Analysing Periodic Phenomena by Circular PCA

  • Matthias Scholz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4414)


Experimental time courses often reveal a nonlinear behaviour. Analysing these nonlinearities is even more challenging when the observed phenomenon is cyclic or oscillatory. This means, in general, that the data describe a circular trajectory which is caused by periodic gene regulation.

Nonlinear PCA (NLPCA) is used to approximate this trajectory by a curve referred to as nonlinear component. Which, in order to analyse cyclic phenomena, must be a closed curve hence a circular component. Here, a neural network with circular units is used to generate circular components.

This circular PCA is applied to gene expression data of a time course of the intraerythrocytic developmental cycle (IDC) of the malaria parasite Plasmodium falciparum. As a result, circular PCA provides a model which describes continuously the transcriptional variation throughout the IDC. Such a computational model can then be used to comprehensively analyse the molecular behaviour over time including the identification of relevant genes at any chosen time point.


gene expression nonlinear PCA neural networks  nonlinear dimensionality reduction  Plasmodium falciparum 


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  1. 1.
    Scholz, M., Kaplan, F., Guy, C.L., Kopka, J., Selbig, J.: Non-linear PCA: a missing data approach. Bioinformatics 21(20), 3887–3895 (2005)CrossRefGoogle Scholar
  2. 2.
    Jolliffe, I.T.: Principal Component Analysis. Springer, New York (1986)Google Scholar
  3. 3.
    Diamantaras, K.I., Kung, S.Y.: Principal Component Neural Networks. Wiley, New York (1996)zbMATHGoogle Scholar
  4. 4.
    Kramer, M.A.: Nonlinear principal component analysis using auto-associative neural networks. AIChE Journal 37(2), 233–243 (1991)CrossRefGoogle Scholar
  5. 5.
    DeMers, D., Cottrell, G.W.: Nonlinear dimensionality reduction. In: Hanson, D., Cowan, J., Giles, L. (eds.) Advances in Neural Information Processing Systems 5, pp. 580–587. Morgan Kaufmann, San Mateo (1993)Google Scholar
  6. 6.
    Hecht-Nielsen, R.: Replicator neural networks for universal optimal source coding. Science 269, 1860–1863 (1995)CrossRefGoogle Scholar
  7. 7.
    Scholz, M., Vigário, R.: Nonlinear PCA: a new hierarchical approach. In: Verleysen, M. (ed.) Proceedings ESANN, pp. 439–444 (2002)Google Scholar
  8. 8.
    Kirby, M.J., Miranda, R.: Circular nodes in neural networks. Neural Computation 8(2), 390–402 (1996)CrossRefGoogle Scholar
  9. 9.
    Hsieh, W.W.: Nonlinear multivariate and time series analysis by neural network methods. Reviews of Geophysics 42(1), RG1003.1–RG1003.25 (2004)Google Scholar
  10. 10.
    MacDorman, K.F., Chalodhorn, R., Asada, M.: Periodic nonlinear principal component neural networks for humanoid motion segmentation, generalization, and generation. In: Proceedings of the Seventeenth International Conference on Pattern Recognition (ICPR), Cambridge, UK, pp. 537–540 (2004)Google Scholar
  11. 11.
    Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)Google Scholar
  12. 12.
    Haykin, S.: Neural Networks - A Comprehensive Foundation, 2nd edn. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
  13. 13.
    Kissinger, J.C., Brunk, B.P., Crabtree, J., Fraunholz, M.J., et al.: The plasmodium genome database. Nature 419(6906), 490–492 (2002)CrossRefGoogle Scholar
  14. 14.
    Gardner, M.J., Hall, N., Fung, E., et al.: Genome sequence of the human malaria parasite plasmodium falciparum. Nature 419(6906), 498–511 (2002)CrossRefGoogle Scholar
  15. 15.
    Bozdech, Z., Llinas, M., Pulliam, B.L., Wong, E.D., Zhu, J., DeRisi, J.L.: The Transcriptome of the Intraerythrocytic Developmental Cycle of plasmodium falciparum. PLoS Biology 1(1), E5 (2003)Google Scholar
  16. 16.
    Verbeek, J.J., Vlassis, N., Kröse, B.: Procrustes analysis to coordinate mixtures of probabilistic principal component analyzers. Technical report, Computer Science Institute, University of Amsterdam, The Netherlands (2002)Google Scholar
  17. 17.
    Roweis, S.T., Saul, L.K., Hinton, G.E.: Global coordination of locally linear models. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems 14, pp. 889–896. MIT Press, Cambridge (2002)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Matthias Scholz
    • 1
  1. 1.Competence Centre for Functional Genomics (CC-FG), Institute for Microbiology, Ernst-Moritz-Arndt-University GreifswaldGermany

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