Skip to main content
SpringerLink
Log in

Sensitivity analysis of fuzzy rule-based classification systems by means of the Lipschitz condition

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

The fuzzy rule-based classifier can be taken as a function that assigns to a point from the feature space a class, or a class with an association degree. Under this assumption, the robustness of fuzzy rule-based classifiers is investigated by means of the Lipschitz condition. The Lipschitz continuity of fuzzy sets, fuzzy rules and whole fuzzy rule-based classifiers is examined for multi-polar outputs, extended multi-polar outputs and outputs in the form of a class. Related performance of a fuzzy rule-based classifier is also discussed. All studied concepts are shown on an exemplar fuzzy rule-based classifier.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Anderson E (1935) The irises of the Gaspe Peninsula. Bull Am Iris Soc 59:2–5

    Google Scholar 

  • Beliakov G, Calvo T, James S (2010) On Lipschitz properties of generated aggregation functions. Fuzzy Sets Syst 161(10):1437–1447

    Article  MathSciNet  MATH  Google Scholar 

  • Calvo T, Mesiar R (2001) Stability of aggregation operators. In: Proceedings of second conference European Society for fuzzy logic and technology. Leicester, pp 475–478

  • Carmona P, Castro JL, Zurita JM (2004) Fuzzy rule identification with exceptions. IEEE Trans Fuzzy Syst 12(1):140–151

    Article  MathSciNet  Google Scholar 

  • Casillas J, Cordón O, Del Jesus MJ, Herrera F (2001) Genetic feature selection in a fuzzy rule-based classification system learning process for high-dimensional problems. Inf Sci 136(1–4):135–157

    Article  MATH  Google Scholar 

  • Castro JL, Castro-Schez JJ, Zurita JM (1999) Learning maximal structure rules in fuzzy logic for knowledge acquisition in expert systems. Fuzzy Sets Syst 101(3):331–342

    Article  MathSciNet  MATH  Google Scholar 

  • Cordón O, del Jesus MJ, Herrera F (1998) Genetic learning of fuzzy rule-based classification systems cooperating with fuzzy reasoning methods. Int J Intell Syst 13(1011):1025–1053

    Article  Google Scholar 

  • 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):21–45

    Article  Google Scholar 

  • De Cock M, Cornelis C, Kerre EE (2005) Elicitation of fuzzy association rules from positive and negative examples. Fuzzy Sets Syst 149(1):73–85

    Article  MATH  Google Scholar 

  • Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Eugen 7(Part II):179–188

    Article  Google Scholar 

  • Grabisch M, Marichal J-L, Mesiar R, Pap E (2009) Aggregation functions. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hu YC, Tzeng GH (2003) Elicitation of classification rules by fuzzy data mining. Eng Appl Artif Intell 16(7–8):709–716

    Article  Google Scholar 

  • Ishibuchi H, Nozaki K, Tanaka H (1992) Distributed representation of fuzzy rules and its application to pattern classification. Fuzzy Sets Syst 52(1):21–32

    Article  Google Scholar 

  • Ishibuchi H, Nozaki K, Yamamoto N, Tanaka H (1994) Construction of fuzzy classification systems with rectangular fuzzy rules using genetic algorithms. Fuzzy Sets Syst 65:237–253

    Article  MathSciNet  Google Scholar 

  • Ishibuchi H, Nakashima T, Murata T (2001) Three-objective genetics-based machine learning for linguistic rule extraction. Inf Sci 136(1–4):109–133

    Article  MATH  Google Scholar 

  • Ishibuchi H, Morisawa T, Nakashima T (1996) Voting schemes for fuzzy-rule-based classification systems. In: Proceedings of the Fifth IEEE international conference on fuzzy systems (FUZZ-IEEE’96), pp 614–620

  • Ishibuchi H, Yamamoto T (2005) Rule weight specification in fuzzy rule-based classification systems. IEEE Trans Fuzzy Syst 13(4):428–435

    Article  Google Scholar 

  • Isibuchi H, Yamamoto T, Nakashima T (2005) Hybridization of fuzzy GBML approaches for pattern classification problems. IEEE Trans Syst Man Cybern Part B Cybern 35(2):359–365

    Article  Google Scholar 

  • Kuncheva LI (2000) How good are fuzzy If-Then classifiers? IEEE Trans Syst Man Cybern Part B Cybern 30(4):501–509

    Article  Google Scholar 

  • Lee K-M (2004) Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets and bipolar-valued fuzzy sets. J Fuzzy Log Intell Syst 14(2):125–129

    Google Scholar 

  • López V, Fernández A, del Jesus MJ, Herrera F (2013) A hierarchical genetic fuzzy system based on genetic programming for addressing classification with highly imbalanced and borderline data-sets. Knowl Base Syst 38:85–104

    Article  Google Scholar 

  • Mesiarová-Zemánková A (2014) Multi-polar aggregation operators in reasoning methods for fuzzy rule-based classification systems. IEEE Trans Fuzzy Syst 22(6):1569–1584

    Article  Google Scholar 

  • Mesiarová-Zemánková A, Ahmad K (2012) Multi-polar aggregation. In: Greco S et al (ed) Proceedings of IPMU’2012. Catania, pp 379–387

  • Mesiarová-Zemánková A, Ahmad K (2013) Multi-polar Choquet integral. Fuzzy Sets Syst 220:1–20

    Article  MATH  Google Scholar 

  • Mesiarová-Zemánková A, Ahmad K (2014) Extended multi-polarity and multi-polar-valued fuzzy sets. Fuzzy Sets Syst 234:61–78

    Article  Google Scholar 

  • Mesiarová-Zemánková A, Hyčko M (2014) Multi-capacities and multi-polar capacities. Fuzzy Sets Syst. doi:10.1016/j.fss.2014.12.005

  • Mitra S, Kuncheva LI (1995) Improving classification performance using fuzzy MLP and two-level selective partitioning of the feature space. Fuzzy Sets Syst 70(1):1–13

    Article  Google Scholar 

  • Nauck D, Kruse R (1997) A neuro-fuzzy method to learn fuzzy classification rules from data. Fuzzy Sets Syst 89(3):277–288

    Article  MathSciNet  Google Scholar 

  • Pitambare DP, Kamde PM (2013) Fuzzy classification system for glass data classification. Int J Eng Res 2(2):33–37

    Google Scholar 

  • Roubos JA, Setnes M, Abonyi J (2003) Learning fuzzy classification rules from labeled data. Inf Sci 150(1–2):77–93

    Article  MathSciNet  Google Scholar 

  • Sánchez L, Couso I, Corrales JA, Cordón O, Del Jesus MJ, Herrera F (2001) Combining GP operators with SA search to evolve fuzzy rule based classifiers. Inf Sci 136(1–4):175–191

    Article  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank the editor and two anonymous reviewers for their valuable comments. This work was supported by grants VEGA 2/0049/14, APVV-0073-10 and Program Fellowship of SAS.

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Andrea Mesiarová-Zemánková

Authors
  1. Andrea Mesiarová-Zemánková
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Mesiarová-Zemánková.

Additional information

Communicated by M. Navara.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mesiarová-Zemánková, A. Sensitivity analysis of fuzzy rule-based classification systems by means of the Lipschitz condition. Soft Comput 20, 103–113 (2016). https://doi.org/10.1007/s00500-015-1744-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-015-1744-z

Keywords

Search

Navigation