In environmental sciences, one often encounters large datasets with many variables. For instance, one may have a dataset of the monthly sea surface temperature (SST) anomalies (“anomalies” are the departures from the mean) collected at l = 1,000 grid locations over several decades, i.e. the data are of the form x = [x 1 , …, xl ], where each variable xi (i = 1, …, l) has n samples. The samples may be collected at times tk (k = 1, …, n), so each xi is a time series containing n observations. Since the SST of neighboring grids are correlated, and a dataset with 1,000 variables is quite unwieldy, one looks for ways to condense the large dataset to only a few principal variables. The most common approach is via principal component analysis (PCA), also known as empirical orthogonal function (EOF) analysis (Jolliffe 2002).

In this chapter, we examine the use of MLP NN models for nonlinear PCA (NLPCA) in Section 8.2, the overfitting problem associated with NLPCA in Section 8.3, and the extension of NLPCA to closed curve solutions in Section 8.4. MATLAB codes for NLPCA are downloadable from http://www.ocgy. ubc.ca/projects/clim.pred/download.html.The discrete approach by self-organizing maps is presented in Sections 8.5, and the generalization of NLPCA to complex variables in Section 8.6.

Keywords

Migration Convection Covariance Beach Bors 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baldwin, M., Gray, L., Dunkerton, T., Hamilton, K., Haynes, P., Randel, W., Holton, J., Alexander, M., Hirota, I., Horinouchi, T., Jones, D., Kinnersley, J., Marquardt, C., Sato, K., & Takahashi, M. (2001). The quasi-biennial oscillation. Reviews of Geophysics, 39, 179–229CrossRefGoogle Scholar
  2. Bishop, C. M. (1995). Neural networks for pattern recognition. (482 pp.) Oxford: Oxford University PressGoogle Scholar
  3. Cavazos, T. (1999). Large-scale circulation anomalies conducive to extreme precipitation events and derivation of daily rainfall in northeastern Mexico and southeastern Texas. Journal of Climate, 12, 1506–1523CrossRefGoogle Scholar
  4. Cavazos, T. (2000). Using self-organizing maps to investigate extreme climate events: An application to wintertime precipitation in the Balkans. Journal of Climate, 13, 1718–1732CrossRefGoogle Scholar
  5. Cavazos, T., Comrie, A. C., & Liverman, D. M. (2002). Intrasea-sonal variability associated with wet monsoons in southeast Arizona. Journal of Climate, 15, 2477–2490CrossRefGoogle Scholar
  6. Cherkassky, V., & Mulier, F. (1998). Learning from data (441 pp.). New York: WileyGoogle Scholar
  7. Christiansen, B. (2005). The shortcomings of nonlinear principal component analysis in identifying circulation regimes. Journal of Climate, 18, 4814–4823CrossRefGoogle Scholar
  8. Christiansen, B. (2007). Reply to Monahan and Fyfe's comment on “The shortcomings of nonlinear principal component analysis in identifying circulation regimes”. Journal of Climate, 20, 378–379. DOI: 10.1175/JCLI4006.1CrossRefGoogle Scholar
  9. Clarke, T. (1990). Generalization of neural network to the complex plane. Proceedings of International Joint Conference on Neural Networks, 2, 435–440CrossRefGoogle Scholar
  10. Del Frate, F., & Schiavon, G. (1999). Nonlinear principal component analysis for the radiometric inversion of atmospheric profiles by using neural networks. IEEE Transactions on Geoscience and Remote Sensing, 37, 2335–2342CrossRefGoogle Scholar
  11. Diaz, H. F., & Markgraf, V. (Eds.) (2000) El Nino and the southern oscillation: Multiscale variability and global and regional impacts (496 pp.). Cambridge: Cambridge University PressGoogle Scholar
  12. Georgiou, G., & Koutsougeras, C. (1992). Complex domain backpropagation. IEEE Transactions on Circults and Systems II, 39, 330–334CrossRefGoogle Scholar
  13. Hamilton, K. (1998). Dynamics of the tropical middle atmosphere: A tutorial review. Atmosphere-Ocean, 36, 319– 354Google Scholar
  14. Hamilton, K., & Hsieh, W. W. (2002). Representation of the QBO in the tropical stratospheric wind by nonlinear principal component analysis. Journal of Geophysical Research, 107. DOI: 10.1029/2001JD001250Google Scholar
  15. Hardman-Mountford, N. J., Richardson, A. J., Boyer, D. C., Kreiner, A., & Boyer, H. J. (2003). Relating sardine recruitment in the Northern Benguela to satellite-derived sea surface height using a neural network pattern recognition approach. Progress in Oceanograply, 59, 241–255CrossRefGoogle Scholar
  16. Hastie, T., & Stuetzle, W. (1989). Principal curves. Journal of the American Statistical Association, 84, 502–516CrossRefGoogle Scholar
  17. Hastie, T., Tibshirani, R., & Friedman, J. (2001). Elements of statistical learning: Data mining, inference and prediction (552 pp.). New York: SpringerGoogle Scholar
  18. Hirose, A. (1992). Continuous complex-valued backpropagation learning. Electronic Letters, 28, 1854–1855CrossRefGoogle Scholar
  19. Hoerling, M. P., Kumar, A., & Zhong, M. (1997). El Nino, La Nina and the nonlinearity of their teleconnections. Journal of Climate, 10, 1769–1786CrossRefGoogle Scholar
  20. Holton, J. R., & Tan, H.-C. (1980). The influence of the equatorial quasi-biennial oscillation on the global circulation at 50 mb. Journal of the Atmospheric Sciences, 37, 2200– 2208CrossRefGoogle Scholar
  21. Hsieh, W. W. (2001). Nonlinear principal component analysis by neural networks. Tellus, 53A, 599–615Google Scholar
  22. Hsieh, W. W. (2004). Nonlinear multivariate and time series analysis by neural network methods. Reviews of Geophysics, 42, RG1003. DOI: 10.1029/2002RG000112CrossRefGoogle Scholar
  23. Hsieh, W. W. (2007). Nonlinear principal component analysis of noisy data. Neural Networks, 20, 434–443. DOI 10.1016/j.neunet.2007.04.018CrossRefGoogle Scholar
  24. Hsieh, W. W., & Wu, A. (2002). Nonlinear multichannel singular spectrum analysis of the tropical Pacific climate variability using a neural network approach. Journal of Geophysical Research, 107. DOI: 10.1029/2001JC000957Google Scholar
  25. Jolliffe, I. T. (2002). Principal component analysis (502 pp.) Berlin: SpringerGoogle Scholar
  26. Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200CrossRefGoogle Scholar
  27. Kim, T., & Adali, T. (2002). Fully complex multi-layer per-ceptron network for nonlinear signal processing. Journal of VLSI Signal Processing, 32, 29–43CrossRefGoogle Scholar
  28. Kirby, M. J., & Miranda, R. (1996). Circular nodes in neural networks. Neural Computation, 8, 390–402CrossRefGoogle Scholar
  29. Kohonen, T. (1982). Self-organzing formation of topologi-cally correct feature maps. Biological Cybernetics, 43, 59–69CrossRefGoogle Scholar
  30. Kohonen, T. (2001). Self-Organizing maps (3rd ed., 501 pp.) Berlin: SpringerGoogle Scholar
  31. Kramer, M. A. (1991). Nonlinear principal component analysis using autoassociative neural networks. AIChE Journal 37233–243CrossRefGoogle Scholar
  32. Liu, Y., Wieisberg, R. H., & Mooers, C. N. K. (2006). Performance evaluation of the self-organizing map for feature extraction. Journal of Geophysical Research 111. DOI: 10.1029/2005JC003117Google Scholar
  33. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20130–141CrossRefGoogle Scholar
