Advertisement

A proposal for tuning the \(\alpha \) parameter in \(C_{\alpha }C\)-integrals for application in fuzzy rule-based classification systems

  • Giancarlo Lucca
  • José A. Sanz
  • Graçaliz P. Dimuro
  • Benjamín Bedregal
  • Humberto Bustince
Article

Abstract

In this paper, we consider the concept of extended Choquet integral generalized by a copula, called CC-integral. In particular, we adopt a CC-integral that uses a copula defined by a parameter \(\alpha \), which behavior was tested in a previous work using different fixed values. In this contribution, we propose an extension of this method by learning the best value for the parameter \(\alpha \) using a genetic algorithm. This new proposal is applied in the fuzzy reasoning method of fuzzy rule-based classification systems in such a way that, for each class, the most suitable value of the parameter \(\alpha \) is obtained, which can lead to an improvement on the system’s performance. In the experimental study, we test the performance of 4 different so called \(C_{\alpha }C\)-integrals, comparing the results obtained when using fixed values for the parameter \(\alpha \) against the results provided by our new evolutionary approach. From the obtained results, it is possible to conclude that the genetic learning of the parameter \(\alpha \) is statistically superior than the fixed one for two copulas. Moreover, in general, the accuracy achieved in test is superior than that of the fixed approach in all functions. We also compare the quality of this approach with related approaches, showing that the methodology proposed in this work provides competitive results. Therefore, we demonstrate that \(C_{\alpha }C\)-integrals with \(\alpha \) learned genetically can be considered as a good alternative to be used in fuzzy rule-based classification systems.

Keywords

Aggregation functions Choquet integral Fuzzy rule-based classification systems Fuzzy reasoning method Genetic algorithms Evolutionary fuzzy systems 

Notes

Acknowledgements

The authors would like to thank the Brazilian National Counsel of Technological and Scientific Development CNPq (Proc. 233950/2014-1, 481283/2013-7, 306970/ 2013-9, 307681/2012-2) and the Spanish Ministry of Science and Technology under project TIN2016-77356-P (AEI/FEDER, UE). G.P. Dimuro is also supported by Caixa and Fundación Caja Navarra of Spain.

