal concept," Annals of Mathematics and Artificial Intelligence, vol. 49, no. 1-4, pp. 101-115, 2007.

41.

E. Bartl, R. Belohlavek, and V. Vychodil, "Bivalent and other solutions of fuzzy relations equations via linguistic hedges," Fuzzy Sets and Systems, vol. 187, no. 1, pp. 103-112, 2012.

42.

E. K. Horvath, B. Seselja, and A. Tepavcevic, "Cut approach to islands in rectangular fuzzy relations," Fuzzy Sets and Systems, vol. 161, no. 24, pp. 3114-3126, 2010.

43.

H. Liu, S. Xiaong, and Z. Fang, "FL-GrCCA: a granular computing classification algorithm based on fuzzy lattices," Computers and Mathematics with Applications, vol. 61, no. 1, pp. 138-147, 2011.

44.

A. Skowron, J. Stepaniuk, and R. Swiniarski, "Modelling rough granular computing based on approximation spaces," Information Sciences, vol. 184, no. 1, pp. 20-43, 2012.

45.

C. Aswani Kumar and S. Srinivas, "Mining associations in health care data using formal concept analysis and singular value decomposition," Journal of Biological Systems, vol. 18, no. 4, pp. 787-807, 2010.

46.

C. Aswani Kumar, "Fuzzy clustering based formal concept analysis for association rules mining," Applied Artificial Intelligence, vol. 26, no. 3, pp. 274-301, 2012.

47.

ConExp tool, http://conexp.sourceforge.net/

48.

J. Li, C. Mei, and Y. Lv, "Knowledge reduction in real decision formal contexts," Information Sciences, vol.189, pp. 191-207, 2012.

49.

J. Li, Mei, C., C. Aswani Kumar, and X. Zhang, "On rule acquisition in decision formal contexts," International Journal of Machine Learning and Cybernetics, vol. 4, no. 6, pp. 721-731, 2013.

50.

H. Z. Yang, Y. Leung, and M. W. Shao, "Rule acquisition and attribute reduction in real decision formal context," Soft Computing, vol. 15, no. 6, pp. 1115-1128, 2011.

51.

Prem Kumar Singh and Ch. Aswani Kumar, "Interval-valued fuzzy graph representation of concept lattice," in Proceedings of the 12th International Conference on Intelligent Systems Design and Applications (ISDA), Kochi, India, 2012, pp. 604-609.

52.

P. K. Singh and C. Aswani Kumar, "A note on constructing fuzzy homomorphism map for a given fuzzy formal context," in Proceedings of the 3rd International Conference on Soft Computing for Problem Solving, 2013, pp. 845-855.

53.

C. Aswani Kumar and P. K. Singh, "Knowledge representation using formal concept analysis: a study on concept generation," in Global Trends in Intelligent Computing Research and Development, Hershey, PA: IGI Global Publishers, 2014, pp. 306-336.

54.

C. Aswani Kumar, M. Radvansky, and J. Annapurna, "Analysis of a vector space model, latent semantic indexing and formal concept analysis for information retrieval," Cybernetics and Information Technologies, vol. 12, no. 1, pp. 34-48, 2012.