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

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## Abstract

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.

## Keywords

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

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## References

- 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 - 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 - Bennett, G. W. (1988) Determination of anaerobic threshold.
*Canadian Journal of Statistics*,**16**(3), 307–16.Google Scholar - Broomhead, D. S. and Lowe, D. (1988) Multi-variable functional interpolation and adaptive networks.
*Complex Systems*,**2**(3), 269–303.Google Scholar - de Leeuw, J. (1982) Nonlinear principal components analysis. In H. Caussinus, ed.,
*COMPSTAT '82. Proceedings in Computational Statistics*. Physica-Verlag, Vienna.Google Scholar - Flury, B. D. (1993) Estimation of principal points.
*Applied Statistics*,**42**(1), 139–51.Google Scholar - Gifi, A. (1990)
*Nonlinear Multivariate Analysis*. Wiley, New York.Google Scholar - Hand, D. J. (1981)
*Discrimination and Classification*. Wiley, New York.Google Scholar - 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 - 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 - Hastie, T. and Stuetzle, W. (1989) Principal curves.
*Journal of the American Statistical Association*,**84**(406), 502–16.Google Scholar - Hotelling, H. (1933) Analysis of a complex of statistical variables into principal components.
*Journal of Educational Psychology*,**24**, 417–41, 498–520.Google Scholar - Kramer, M. A. (1991) Nonlinear principal component analysis using autoassociative neural networks.
*American Institute of Chemical Engineers Journal*,**37**(2), 233–43.Google Scholar - LeBlanc, M. and Tibshirani, R. (1994) Adaptive principal surfaces.
*Journal of the American Statistical Association*,**89**(425), 53–64.Google Scholar - 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 - 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 - 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 - Pearson, K. (1901) On lines and planes of closest fit.
*Philosophical Magazine*,**6**, 559–72.Google Scholar - 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
- Silverman, B. W. (1986)
*Density Estimation for Statistics and Data Analysis*. Chapman & Hall, London.Google Scholar - Stewart, G. W. (1973)
*Introduction to Matrix Computations*. Academic Press, London.Google Scholar - Tibshirani, R. (1992) Principal curves revisited.
*Statistics and Computing*,**2**(4), 183–90.Google Scholar - 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 - 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 - 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