References

  1. Agrawal R, Srikant R (1994) Fast algorithms for mining association rules in large databases. In: Proceedings of the 20th international conference on very large data bases, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, VLDB ’94, pp 487–499Google Scholar
  2. Alcalá R, Alcalá-Fdez J, Herrera F (2007) A proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection. IEEE Trans Fuzzy Syst 15(4):616–635CrossRefzbMATHGoogle Scholar
  3. Alcalá-Fdez J, Sánchez L, García S, Jesus M, Ventura S, Garrell J, Otero J, Romero C, Bacardit J, Rivas V, Fernández J, Herrera F (2009) Keel: a software tool to assess evolutionary algorithms for data mining problems. Soft Comput 13(3):307–318CrossRefGoogle Scholar
  4. Alcalá-Fdez J, Alcalá R, Herrera F (2011) A fuzzy association rule-based classification model for high-dimensional problems with genetic rule selection and lateral tuning. IEEE Trans Fuzzy Syst 19(5):857–872CrossRefGoogle Scholar
  5. Alsina C, Frank MJ, Schweizer B (2006) Associative functions: triangular norms and copulas. World Scientific Publishing Company, SingaporeCrossRefzbMATHGoogle Scholar
  6. Barrenechea E, Bustince H, Fernandez J, Paternain D, Sanz JA (2013) Using the Choquet integral in the fuzzy reasoning method of fuzzy rule-based classification systems. Axioms 2(2):208–223CrossRefzbMATHGoogle Scholar
  7. Bedregal BC, Dimuro GP, Reiser RHS (2009) An approach to interval-valued R-implications and automorphisms. In: Proceedings of the joint 2009 international fuzzy systems association world congress and 2009 European society of fuzzy logic and technology conference, IFSA/EUSFLAT, 2009Google Scholar
  8. Bedregal BC, Dimuro GP, Santiago RHN, Reiser RHS (2010) On interval fuzzy S-implications. Inf Sci 180(8):1373–1389MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bedregal BC, Dimuro GP, Bustince H, Barrenechea E (2013) New results on overlap and grouping functions. Inf Sci 249:148–170MathSciNetCrossRefzbMATHGoogle Scholar
  10. Beliakov G, Pradera A, Calvo T (2007) Aggregation functions: a guide for practitioners. Springer, BerlinzbMATHGoogle Scholar
  11. Bustince H, Fernández J, Mesiar R, Montero J, Orduna R (2009) Overlap index, overlap functions and migrativity. In: Proceedings of IFSA/EUSFLAT conference, pp 300–305Google Scholar
  12. Bustince H, Fernandez J, Mesiar R, Montero J, Orduna R (2010) Overlap functions. Nonlinear Anal Theory Methods Appl 72(3–4):1488–1499MathSciNetCrossRefzbMATHGoogle Scholar
  13. Bustince H, Pagola M, Mesiar R, Hüllermeier E, Herrera F (2012) Grouping, overlaps, and generalized bientropic functions for fuzzy modeling of pairwise comparisons. IEEE Trans Fuzzy Syst 20(3):405–415CrossRefGoogle Scholar
  14. Bustince H, Fernandez J, Kolesárová A, Mesiar R (2015) Directional monotonicity of fusion functions. Eur J Oper Res 244(1):300–308MathSciNetCrossRefzbMATHGoogle Scholar
  15. Choquet G (1953–1954) Theory of capacities. Annales de línstitut Fourier 5: 131–295Google Scholar
  16. Cordón O, del Jesus MJ, Herrera F (1999) A proposal on reasoning methods in fuzzy rule-based classification systems. Int J Approx Reason 20(1):21–45CrossRefGoogle Scholar
  17. Dimuro GP, (2011) On interval fuzzy numbers. In: 2011 Workshop-school on theoretical computer science, WEIT 2011, IEEE, Los Alamitos, 2011Google Scholar
  18. Dimuro GP, Bedregal B (2014) Archimedean overlap functions: the ordinal sum and the cancellation, idempotency and limiting properties. Fuzzy Sets Syst 252:39–54MathSciNetCrossRefzbMATHGoogle Scholar
  19. Dimuro GP, Bedregal B (2015) On residual implications derived from overlap functions. Inf Sci 312:78–88MathSciNetCrossRefGoogle Scholar
  20. Dimuro GP, Bedregal B, Bustince H, Asiáin MJ, Mesiar R (2016a) On additive generators of overlap functions. Fuzzy Sets Syst 287:76–96MathSciNetCrossRefGoogle Scholar
  21. Dimuro GP, Bedregal B, Bustince H, Fernandez J, Lucca G, Mesiar R (2016b) New results on pre-aggregation functions. In: Uncertainty modelling in knowledge engineering and decision making, proceedings of the 12th international FLINS conference (FLINS 2016), world scientific proceedings series on computer engineering and information science 10, Singapura, pp 213–219Google Scholar
  22. Dimuro GP, Bedregal B, Bustince H, Jurio A, Baczyński M, Miś K (2017) $QL$-operations and $QL$-implication functions constructed from tuples $(O, G, N)$ and the generation of fuzzy subsethood and entropy measures. Int J Approx Reason 82:170–192MathSciNetCrossRefzbMATHGoogle Scholar
  23. Duda R, Hart P, Stork D (2001) Pattern classification. Wiley, HobokenzbMATHGoogle Scholar
  24. Elkano M, Galar M, Sanz J, Fernández A, Barrenechea E, Herrera F, Bustince H (2015) Enhancing multi-class classification in FARC-HD fuzzy classifier: on the synergy between n-dimensional overlap functions and decomposition strategies. IEEE Trans Fuzzy Syst 23(5):1562–1580CrossRefGoogle Scholar
  25. Elkano M, Galar M, Sanz J, Bustince H (2016) Fuzzy rule-based classification systems for multi-class problems using binary decomposition strategies: on the influence of n-dimensional overlap functions in the fuzzy reasoning method. Inf Sci 332:94–114CrossRefGoogle Scholar
  26. Eshelman LJ (1991) The CHC adaptive search algorithm: how to have safe search when engaging in nontraditional genetic recombination. In: Rawlings GJE (ed) Foundations of genetic algorithms. Morgan Kaufmann, San Francisco, pp 265–283Google Scholar
