Combining Functional Data Projections for Time Series Classification

  • Alberto Muñoz
  • Javier González
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5856)


We afford the classification of time series in the Functional Data Analysis (FDA) context. To this aim we introduce projections methods for the time series onto appropriate Reproducing Kernel Hilbert Spaces (RKHSs) with the aid of Regularization Theory. Next we project the curves onto a set of different RKHSs. Then we consider the induced Euclidean metrics in these spaces and combine them in order to obtain a single kernel valid for classification purposes. The methodology is tested on some real and simulated classification examples.


Functional data Regularization Theory Reproducing Kernel Hilbert Spaces Kernel Combination Classifier Fusion 


  1. 1.
    Aroszajn, N.: Theory of Reproducing Kernels. Transactions of the American Mathematical Society 68(3), 337–404 (1950)CrossRefMathSciNetGoogle 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)zbMATHCrossRefGoogle Scholar
  3. 3.
    Boser, B.E., Guyon, I., Vapnik, V.: A training algorithm for optimal margin classifiers. In: Proc. Fifth ACM Workshop on Computational Learning Theory (COLT), pp. 144–152. ACM Press, New York (1992)CrossRefGoogle Scholar
  4. 4.
    Cucker, F., Smale, S.: On the Mathematical Foundations of Learning. Bulletin of the American Mathematical Society 39(1), 1–49 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    de Diego, I.M.n., Moguerza, J.M., Muñoz, A.: Combining Kernel Information for Support Vector Classification. In: Roli, F., Kittler, J., Windeatt, T. (eds.) MCS 2004. LNCS, vol. 3077, pp. 102–111. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Ferraty, F., Vieu, P.: Curves discrimination: a nonparametric functional approach. Computational Statistics & Data Analysis 44, 161–173 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    González, J., Muñoz, A.: Representing Functional Data using Support Vector Machines. In: Ruiz-Shulcloper, J., Kropatsch, W.G. (eds.) CIARP 2008. LNCS, vol. 5197, pp. 332–339. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Marx, B., Eilers, P.: Generalized linear regression on sampled signals and curves: a P-spline approach. Technometrics 41(1), 1–13 (1999)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Moguerza, J.M., Muñoz, A.: Support Vector Machines with Applications. Statistical Science 21(3), 322–357 (2006)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Muñoz, A., González, J.: Functional Learning of Kernels for Information Fusion Purposes. In: Ruiz-Shulcloper, J., Kropatsch, W.G. (eds.) CIARP 2008. LNCS, vol. 5197, pp. 277–283. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Parzen, E.: On Recent Advances in Time Series Modelling. IEEE Transactions on Automatic Control AC-19, 723–730 (1977)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, New York (2006)Google Scholar
  13. 13.
    Schlesinger, S.: Approximating Eigenvalues and Eigenfunctions of Symmetric Kernels. Journal of the Society for Industrial and Applied Mathematics 6(1), 1–14 (1957)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Wahba, G.: Spline Models for Observational Data. Series in Applied Mathematics, vol. 59. SIAM, Philadelphia (1990)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

Personalised recommendations