Nonlinear Principal Component Analysis: Neural Network Models and Applications

  • Matthias Scholz
  • Martin Fraunholz
  • Joachim Selbig
Part of the Lecture Notes in Computational Science and Enginee book series (LNCSE, volume 58)

Nonlinear principal component analysis (NLPCA) as a nonlinear generalisation of standard principal component analysis (PCA) means to generalise the principal components from straight lines to curves. This chapter aims to provide an extensive description of the autoassociative neural network approach for NLPCA. Several network architectures will be discussed including the hierarchical, the circular, and the inverse model with special emphasis to missing data. Results are shown from applications in the field of molecular biology. This includes metabolite data analysis of a cold stress experiment in the model plant Arabidopsis thaliana and gene expression analysis of the reproductive cycle of the malaria parasite Plasmodium falciparum within infected red blood cells.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kramer, M. A.: Nonlinear principal component analysis using auto-associative neural networks. AIChE Journal, 37(2), 233-243 (1991)CrossRefGoogle Scholar
  2. 2.
    DeMers, D., Cottrell, G. W.: Nonlinear dimensionality reduction. In: Hanson, D., Cowan, J., Giles, L., eds.: Advances in Neural Information Processing Systems 5, San Mateo, CA, Morgan Kaufmann, 580-587 (1993)Google Scholar
  3. 3.
    Hecht-Nielsen, R.: Replicator neural networks for universal optimal source cod-ing. Science, 269, 1860-1863 (1995)CrossRefGoogle Scholar
  4. 4.
    Malthouse, E. C.: Limitations of nonlinear PCA as performed with generic neural networks. IEEE Transactions on Neural Networks, 9(1), 165-173 (1998)CrossRefGoogle Scholar
  5. 5.
    Kirby, M. J., Miranda, R.: Circular nodes in neural networks. Neural Compu-tation, 8(2), 390-402 (1996)CrossRefGoogle Scholar
  6. 6.
    Hsieh, W. W., Wu, A., Shabbar, A.: Nonlinear atmospheric teleconnections. Geo-physical Research Letters, 33(7), L07714 (2006)CrossRefGoogle Scholar
  7. 7.
    Herman, A.: Nonlinear principal component analysis of the tidal dynamics in a shallow sea. Geophysical Research Letters, 34, L02608 (2007)CrossRefGoogle Scholar
  8. 8.
    MacDorman, K., Chalodhorn, R., Asada, M.: Periodic nonlinear principal com-ponent neural networks for humanoid motion segmentation, generalization, and generation. In: Proceedings of the Seventeenth International Conference on Pattern Recognition (ICPR), Cambridge, UK, 537-540 (2004)Google Scholar
  9. 9.
    Scholz, M.: Analysing periodic phenomena by circular PCA. In: Hochreiter, M., Wagner, R. (eds. ) Proceedings BIRD conference. LNBI 4414, Springer-Verlag Berlin Heidelberg, 38-47 (2007)Google Scholar
  10. 10. Scholz, M., Vigário, R.: Nonlinear PCA: a new hierarchical approach. In: Verleysen, M., ed.: Proceedings ESANN, 439-444 (2002)Google Scholar
  11. 11. Hassoun, M. H., Sudjianto, A.: Compression net-free autoencoders. Workshop on Advances in Autoencoder/Autoassociator-Based Computations at the NIPS’97 Conference (1997)Google Scholar
  12. 12.
    Oh, J. H., Seung, H.: Learning generative models with the up-propagation al-gorithm. In: Jordan, M. I., Kearns, M. J., Solla, S. A., eds.: Advances in Neural Information Processing Systems. Vol. 10., The MIT Press, 605-611 (1998)Google Scholar
  13. 13.
    Lappalainen, H., Honkela, A.: Bayesian nonlinear independent component analysis by multi-layer perceptrons. In: Girolami, M. (ed. ) Advances in In-dependent Component Analysis. Springer-Verlag, 93-121 (2000)Google Scholar
  14. Honkela, A., Valpola, H.: Unsupervised variational bayesian learning of non-linear models. In: Saul, L., Weis, Y., Bottous, L. (eds. ) Advances in Neural Information Processing Systems, 17 (NIPS’04), 593-600 (2005)Google Scholar
  15. 15.
    Scholz, M., Kaplan, F., Guy, C., Kopka, J., Selbig, J.: Non-linear PCA: a missing data approach. Bioinformatics, 21(20), 3887-3895 (2005)CrossRefGoogle Scholar
  16. 16.
    Hinton, G. E., Salakhutdinov, R. R.: Reducing the dimensionality of data with neural networks. Science, 313 (5786), 504-507 (2006)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Roweis, S. T., Saul, L. K.: Nonlinear dimensionality reduction by locally linear embedding. Science, 290 (5500), 2323-2326 (2000)CrossRefGoogle Scholar
  18. 18.
    Saul, L. K., Roweis, S. T.: Think globally, fit locally: Unsupervised learning of low dimensional manifolds. Journal of Machine Learning Research, 4 (2), 119-155 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Tenenbaum, J., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science, 290 (5500), 2319-2323 (2000)CrossRefGoogle Scholar
