Skip to main content
SpringerLink
Log in

Use of the q-Gaussian Function in Radial Basis Function Networks

  • Chapter
Foundations of Computational Intelligence Volume 5

Part of the book series: Studies in Computational Intelligence ((SCI,volume 205))

Summary

Radial Basis Function Networks (RBFNs) have been successfully employed in several function approximation and pattern recognition problems. In RBFNs, radial basis functions are used to compute the activation of artificial neurons. The use of different radial basis functions in RBFN has been reported in the literature. Here, the use of the q-Gaussian function as a radial basis function in RBFNs is investigated. An interesting property of the q-Gaussian function is that it can continuously and smoothly reproduce different radial basis functions, like the Gaussian, the Inverse Multiquadratic, and the Cauchy functions, by changing a real parameter q. In addition, the mixed use of different shapes of radial basis functions in only one RBFN is allowed. For this purpose, a Genetic Algorithm is employed to select the number of hidden neurons, and center, width and q parameter of the q-Gaussian radial basis function associated with each radial unit. The RBF Network with the q-Gaussian RBF is compared to RBF Networks with Gaussian, Cauchy, and Inverse Multiquadratic RBFs in problems in the Medical Informatics domain.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atakishiyev, N.M.: On a one-parameter family of q-exponential functions. Journal of Physics A: Mathematical and General 29(10), L223–L227 (1996)

    Article  MathSciNet  Google Scholar 

  2. Atakishiyev, N.M.: On the fourier-gauss transforms of some q-exponential and q-trigonometric functions. Journal of Physics A: Mathematical and General 29(22), 7177–7181 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Atakishiyev, N.M., Feinsilver, P.: On the coherent states for the q-Hermite polynomials and related Fourier transformation. Journal of Physics A: Mathematical and General 29(8), 1659–1664 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Biedenharn, L.C.: The quantum group suq(2) and a q-analogue of the boson operators. Journal of Physics A: Mathematical and General 22(18), I873–I878 (1989)

    Article  MathSciNet  Google Scholar 

  5. Billings, S., Wei, H.-L., Balikhin, M.A.: Generalized multiscale radial basis function networks. Neural Networks 20, 1081–1094 (2007)

    Article  Google Scholar 

  6. Billings, S., Zheng, G.: Radial basis function network configuration using genetic algorithms. Neural Networks 8(6), 877–890 (1995)

    Article  Google Scholar 

  7. Borges, E.P.: A possible deformed algebra and calculus inspired in nonextensive thermostatistics. Physica A: Statistical Mechanics and its Applications 340(1-3), 95–101 (2004)

    Article  MathSciNet  Google Scholar 

  8. Chen, S., Cowan, C.F.N., Grant, P.M.: Orthogonal least squares learning algorithm for radial basis function networks. IEEE Transactions on Neural Networks 2(2), 302–309 (1991)

    Article  Google Scholar 

  9. Floreanini, R., Vinet, L.: Q-orthogonal polynomials and the oscillator quantum group. Letters in Mathematical Physics 22(1), 45–54 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Floreanini, R., Vinet, L.: Quantum algebras and q-special functions. Annals of Physics 221(1), 53–70 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fogel, D.B., Wasson, E.D., Boughton, E.M.: Evolving neural networks for detecting breast cancer. Cancer Letters 96, 49–53 (1995)

    Article  Google Scholar 

  12. Gell-Mann, M., Tsallis, C.: Nonextensive Entropy - Interdisciplinary Applications. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  13. Harpham, C., Dawson, C.W.: The effect of different basis functions on a radial basis function network for time series prediction: A comparative study. Neurocomputing 69, 2161–2170 (2006)

    Article  Google Scholar 

  14. Harpham, C., Dawson, C.W., Brown, M.R.: A review of genetic algorithms applied to training radial basis function networks. Neural Computing and Applications 13(3), 193–201 (2004)

    Article  Google Scholar 

  15. Liu, Y., Yao, X.: Evolutionary design of artificial neural networks with different nodes. In: Proc. of the IEEE Conference on Evolutionary Computation, ICEC, pp. 670–675 (1996)

    Google Scholar 

  16. Maillard, E.P., Gueriot, D.: Rbf neural network, basis functions and genetic algorithms. In: Proc. of the IEEE International Conference on Neural Networks, vol. 4, pp. 2187–2190 (1997)

    Google Scholar 

  17. Mangasarian, O.L., Wolberg, W.H.: Cancer diagnosis via linear programming. SIAM News 23(5), 1–18 (1990)

    Google Scholar 

  18. McAnally, D.S.: Q-exponential and q-gamma functions. i. q-exponential functions. Journal of Mathematical Physics 36(1), 546–573 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1996)

    Google Scholar 

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

    Article  Google Scholar 

  21. Nivanen, L., Le Mehaute, A., Wang, Q.A.: Generalized algebra within a nonextensive statistics. Reports on Mathematical Physics 52(3), 437–444 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Orr, M.: Introduction to radial basis function networks. Center for Cognitive Science, Edinburgh University, Scotland, U. K (1996)

    Google Scholar 

  23. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proceedings of London Mathematical Society 25, 318–343 (1894)

    Article  Google Scholar 

  24. Smith, J.W., Everhart, J.E., Dickson, W.C., Knowler, W.C., Johannes, R.S.: Using the adap learning algorithm to forecast the onset of diabetes mellitus. In: Proc. of the Symposium on Computer Applications and Medical Care, pp. 261–265 (1988)

    Google Scholar 

  25. Tinós, R., Terra, M.H.: Fault detection and isolation in robotic manipulators using a multilayer perceptron and a rbf network trained by kohonen’s self-organizing map. Rev. Controle e Automação 12(1), 11–18 (2001)

    Google Scholar 

  26. Tsallis, C.: Possible generalization of boltzmann-gibbs statistics. Journal of Statistical Physics 52, 479–487 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tsallis, C.: What are the number that experiments provide? Química Nova 17, 468 (1994)

    Google Scholar 

  28. Umarov, S., Tsallis, C., Steinberg, S.: On a q-central limit theorem consistent with nonextensive statistical mechanics. Milan Journal of Mathematic (2008), doi:10.1007/s00032-008-0087-y

    Google Scholar 

  29. Yamano, T.: Some properties of q-logarithm and q-exponential functions in tsallis statistics. Physica A 305, 486–496 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física e Matemática Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP), Universidade de São Paulo (USP), Av. Bandeirantes, 3900, 14040-901, Ribeirão Preto, SP, Brazil

    Renato Tinós & Luiz Otávio Murta Júnior

Authors
  1. Renato Tinós
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luiz Otávio Murta Júnior
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Machine Intelligence Research Labs (MIR Labs), Scientific Network for Innovation and Research Excellence, P.O. Box 2259, 98071-2259, Washington, USA

    Ajith Abraham

  2. Faculty of Computers and Information, Information Technology Department, Cairo University, 5 Ahmed Zewal St., Orman, Giza

    Aboul-Ella Hassanien

  3. Dept. Computer Science, Technical University Ostrava, Tr. 17. Listopadu 15, 708 33, Ostrava, Czech Republic

    Václav Snášel

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Tinós, R., Júnior, L.O.M. (2009). Use of the q-Gaussian Function in Radial Basis Function Networks. In: Abraham, A., Hassanien, AE., Snášel, V. (eds) Foundations of Computational Intelligence Volume 5. Studies in Computational Intelligence, vol 205. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01536-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-01536-6_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-01535-9

  • Online ISBN: 978-3-642-01536-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation