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


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.


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



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.


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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

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