  20. 20.
    Hastie, T., Stuetzle, W.: Principal curves. Journal of the American Statistical Association, 84, 502-516 (1989)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kohonen, T.: Self-Organizing Maps. 3rd edn. Springer (2001)MATHGoogle Scholar
  22. 22.
    Schölkopf, B., Smola, A., Müller, K. R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10, 1299-1319 (1998)CrossRefGoogle Scholar
  23. 23.
    Mika, S., Schölkopf, B., Smola, A., Müller, K. R., Scholz, M., Rätsch, G.: Kernel PCA and de-noising in feature spaces. In: Kearns, M., Solla, S., Cohn, D., eds.: Advances in Neural Information Processing Systems 11, MIT Press, 536-542 (1999)Google Scholar
  24. 24.
    Harmeling, S., Ziehe, A., Kawanabe, M., Müller, K. R.: Kernel-based nonlinear blind source separation. Neural Computation, 15, 1089-1124 (2003)MATHCrossRefGoogle Scholar
  25. 25.
    Jutten, C., Karhunen, J.: Advances in nonlinear blind source separation. In: Proc. Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), Nara, Japan, 245-256 (2003)Google Scholar
  26. 26.
    Cichocki, A., Amari, S.: Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications. Wiley, New York (2003)Google Scholar
  27. 27. Scholz, M.: Approaches to analyse and interpret biological profile data. PhD thesis, University of Potsdam, Germany (2006) URN: urn:nbn:de:kobv:517-opus-7839, URL: http://opus. kobv. de/ubp/volltexte/2006/783/.
  28. 28.
    Bishop, C.: Neural Networks for Pattern Recognition. Oxford University Press (1995)Google Scholar
  29. 29.
    Bourlard, H., Kamp, Y.: Auto-association by multilayer perceptrons and singu-lar value decomposition. Biological Cybernetics, 59 (4-5), 291-294, (1988)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Scholz, M.: Nonlinear PCA based on neural networks. Master’s thesis, Dep. of Computer Science, Humboldt-University Berlin (2002) (in German) Google Scholar
  31. 31.
    Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409-436(1952)MATHMathSciNetGoogle Scholar
  32. 32.
    Little, R. J. A., Rubin, D. B.: Statistical Analysis with Missing Data. 2nd edn. John Wiley & Sons, New York (2002)MATHGoogle Scholar
  33. 33. Ghahramani, Z., Jordan, M.: Learning from incomplete data. Technical Report AIM-1509 (1994)Google Scholar
  34. 34.
    Vesanto, J.: Neural network tool for data mining: SOM toolbox. In: Proceed-ings of Symposium on Tool Environments and Development Methods for Intelli-gent Systems (TOOLMET2000), Oulu, Finland, Oulun yliopistopaino, 184-196 (2000)Google Scholar
  35. 35.
    Oba, S., Sato, M., Takemasa, I., Monden, M., Matsubara, K., Ishii, S.: A bayesian missing value estimation method for gene expression profile data. Bioinformatics, 19(16), 2088-2096 (2003)CrossRefGoogle Scholar
  36. 36. Bishop, C.: Variational principal components. In: Proceedings Ninth Interna-tional Conference on Artificial Neural Networks, ICANN’99, 509-514 (1999)Google Scholar
  37. 37.
    Stock, J., Stock, M.: Quantitative stellar spectral classification. Revista Mexi-cana de Astronomia y Astrofisica, 34, 143-156 (1999)Google Scholar
  38. 38.
    Webber Jr., C., Zbilut, J.: Dynamical assessment of physiological systems and states using recorrence plot strategies. Journal of Applied Physiology, 76, 965-973(1994)Google Scholar
  39. 39.
    Mewett, D. T., Reynolds, K. J., Nazeran, H.: Principal components of recur-rence quantification analysis of EMG. In: Proceedings of the 23rd Annual IEEE/EMBS Conference, Istanbul, Turkey (2001)Google Scholar
  40. 40.
    Kaplan, F., Kopka, J., Haskell, D., Zhao, W., Schiller, K., Gatzke, N., Sung, D., Guy, C.: Exploring the temperature-stress metabolome of Arabidopsis. Plant Physiology, 136(4), 4159-4168 (2004)CrossRefGoogle Scholar
  41. 41.
    Bozdech, Z., Llinas, M., Pulliam, B., Wong, E., Zhu, J., DeRisi, J.: The tran-scriptome of the intraerythrocytic developmental cycle of Plasmodium falci-parum. PLoS Biology, 1 (1), E5 (2003)CrossRefGoogle Scholar
  42. 42.
    Kissinger, J., Brunk, B., Crabtree, J., Fraunholz, M., Gajria B., et al., : The plasmodium genome database. Nature, 419 (6906), 490-492 (2002)CrossRefGoogle Scholar
  43. 43.
    Fridman, E., Carrari, F., Liu, Y. S., Fernie, A., Zamir, D.: Zooming in on a quantitative trait for tomato yield using interspecific introgressions. Science, 305 (5691), 1786-1789 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Matthias Scholz
    • 1
  • Martin Fraunholz
    • 1
  • Joachim Selbig
    • 2
  1. 1.Competence Centre for Functional Genomics, Institute for MicrobiologyErnst-Moritz-Arndt-University GreifswaldGreifswaldGermany
  2. 2.Institute for Biochemistry and BiologyUniversity of PotsdamPotsdamGermany

Personalised recommendations