  34. Monahan, A. H. (2000). Nonlinear principal component analysis by neural networks: Theory and application to the Lorenz system. Journal of Climate 13821–835CrossRefGoogle Scholar
  35. Monahan, A. H. (2001). Nonlinear principal component analysis: Tropical Indo-Pacific sea surface temperature and sea level pressure. Journal of Climate 14219–233CrossRefGoogle Scholar
  36. Monahan, A. H., & Fyfe, J. C. (2007). Comment on “The shortcomings of nonlinear principal component analysis in identifying circulation regimes”. Journal of Climate 20375–377. DOI: 10.1175/JCLI4002.1CrossRefGoogle Scholar
  37. Monahan, A. H., Fyfe, J. C., & Flato, G. M. (2000). A regime view of northern hemisphere atmospheric variability and change under global warming. Geophysics Research Letters 271139–1142CrossRefGoogle Scholar
  38. Monahan, A. H., Pandolfo, L., & Fyfe, J. C. (2001). The preferred structure of variability of the northern hemisphere atmospheric circulation. Geophysical Research Letters28, 1019–1022CrossRefGoogle Scholar
  39. Newbigging, S. C., Mysak, L. A., & Hsieh, W. W. (2003). Improvements to the non-linear principal component analysis method, with applications to ENSO and QBO. Atmosphere-Ocean 41290–298Google Scholar
  40. Nitta, T. (1997). An extension of the back-propagation algo-rtihm to complex numbers. Neural Networks 101391– 1415CrossRefGoogle Scholar
  41. Oja, E. (1982). A simplified neuron model as a principal component analyzer. Journal of Mathematical Biology 15267– 273CrossRefGoogle Scholar
  42. Philander, S. G. (1990). El Niño, La Niña, and the southern oscillation (293 pp.). San Diego, CA: AcademicGoogle Scholar
  43. Preisendorfer, R. W. (1988). Principal component analysis in meteorology and oceanography (425 pp.). Amsterdam: ElsevierGoogle Scholar
  44. Rattan, S. S. P., & Hsieh, W. W. (2004), Nonlinear complex principal component analyis of the tropical Pacific interan-nual wind variability. Geophysical Research Letters 31 (21), L21201. DOI: 10.1029/2004GL020446Google Scholar
  45. Rattan, S. S. P., & Hsieh, W. W. (2005). Complex-valued neural networks for nonlinear complex principal component analysis. Neural Networks 1861–69. DOI: 10.1016/j.neunet.2004.08.002CrossRefGoogle Scholar
  46. Rattan, S. S. P., Ruessink, B. G., & Hsieh, W. W. (2005). Nonlinear complex principal component analysis of nearshore bathymetry. Nonlinear Processes in Geophysics12, 661– 670Google Scholar
  47. Richardson, A. J., Risien, C., & Shillington, F. A. (2003). Using self-organizing maps to identify patterns in satellite imagery. Progress in Oceanography 59223–239CrossRefGoogle Scholar
  48. Richman, M. B. (1986). Rotation of principal components. Journal of Climatology 6293–335CrossRefGoogle Scholar
  49. Rojas, R. (1996). Neural networks – A systematic introduction (502 pp.). Berlin: SpringerGoogle Scholar
  50. Ruessink, B. G., van Enckevort, I. M. J., & Kuriyama, Y. (2004). Non-linear principal component analysis of nearshore bathymetry. Marine Geology 203185– 197CrossRefGoogle Scholar
  51. Saff, E. B., & Snider, A. D. (2003). Fundamentals of complex analysis with applications to engineering and science (528 pp.). Englewood Cliffs, NJ: Prentice-HallGoogle Scholar
  52. Sanger, T. D. (1989). Optimal unsupervised learning in a single-layer linear feedforward neural network. Neural Networks 2459–473CrossRefGoogle Scholar
  53. Schölkopf, B., Smola, A., & Muller, K.-R. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 101299–1319CrossRefGoogle Scholar
  54. Tang, Y., & