Skip to main content

Advertisement

Springer Nature Link
Log in

A new hybrid methodology for cooperative-coevolutionary optimization of radial basis function networks

  • Original Article
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

This paper presents a new multiobjective cooperative–coevolutive hybrid algorithm for the design of a Radial Basis Function Network (RBFN). This approach codifies a population of Radial Basis Functions (RBFs) (hidden neurons), which evolve by means of cooperation and competition to obtain a compact and accurate RBFN. To evaluate the significance of a given RBF in the whole network, three factors have been proposed: the basis function’s contribution to the network’s output, the error produced in the basis function radius, and the overlapping among RBFs. To achieve an RBFN composed of RBFs with proper values for these quality factors our algorithm follows a multiobjective approach in the selection process. In the design process, a Fuzzy Rule Based System (FRBS) is used to determine the possibility of applying operators to a certain RBF. As the time required by our evolutionary algorithm to converge is relatively small, it is possible to get a further improvement of the solution found by using a local minimization algorithm (for example, the Levenberg–Marquardt method). In this paper the results of applying our methodology to function approximation and time series prediction problems are also presented and compared with other alternatives proposed in the bibliography.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Angeline P, Saunders G, Pollack J (1994) An evolutionary algorithm that constructs recurrent neural networks. IEEE Trans Neural Netw 5(1):54–65

    Article  Google Scholar 

  • Bersini H, Duchateau A, Bradshaw N (1997) Using incremental learning algorithms in the search for minimal and effective models. In: Proceedings of the 6th IEEE international conference on fuzzy system. IEEE Computer Society Press, Barcelona, pp 1417–1422

  • Broomhead D, Lowe D (1988) Multivariable functional interpolation and adaptive networks. Complex Syst 2:321–355

    MATH  MathSciNet  Google Scholar 

  • Buchtala O, Klimek M, Sick B (2005) Evolutionary optimization of radial basis function classifiers for data mining applications. IEEE Trans Syst Man Cybern B 35(5):928–947

    Article  Google Scholar 

  • Castillo P (2000) Optimización de perceptrones multicapa mediante algoritmos evolutivos. Ph.D. dissertation, University of Granada, Spain

    Google Scholar 

  • Chen S, Woo Y, Luk B (1999) Combined genetic algorithms optimization and regularized orthogonal least squares learning for radial basis function networks. IEEE Trans Neural Netw 10(5):1239–1243

    Article  Google Scholar 

  • Cherkassky V, Lari-Najafi H (1991) Constrained topological mapping for non-parametric regression analysis. Neural Netw 4(1):27–40

    Article  Google Scholar 

  • Cherkassky V, Gehring D, Mulier F (1996) Comparison of adaptive methods for function estimation from samples. IEEE Trans Neural Netw 7(4):969–984

    Article  Google Scholar 

  • Coello CA, Van Veldhuizen DA, Lamont GB (2002) Evolutionary algorithms for solving multi-objective problems. Kluwer, New York

    MATH  Google Scholar 

  • De Falco I, Della Cioppa A, Iazzetta A, Natale P, Tarantino E (1998) Optimizing neural networks for time series prediction. In: Roy R et al (eds) Proceedings of the 3rd on line world conference on soft computing (WSC3). Advances in Soft Computing – Engeenering design and Manufacturing Internet

  • Dickerson J, Kosko B (1996) Fuzzy function approximation with ellipsoidal rules. IEEE Trans Syst Man Cybern B 26(4):542–560

    Article  Google Scholar 

  • Friedman J (1981) Projection pursuit regression. J Am Stat Assoc 76:817–823

    Article  Google Scholar 

  • Friedman J (1991) Multivariate adaptive regression splines (with discussion). Ann Stat 19:1–141

    MATH  Google Scholar 

  • García N, Hervás C, Muñoz J (2002) Multi-objective cooperative coevolution of artificial neural networks. Neural Netw 15:1259–1278

    Article  Google Scholar 

  • Goldberg D, Richardson J (1987) Genetic algorithms with sharing for multimodal function optimization. In: Grefenstette J (ed) Proceedings of the second international conference on genetic algorithms. Lawrence Erlbaum Associates, pp 41–49

  • Golub G, Van Loan C (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • González J, Rojas I, Ortega J, Pomares H, Fernández F, Diaz A (2003) Multiobjective evolutionary optimization of the size, shape and position parameters of radial basis function networks for function approximation. IEEE Trans Neural Netw 14(6):1478–1495

    Article  Google Scholar 

  • Harpham C, Dawson C, Brown M (2004) A review of genetic algorithms applied to training radial basis function networks. Neural Comput Appl 13:193–201

    Article  Google Scholar 

  • Haykin S (1999) Neural Networks. A comprehensive foundation, 2nd edn. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  • Howlett RJ, Jain LD (2001a) Radial basis function networks 1—recent developments in theory and applications. Studies in fuzzyness and soft computing. Physica-Verlag, New York

    Google Scholar 

  • Howlett RJ, Jain LD (2001b) Radial basis function networks 2—New avances in design. Studies in fuzzyness and soft computing. Physica-Verlag, New York

