On the Use of the GTM Algorithm for Mode Detection

  • Susana Vegas-Azcárate
  • Jorge Muruzábal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2810)

Abstract

The problem of detecting the modes of the multivariate continuous distribution generating the data is of central interest in various areas of modern statistical analysis. The popular self-organizing map (SOM) structure provides a rough estimate of that underlying density and can therefore be brought to bear with this problem. In this paper we consider the recently proposed, mixture-based generative topographic mapping (GTM) algorithm for SOM training. Our long-term goal is to develop, from a map appropriately trained via GTM, a fast, integrated and reliable strategy involving just a few key statistics. Preliminary simulations with Gaussian data highlight various interesting aspects of our working strategy.

Keywords

Mode Detection Pointer Concentration Training Parameter Generative Topographic Mapping Gaussian Centre 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: GTM: The Generative Topographic Mapping. Neural Computation 10, 215–235 (1997a)CrossRefGoogle Scholar
  2. 2.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: GTM: A principle alternative to the Self-Organizing Map. In: Advances in Neural Information Processing Systems, vol. 9. MIT Press, Cambridge (1997)Google Scholar
  3. 3.
    Bishop, C.M., Svensén, M., Williams, C.K.I.: Magnification Factors for the SOM and GTM Algorithms. In: Proccedings 1997 Workshop on Self-Organizing Maps, Helsinki, Finland (1997)Google Scholar
  4. 4.
    Bodt, E., Verleysen, M., Cottrell, M.: Kohonen Maps vs. Vector Quantization for Data Analysis. In: Proceedings of the European Symposium on Artificial Neural Networks, pp. 220–221. D-Facto Publications, Brusseels (1997)Google Scholar
  5. 5.
    Carreira-Perpiñán, M.A.: Mode-Finding for Mixtures of Gaussians Distributions. IEEE Trans. Pattern Analysis and Machine Intelligence 22(11), 1318–1323 (2000)CrossRefGoogle Scholar
  6. 6.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistics Society, B 39(1), 1–38 (1977)MathSciNetMATHGoogle Scholar
  7. 7.
    Good, I.J., Gaskins, R.A.: Density Estimation and Bump-Hunting by Penalized Likelihood Method Exemplified by Scattering and Meteorite Data. Journal of the American Statistical Association 75, 42–73 (1980)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Hartigan, J.A., Hartigan, P.M.: The DIP test of unimodality. Ann. Statist. 13, 70–84 (1985)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Izenman, A.J., Sommer, C.J.: Philatelic mixtures and multimodal densities. J. Am. Statist. Assoc. 83, 941–953 (1988)CrossRefGoogle Scholar
  10. 10.
    Kiviluoto, K., Oja, E.: S-map: A Network with a Simple Self-Organization Algorithm for Generative Topographic Mappings. In: Jordan, M.I., Kearns, M.J., Solla, S.A. (eds.) Advances in Neural Information Processing Systems, vol. 10, pp. 549–555. MIT Press, Cambridge (1998)Google Scholar
  11. 11.
    Kohonen, T.: Self-Organizing Maps. Springer, Berlin (2001)MATHGoogle Scholar
  12. 12.
    Muruzábal, J., Muñoz, A.: On the Visualization of Outliers via Self-Organizing Maps. Journal of Computational and Graphical Statistics 6(4), 355–382 (1997)CrossRefGoogle Scholar
  13. 13.
    Ritter, H., Schulten, K.: On the Stationary State of Kohonen’s Self-Organizing Sensory Mapping. Biological Cybernetics 54, 99–106 (1986)MATHCrossRefGoogle Scholar
  14. 14.
    Sammon, J.W.: A Nonlinear Mapping for Data Structure Analysis. IEEE Transactions on Computers 18(5), 401–409 (1969)CrossRefGoogle Scholar
  15. 15.
    Scott, D.W.: Multivariate Density Estimation. John Wiley and Sons, Inc., New York (1992)MATHCrossRefGoogle Scholar
  16. 16.
    Scott, D.W., Szewczyk, W.F.: The Stochastic Mode Tree and Clustering. Journal of Computational and Graphical Statistics (2000) (to appear)Google Scholar
  17. 17.
    Svensén, M.: The GTM: Toolbox-User’s Guide (1999), http://www.ncrg.aston.ac.uk/GTM
  18. 18.
    Terrell, G.R., Scott, D.W.: Variable kernel density estimation. Ann. Statist. 20, 1236–1265 (1992)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Ultsch, A.: Self-Organizing Neural Networks for Visualization and Classification. In: Opitz, O., Lausen, B., Klar, R. (eds.) Information and Classification, pp. 307–313. Springer, Berlin (1993)Google Scholar
  20. 20.
    Utsugi, A.: Bayesian Sampling and Ensemble Learning in Generative Topographic Mapping. Neural Processing Letters 12(3), 277–290 (2000)MATHCrossRefGoogle Scholar
  21. 21.
    Zheng, Y., Greenleaf, J.F.: The Effect of Concave and Convex Weight Adjustments on Self-Organizing Maps. IEEE Transactions on Neural Networks 7(1), 87–96 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Susana Vegas-Azcárate
    • 1
  • Jorge Muruzábal
    • 1
  1. 1.Statistics and Decision Sciences GroupUniversity Rey Juan CarlosMóstolesSpain

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