Advertisement

Fuzzy Optimization and Decision Making

, Volume 3, Issue 1, pp 63–82 | Cite as

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

  • Mohit Kumar
  • Regina Stoll
  • Norbert Stoll
Article

Abstract

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.

fuzzy-modelling least-squares problems uncertainty robustness second-order cone programming regularization robust identification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Babuska, R. (2000). ''Construction of Fuzzy Systems-interplay between Precision and Transparency,'' In Proc. ESIT 2000, Aachen, pp. 445-452.Google Scholar
  2. Bodenhofer, U. and P. Bauer. (2000). ''Towards an Axiomatic Treatment of Interpretability,'' In Proc. IIZUKA2000, Iizuka, pp. 334-339.Google Scholar
  3. Burger, M., H. W. Engl, J. Haslinger, and U. Bodenhofer. (2002). ''Regularized Data-driven Construction of Fuzzy Controllers,'' Journal of Inverse and Ill-posed Problems 10, 319–344.Google Scholar
  4. Chandrasekaran, S., G. H. Golub, M. Gu, and A. H. Sayed. (1998). ''Parameter Estimation in the Presence of Bounded Data Uncertainties,'' SIMAX, 19, 235–252.Google Scholar
  5. Demoment, G. (1989). ''Image Reconstruction and Restoration: Overview of Common Estimation Problems,'' IEEE Transaction on Acoustic Speech and Signal Processing 37, 3667–3672.Google Scholar
  6. Elden, L. (1977). ''Algorithm for Regularization of Ill-conditioned Least Squares Problems,'' BIT 17, 134–145.Google Scholar
  7. Espinosa, J. and J. Vandewalle. (2000). ''Constructing Fuzzy Models with Linguistic Integrity from Numerical Data-AFRELI Algorithm,'' IEEE Transactions on Fuzzy Systems 8(5), 591–600.Google Scholar
  8. Furuya, M. (1989). ''Optimization of Weighting Constant for Regularization in Least Squares System Identification,'' Transactions on IEICE A J72A, 1012-1015.Google Scholar
  9. Ghaoui, L. El. and H. Lebret. (1997). ''Robust Solutions to Least-Squares Problems with Uncertain Data,'' SIAM Journal of Matrix Analysis and Applications 18, 1035–1064.Google Scholar
  10. Golub, G. H. and C. F. Van Loan. (1989). ''Matrix Computations'', 2nd Edition ''Baltimore, MD: Johns Hopkins University Press.Google Scholar
  11. Hanke M. and P. C. Hansen. Regularization Methods for Large-scale Problems. Surveys on Mathematics for Industry 3, 253–315.Google Scholar
  12. Hunt, B. R. (1973). ''The Application of Constrained Least Square Estimation to Image Restoration by Digital Computer,'' IEEE Transactions on Computers C-22, 805–812.Google Scholar
  13. Lawson, C. L. and R. J. Hanson. (1995). ''Solving Least Squares Problems'', Philadelphia: SIAM Publications.Google Scholar
  14. Nashed, M. Z. (1981). ''Operator-theoretic and Computational Approaches to Ill-posed Problems with Applications to Antenna Theory,'' IEEE Transactions on Antenna and Propagation 29, 220–231.Google Scholar
  15. Setnes, M., R. Babuka, and H. B. Verbruggen. (1998). ''Rule-based Modeling: Precision, and Transparency,'' IEEE Transactions on Systems Man and Cybernetics Part C: Applications and Reviews, 28, 165–169.Google Scholar
  16. Takagi, T. and M. Sugeno. (1985). ''Fuzzy Identification of Systems and its Applications to Modeling and Control,'' IEEE Transactions on Systems Man and Cybernetics 15, 116–132.Google Scholar
  17. Zadeh, L. A. (1973). ''Outline of a new Approach to the Analysis of Complex Systems and Decision Processes,'' IEEE Transactions on Systems Man and Cybernetics 3(1), 28–44.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Mohit Kumar
    • 1
  • Regina Stoll
    • 1
  • Norbert Stoll
    • 2
  1. 1.Institute of Occupational and Social Medicine, Faculty of MedicineUniversity of RostockRostockGermany
  2. 2.Institute of Automation, Department of Electrical Engineering and Information TechnologyUniversity of RostockRostock-WarnemündeGermany

Personalised recommendations