    Google Scholar 

  • Lacerda E, de Carvalho A, Ludermir T (2001). Evolutionary optimization of RBF networks. In: Howlett RJ, Jain LC (eds). Radial basis. function networks 1—Recent developments in theory and applications, vol 66. Studies in fuzzyness and soft computing. Physica-Verlag, New York, pp. 281–309

    Google Scholar 

  • Lacerda E, Carvalho A, Padua A, Bernarda T (2005) Evolutionary radial basis functions for credit assessment. Appl Intell 22:167–181

    Article  Google Scholar 

  • Lipmann R (1987) An introduction to computing with neural nets. IEEE Trans Acoust Speech Signal Process 2(2):4–22

    Google Scholar 

  • Mackey M, Glass L (1977) Oscillation and chaos in physiological control system. Science 197:287–289

    Article  Google Scholar 

  • Mandani E, Assilian S (1975) An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man-Mach Stud 7(1):1–13

    Article  Google Scholar 

  • Marquardt DW (1963) An algorithm for least-squares estimation of non-linear inequalities. SIAM J Appl Math 11:431–441

    Article  MATH  MathSciNet  Google Scholar 

  • Mendel J (1995) Fuzzy logic system for engineering: a tutorial. Proc IEEE 83(3): 345–377

    Article  Google Scholar 

  • Moody J, Darken CJ (1989) Fast learning in networks of locally-tuned processing units. Neural Comput 1:281–294

    Google Scholar 

  • Moriarty D, Miikkulainen R (1997) Forming neural networks through efficient and adaptive coevolution. Evol Comput 5(4):373–399

    Google Scholar 

  • Platt J (1991) A resource-allocating network for function interpolation. Neural Comput 3:213–225

    MathSciNet  Google Scholar 

  • Pomares H, Rojas I, Ortega J, González J, Prieto A (2000) A systematic approach to self-generating fuzzy rule-table for function approximation. IEEE Trans Syst Man Cybern B 30(3):431–447

    Article  Google Scholar 

  • Potter M, De Jong K (2000) Cooperative coevolution: an architecture for evolving coadapted subcomponents. Evol Comput 8(1):1–29

    Article  Google Scholar 

  • Powell M (1985) Radial basis functions for multivariable interpolation: a review. In: IMA. Conference on Algorithms for the approximation of functions and data, pp 143–167

  • Rechenberg I (1973) Evolutionsstrategie: Optimierung Technischer Systeme nach Prinzipien der biologischen Evolution. Frommann-Holzboog, Stuttgart

    Google Scholar 

  • Rivas V, Castillo PA, Merelo-Guervós JJ (2002) Evolved rbf networks for time-series forecasting and function approximation. LNCS 2439:505

    Google Scholar 

  • Rivera AJ, Ortega J, Prieto A (2001) Design of rbf networks by cooperative/competitive evolution of units. In: International conference on artificial neural networks and genetic algorithms. ICANNGA 2001, pp 375–378

  • Rivera AJ, Ortega J, Rojas I, del Jesus M (2003) Co-evolutionary algorithm for RBF by self-organizing population of neurons. LNCS 2686:470–477

    Google Scholar 

  • Rivera AJ (2003) Diseño y optimización de redes de funciones de base radial mediante técnicas bioinspiradas. Ph.D. dissertation, University Granada, Spain

    Google Scholar 

  • Rojas I, Valenzuela O, Prieto A (1997) Statistical analysis of the main parameters in the definition of radial basic function networks. LNCS 1240:882–891

    Google Scholar 

  • Rosin C, Belew R (1997) New methods for competitive coevolution. Evol comput 5(1):1–29

    Google Scholar 

  • Rosipal R, Koska M, Farkaš I (1998) Prediction of chaotic time series with a resource allocating RBF networks. Neural Process Lett 7:185–197

    Article  Google Scholar 

  • Schwefel HP (1995) Evolution and optimum seeking. Wiley, New York

    Google Scholar 

  • Smalz R, Conrad M (1994) Combining evolution with credit apportionment: a new learning algorithm for neural nets. Neural Netw 7(2):341–351

    Article  Google Scholar 

  • Tettamanzi A, Tomassini M (2001) Soft computing. Integrating evolutionary, neural and fuzzy system. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Whitehead B, Choate T (1996) Cooperative-competitive genetic evolution of radial basis function centers and widths for time series prediction. IEEE Trans Neural Netw 7(4):869–880

    Article  Google Scholar 

  • Widrow B, Lehr MA (1990) 30 Years of adaptive neural networks: perceptron, madaline and backpropagation. Proc IEEE 78(9):1415–1442

    Article  Google Scholar 

  • Yao X (1999) Evolving artificial neural networks. Proc IEEE 87(9):1423–1447

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Jaén, Jaen, Spain

    A. J. Rivera & M. J. del Jesus

  2. Department of Computer Architecture and Computer Technology, University of Granada, Granada, Spain

    I. Rojas & J. Ortega

Authors
  1. A. J. Rivera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Rojas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. J. del Jesus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. J. Rivera.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rivera, A.J., Rojas, I., Ortega, J. et al. A new hybrid methodology for cooperative-coevolutionary optimization of radial basis function networks. Soft Comput 11, 655–668 (2007). https://doi.org/10.1007/s00500-006-0128-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-006-0128-9

Keywords

Search

Navigation