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.
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References
Kramer, M. A.: Nonlinear principal component analysis using auto-associative neural networks. AIChE Journal, 37(2), 233-243 (1991)
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)
Hecht-Nielsen, R.: Replicator neural networks for universal optimal source cod-ing. Science, 269, 1860-1863 (1995)
Malthouse, E. C.: Limitations of nonlinear PCA as performed with generic neural networks. IEEE Transactions on Neural Networks, 9(1), 165-173 (1998)
Kirby, M. J., Miranda, R.: Circular nodes in neural networks. Neural Compu-tation, 8(2), 390-402 (1996)
Hsieh, W. W., Wu, A., Shabbar, A.: Nonlinear atmospheric teleconnections. Geo-physical Research Letters, 33(7), L07714 (2006)
Herman, A.: Nonlinear principal component analysis of the tidal dynamics in a shallow sea. Geophysical Research Letters, 34, L02608 (2007)
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)
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)
10. Scholz, M., Vigário, R.: Nonlinear PCA: a new hierarchical approach. In: Verleysen, M., ed.: Proceedings ESANN, 439-444 (2002)
11. Hassoun, M. H., Sudjianto, A.: Compression net-free autoencoders. Workshop on Advances in Autoencoder/Autoassociator-Based Computations at the NIPS’97 Conference (1997)
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)
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)
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)
Scholz, M., Kaplan, F., Guy, C., Kopka, J., Selbig, J.: Non-linear PCA: a missing data approach. Bioinformatics, 21(20), 3887-3895 (2005)
Hinton, G. E., Salakhutdinov, R. R.: Reducing the dimensionality of data with neural networks. Science, 313 (5786), 504-507 (2006)
Roweis, S. T., Saul, L. K.: Nonlinear dimensionality reduction by locally linear embedding. Science, 290 (5500), 2323-2326 (2000)
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)
Tenenbaum, J., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science, 290 (5500), 2319-2323 (2000)
Hastie, T., Stuetzle, W.: Principal curves. Journal of the American Statistical Association, 84, 502-516 (1989)
Kohonen, T.: Self-Organizing Maps. 3rd edn. Springer (2001)
Schölkopf, B., Smola, A., Müller, K. R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10, 1299-1319 (1998)
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)
Harmeling, S., Ziehe, A., Kawanabe, M., Müller, K. R.: Kernel-based nonlinear blind source separation. Neural Computation, 15, 1089-1124 (2003)
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)
Cichocki, A., Amari, S.: Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications. Wiley, New York (2003)
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/.
Bishop, C.: Neural Networks for Pattern Recognition. Oxford University Press (1995)
Bourlard, H., Kamp, Y.: Auto-association by multilayer perceptrons and singu-lar value decomposition. Biological Cybernetics, 59 (4-5), 291-294, (1988)
Scholz, M.: Nonlinear PCA based on neural networks. Master’s thesis, Dep. of Computer Science, Humboldt-University Berlin (2002) (in German)
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)
Little, R. J. A., Rubin, D. B.: Statistical Analysis with Missing Data. 2nd edn. John Wiley & Sons, New York (2002)
33. Ghahramani, Z., Jordan, M.: Learning from incomplete data. Technical Report AIM-1509 (1994)
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)
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)
36. Bishop, C.: Variational principal components. In: Proceedings Ninth Interna-tional Conference on Artificial Neural Networks, ICANN’99, 509-514 (1999)
Stock, J., Stock, M.: Quantitative stellar spectral classification. Revista Mexi-cana de Astronomia y Astrofisica, 34, 143-156 (1999)
Webber Jr., C., Zbilut, J.: Dynamical assessment of physiological systems and states using recorrence plot strategies. Journal of Applied Physiology, 76, 965-973(1994)
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)
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)
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)
Kissinger, J., Brunk, B., Crabtree, J., Fraunholz, M., Gajria B., et al., : The plasmodium genome database. Nature, 419 (6906), 490-492 (2002)
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)
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Scholz, M., Fraunholz, M., Selbig, J. (2008). Nonlinear Principal Component Analysis: Neural Network Models and Applications. In: Gorban, A.N., Kégl, B., Wunsch, D.C., Zinovyev, A.Y. (eds) Principal Manifolds for Data Visualization and Dimension Reduction. Lecture Notes in Computational Science and Enginee, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73750-6_2
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