Constrained Variable Clustering and the Best Basis Problem in Functional Data Analysis

Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


Functional data analysis involves data described by regular functions rather than by a finite number of real valued variables. While some robust data analysis methods can be applied directly to the very high dimensional vectors obtained from a fine grid sampling of functional data, all methods benefit from a prior simplification of the functions that reduces the redundancy induced by the regularity. In this paper we propose to use a clustering approach that targets variables rather than individual to design a piecewise constant representation of a set of functions. The contiguity constraint induced by the functional nature of the variables allows a polynomial complexity algorithm to give the optimal solution.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank the anonymous reviewer for the detailed and constructive comments that have significantly improved this paper.


  1. 1.
    Auger, I.E., Lawrence, C.E.: Algorithms for the optimal identification of segment neighborhoods. Bull. Math. Biol. 51(1), 39–54 (1989)MathSciNetMATHGoogle Scholar
  2. 2.
    Bellman, R.: On the approximation of curves by line segments using dynamic programming. Commun. ACM 4(6), 284 (1961). DOI Google Scholar
  3. 3.
    Coifman, R.R., Wickerhauser, M.V.: Entropy-based algorithms for best basis selection. IEEE Trans. Inf. Theory 38(2), 713–718 (1992)MATHCrossRefGoogle Scholar
  4. 4.
    Ferré, L., Yao, A.F.: Functional sliced inverse regression analysis. Statistics 37(6), 475–488 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Françcois, D., Krier, C., Rossi, F., Verleysen, M.: Estimation de redondance pour le clustering de variables spectrales. In: Actes des 10èmes journés Européennes Agro-industrie et Méthodes statistiques (Agrostat 2008), pp. 55–61. Louvain-la-Neuve, Belgique (2008)Google Scholar
  6. 6.
    Hastie, T., Buja, A., Tibshirani, R.: Penalized discriminant analysis. Ann. Stat. 23, 73–102 (1995)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hugueney, B., Hébrail, G., Lechevallier, Y.: Réduction de séries temporelles par classées de Statistique de la SFDS. Clamart, France (2006)Google Scholar
  8. 8.
    Jackson, B., Scargle, J., Barnes, D., Arabhi, S., Alt, A., Gioumousis, P., Gwin, E., Sangtrakulcharoen, P., Tan, L., Tsai, T.T.: An algorithm for optimal partitioning of data on an interval. IEEE Signal Process. Lett. 12(2), 105–108 (2005)CrossRefGoogle Scholar
  9. 9.
    Krier, C., Rossi, F., Franois, D., Verleysen, M.: A data-driven functional projection approach for the selection of feature ranges in spectra with ICA or cluster analysis. Chemom. Intell. Lab. Syst. 91(1), 43–53 (2008)CrossRefGoogle Scholar
  10. 10.
    Lechevallier, Y.: Classification automatique optimale sous contrainte d’ordre total. Rapport de recherche 200, IRIA (1976)Google Scholar
  11. 11.
    Lechevallier, Y.: Recherche d’une partition optimale sous contrainte d’ordre total. Rapport de recherche RR-1247, INRIA (1990).
  12. 12.
    Olsson, R.J.O., Karlsson, M., Moberg, L.: Compression of first-order spectral data using the b-spline zero compression method. J. Chemom. 10(5–6), 399–410 (1996)CrossRefGoogle Scholar
  13. 13.
    Ramsay, J., Silverman, B.: Functional data analysis. Springer Series in Statistics. Springer, New York, NY (1997)Google Scholar
  14. 14.
    Rossi, F., Françcois, D., Wertz, V., Verleysen, M.: Fast selection of spectral variables with b-spline compression. Chemom. Intell. Lab. Syst. 86(2), 208–218 (2007)CrossRefGoogle Scholar
  15. 15.
    Saito, N., Coifman, R.R.: Local discriminant bases and their applications. J. Math Imaging Vis. 5(4), 337–358 (1995)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Stone, H.: Approximation of curves by line segments. Math. Comput. 15, 40–47 (1961)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Institut Télécom, Télécom ParisTech, LTCI – UMR CNRS 5141ParisFrance
  2. 2.Projet AxIS, INRIA Paris RocquencourtDomaine de Voluceau, RocquencourtLe Chesnay CedexFrance

Personalised recommendations