Soft Computing

, Volume 16, Issue 3, pp 451–470 | Cite as

Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection

  • Marta Galende-Hernández
  • Gregorio I. Sainz-Palmero
  • Maria J. Fuente-Aparicio
Original Paper


The aim of this paper is to develop a general post-processing methodology to reduce the complexity of data-driven linguistic fuzzy models, in order to reach simpler fuzzy models preserving enough accuracy and better fuzzy linguistic performance with respect to their initial values. This post-processing approach is based on rule selection via the formulation of a bi-objective problem with one objective focusing on accuracy and the other on interpretability. The latter is defined via the aggregation of several interpretability measures, based on the concepts of similarity and complexity of fuzzy systems and rules. In this way, a measure of the fuzzy model interpretability is given. Two neuro-fuzzy systems for providing initial fuzzy models, Fuzzy Adaptive System ART based and Neuro-Fuzzy Function Approximation and several case studies, data sets from KEEL Project Repository, are used to check this approach. Both fuzzy and neuro-fuzzy systems generate Mamdani-type fuzzy rule-based systems, each with its own particularities and complexities from the point of view of the fuzzy sets and the rule generation. Based on these systems and data sets, several fuzzy models are generated to check the performance of the proposal under different restrictions of complexity and fuzziness.


Fuzzy modeling Accuracy Interpretability Complexity Genetic algorithms 



The authors would like to thank Francisco Herrera and the reviewers for their valuable and useful comments and support in the preparation of this manuscript. This work was supported by the Spanish Ministry of Science and Innovation under Grants no. CIT-460000-2009-46 and DPI2009-14410-C02-02.


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Marta Galende-Hernández
    • 1
  • Gregorio I. Sainz-Palmero
    • 1
    • 2
  • Maria J. Fuente-Aparicio
    • 2
  1. 1.CARTIF Centro TecnológicoBoecilloSpain
  2. 2.Department of Systems Engineering and Control, School of Industrial EngineeringUniversity of ValladolidValladolidSpain

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