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Representing Functional Data Using Support Vector Machines

  • Javier González
  • Alberto Muñoz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5197)

Abstract

Functional data are difficult to manage for many traditional pattern recognition techniques, given the very high (or intrinsically infinite) dimensionality. The reason is that functional data are functions and most algorithms are designed to work with (small) finite-dimensional vectors. In this paper we propose a functional analysis technique to obtain finite-dimensional representations of functional data. The key idea is to consider each functional curve as a point in a general function space and then project these points onto a Reproducing Kernel Hilbert Space with the aid of a Support Vector Machine. We show some theoretical properties of the method and illustrate the performance of the proposed representation in clustering using a real data set.

Keywords

Support Vector Machines Reproducing Kernel Hilbert Spaces Functional Data 

References

  1. 1.
    Aroszajn, N.: Theory of Reproducing Kernels. Transactions of the American Mathematical Society 68(3), 337–404 (1950)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bengio, Y., Delalleau, O., Le Roux, N., Paiement, J.-F., Vincent, P., Ouimet, M.: Learning eigenfunctions links spectral embedding and kernel PCA. Neural Computation 16, 2197–2219 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cucker, F., Smale, S.: On the Mathematical Foundations of Learning. Bulletin of the American Mathematical Society 39(1), 1–49 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kimeldorf, G.S., Wahba, G.: A Correspondence between Bayesian Estimation on Stochastic Processes and Smoothing by Splines. Annals of Mathematical Statistics 2, 495–502 (1971)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Moguerza, J.M., Muñoz, A.: Support Vector Machines with Applications. Statistical Science 21(3), 322–357 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Schlesinger, S.: Approximating Eigenvalues and Eigenfunctions of Symmetric Kernels. Journal of the Society for Industrial and Applied Mathematics 6(1), 1–14 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  8. 8.
    Schölkopf, B., Herbrich, R., Smola, A.J., Williamson, R.C.: A Generalized Representer Theorem. In: Helmbold, D.P., Williamson, B. (eds.) COLT 2001 and EuroCOLT 2001. LNCS (LNAI), vol. 2111, pp. 416–426. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Wahba, G.: Spline Models for Observational Data. Series in Applied Mathematics, vol. 59. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Javier González
    • 1
  • Alberto Muñoz
    • 1
  1. 1.Universidad Carlos III de MadridGetafeSpain

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