Statistics and Computing

, Volume 6, Issue 2, pp 159–168 | Cite as

An approach to non-linear principal components analysis using radially symmetric kernel functions

  • Andrew R. Webb


An approach to non-linear principal components using radially symmetric kernel basis functions is described. The procedure consists of two steps: a projection of the data set to a reduced dimension using a non-linear transformation whose parameters are determined by the solution of a generalized symmetric eigenvector equation. This is achieved by demanding a maximum variance transformation subject to a normalization condition (Hotelling's approach) and can be related to the homogeneity analysis approach of Gifi through the minimization of a loss function. The transformed variables are the principal components whose values define contours, or more generally hypersurfaces, in the data space. The second stage of the procedure defines the fitting surface, the principal surface, in the data space (again as a weighted sum of kernel basis functions) using the definition of self-consistency of Hastie and Stuetzle. The parameters of this principal surface are determined by a singular value decomposition and crossvalidation is used to obtain the kernel bandwidths. The approach is assessed on four data sets.


Principal components analysis principal curves radial basis functions homogeneity analysis functional approximation self-consistency cross-validation least-squares approximation generalized eigenvalue problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Becker, R. A. and Chambers, J. M. (1984) S. An Interactive Environment for Data Analysis and Graphics. Wadsworth Statistics/Probability Series, Belmont, CA.Google Scholar
  2. Bekker, P. and de Leeuw, J. (1988) Relations between variants of non-linear principal components analysis. In J. L. A. van Rijckevorsel and J. de Leeuw, eds, Component and Correspondence Analysis. Dimension Reduction by Function Approximation, pp. 1–31, Wiley, New York.Google Scholar
  3. Bennett, G. W. (1988) Determination of anaerobic threshold. Canadian Journal of Statistics, 16(3), 307–16.Google Scholar
  4. Broomhead, D. S. and Lowe, D. (1988) Multi-variable functional interpolation and adaptive networks. Complex Systems, 2(3), 269–303.Google Scholar
  5. de Leeuw, J. (1982) Nonlinear principal components analysis. In H. Caussinus, ed., COMPSTAT '82. Proceedings in Computational Statistics. Physica-Verlag, Vienna.Google Scholar
  6. Flury, B. D. (1993) Estimation of principal points. Applied Statistics, 42(1), 139–51.Google Scholar
  7. Gifi, A. (1990) Nonlinear Multivariate Analysis. Wiley, New York.Google Scholar
  8. Hand, D. J. (1981) Discrimination and Classification. Wiley, New York.Google Scholar
  9. Hand, D. J. (1982) Kernel Discriminant Analysis. Volume 2 of Pattern Recognition and Image Processing Research Studies Series. Research Studies Press, Letchworth, Herts.Google Scholar
  10. Hand, D. J., Daly, F., Lunn, A. D., McConway, K. J. and Ostrowski, E. (1994) A Handbook of Small Data Sets. Chapman & Hall, London.Google Scholar
  11. Hastie, T. and Stuetzle, W. (1989) Principal curves. Journal of the American Statistical Association, 84(406), 502–16.Google Scholar
  12. Hotelling, H. (1933) Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417–41, 498–520.Google Scholar
  13. Kramer, M. A. (1991) Nonlinear principal component analysis using autoassociative neural networks. American Institute of Chemical Engineers Journal, 37(2), 233–43.Google Scholar
  14. LeBlanc, M. and Tibshirani, R. (1994) Adaptive principal surfaces. Journal of the American Statistical Association, 89(425), 53–64.Google Scholar
  15. Lowe, D. (1995) On the use of nonlocal and non positive definite basis functions in radial basis function networks. Fourth IEE International Conference on Artificial Neural Networks, Cambridge, pp. 206–211. IEE Conference Publication 409.Google Scholar
  16. Martin, J.-F. (1988) On probability coding. In J. L. A. van Rijckevorsel and J. de Leeuw, editors, Component and Correspondence Analysis. Dimension Reduction by Function Approximation, pp. 103–14. Wiley, New York.Google Scholar
  17. Nakagawa, S., Ono, Y. and Hirata, Y. (1991) Dimensionality reduction of dynamical patterns using a neural network. In B. H. Juang, S. Y. Kung, and C. A. Kamm, eds, Neural Networks for Signal Processing, Proceedings of the 1991 IEEE Workshop, pp. 256–65, Princeton, NJ.Google Scholar
  18. Pearson, K. (1901) On lines and planes of closest fit. Philosophical Magazine, 6, 559–72.Google Scholar
  19. Powell, M. J. D. (1990) The theory of radial basis function approximation in 1990. DAMPT Numerical Analysis Report 1990/NA11, University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, UK.Google Scholar
  20. Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.Google Scholar
  21. Stewart, G. W. (1973) Introduction to Matrix Computations. Academic Press, London.Google Scholar
  22. Tibshirani, R. (1992) Principal curves revisited. Statistics and Computing, 2(4), 183–90.Google Scholar
  23. Usui, S., Nakauchi, S. and Nakano, M. (1990) Reconstruction of Munsell color space by a five-layered neural network. In IJCNN Int. Joint Conf. Neural Networks, San Diego, Vol. II, pp. 515–20.Google Scholar
  24. van Rijckevorsel, J. L. A. (1988) Fuzzy coding and B-splines. In J. L. A. van Rijckevorsel and J. de Leeuw, eds, Component and Correspondence Analysis. Dimension Reduction by Function Approximation, pp. 33–54. Wiley, New York.Google Scholar
  25. Wyse, N., Dubes, R. and Jain, A. K. (1980) A critical evaluation of intrinsic dimensionality algorithms. In E. S. Gelsema and L. N. Kanal, eds, Pattern Recognition in Practice, pp. 415–25. North-Holland, Amsterdam.Google Scholar

Copyright information

© Chapman & Hall 1996

Authors and Affiliations

  • Andrew R. Webb
    • 1
  1. 1.Defence Research AgencyMalvernUK

Personalised recommendations