  27. Garcia-Jimenez S, Bustince H, Hüllermeier E, Mesiar R, Pal NR, Pradera A (2015) Overlap indices: construction of and application to interpolative fuzzy systems. IEEE Trans Fuzzy Syst 23(4):1259–1273CrossRefGoogle Scholar
  28. Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175(1):247–286MathSciNetCrossRefzbMATHGoogle Scholar
  29. Herrera F, Lozano M, Sánchez AM (2003) A taxonomy for the crossover operator for real-coded genetic algorithms: an experimental study. Int J Intell Syst 18(3):309–338CrossRefzbMATHGoogle Scholar
  30. Ishibuchi H, Nakashima T (2001) Effect of rule weights in fuzzy rule-based classification systems. IEEE Trans Fuzzy Syst 9(4):506–515CrossRefGoogle Scholar
  31. Ishibuchi H, Nakashima T, Nii M (2005) Classification and modeling with linguistic information granules, advanced approaches to linguistic data mining, advanced information processing. Springer, BerlinzbMATHGoogle Scholar
  32. Jurio A, Bustince H, Pagola M, Pradera A, Yager R (2013) Some properties of overlap and grouping functions and their application to image thresholding. Fuzzy Sets Syst 229:69–90MathSciNetCrossRefzbMATHGoogle Scholar
  33. Kavšek B, Lavrač N, Jovanoski V (2003) Advances in intelligent data analysis V: 5th international symposium on intelligent data analysis, IDA 2003, Berlin, Germany. In: Proceedings, Springer, Berlin, pp 230–241Google Scholar
  34. Lucca G, Dimuro GP, Mattos V, Bedregal B, Bustince H, Sanz JA (2015) A family of Choquet-based non-associative aggregation functions for application in fuzzy rule-based classification systems. In: 2015 IEEE international conference on fuzzy systems (FUZZ-IEEE). IEEE, Los Alamitos, pp 1–8Google Scholar
  35. Lucca G, Dimuro GP, Bedregal B, Sanz J, Bustince H (2016a) A proposal for tuning the $\alpha $ parameter in a copula function applied in fuzzy rule-based classification systems. In: Brazilian conference on intelligent systemsGoogle Scholar
  36. Lucca G, Sanz J, Pereira Dimuro G, Bedregal B, Mesiar R, Kolesárová A, Bustince Sola H (2016b) Pre-aggregation functions: construction and an application. IEEE Trans Fuzzy Syst 24(2):260–272CrossRefzbMATHGoogle Scholar
  37. Lucca G, Sanz JA, Dimuro GP, Bedregal B, Asiain MJ, Elkano M, Bustince H (2017) CC-integrals: Choquet-like copula-based aggregation functions and its application in fuzzy rule-based classification systems. Knowl Based Syst 119:32–43CrossRefGoogle Scholar
  38. Martínez L, Rodríguez RM, Herrera F (2015) The 2-tuple linguistic model. Computing with words in decision making. Springer, BerlinzbMATHGoogle Scholar
  39. Mayor G, Trillas E (1986) On the representation of some aggregation functions. In: Proceedings of IEEE international symposium on multiple-valued logic, IEEE, Los Alamitos, pp 111–114Google Scholar
  40. Mesiar R, Kolesárová A, Bustince H, Dimuro GP, Bedregal BC (2016) Fusion functions based discrete Choquet-like integrals. Eur J Oper Res 252(2):601–609MathSciNetCrossRefzbMATHGoogle Scholar
  41. Murofushi T, Sugeno M, Machida M (1994) Non-monotonic fuzzy measures and the Choquet integral. Fuzzy Sets Syst 64(1):73–86MathSciNetCrossRefzbMATHGoogle Scholar
  42. Paternain D, Bustince H, Pagola M, Sussner P, Kolesárová A, Mesiar R (2016) Capacities and overlap indexes with an application in fuzzy rule-based classification systems. Fuzzy Sets Syst 305:70–94MathSciNetCrossRefzbMATHGoogle Scholar
  43. Samantaray SR, El-Arroudi K, Joos G, Kamwa I (2010) A fuzzy rule-based approach for islanding detection in distributed generation. IEEE Trans Power Deliv 25(3):1427–1433CrossRefGoogle Scholar
  44. Sanz JA, Fernández A, Bustince H, Herrera F (2010) Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning. Inf Sci 180(19):3674–3685CrossRefGoogle Scholar
  45. Sanz J, Fernández A, Bustince H, Herrera F (2013) IVTURS: a linguistic fuzzy rule-based classification system based on a new interval-valued fuzzy reasoning method with tuning and rule selection. IEEE Trans Fuzzy Syst 21(3):399–411CrossRefGoogle Scholar
  46. Sanz JA, Galar M, Jurio A, Brugos A, Pagola M, Bustince H (2014) Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system. Appl Soft Comput 20:103–111CrossRefGoogle Scholar
  47. Sanz J, Bernardo D, Herrera F, Bustince H, Hagras H (2015) A compact evolutionary interval-valued fuzzy rule-based classification system for the modeling and prediction of real-world financial applications with imbalanced data. IEEE Trans Fuzzy Syst 23(4):973–990CrossRefGoogle Scholar
  48. Sola HB, Tartas EB, Pagola M, Soria F (2006) Weak fuzzy s-subsethood measures: overlap index. Int J Uncertain Fuzziness Knowl Based Syst 14(5):537–560MathSciNetCrossRefzbMATHGoogle Scholar
  49. Wilcoxon F (1945) Individual comparisons by ranking methods. Biometrics 1:80–83MathSciNetCrossRefGoogle Scholar
  50. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Giancarlo Lucca
    • 1
  • José A. Sanz
    • 1
    • 2
  • Graçaliz P. Dimuro
    • 2
    • 3
  • Benjamín Bedregal
    • 4
  • Humberto Bustince
    • 1
    • 2
  1. 1.Departamento de Automática y ComputaciónUniversidad Publica de NavarraPamplonaSpain
  2. 2.Institute of Smart CitiesUniversidad Publica de NavarraPamplonaSpain
  3. 3.Centro de Ciências ComputacionaisUniversidade Federal do Rio GrandeRio GrandeBrazil
  4. 4.Departamento de Informatica e Matemática AplicadaUniversidade Federal do Rio Grande do NorteNatalBrazil

Personalised recommendations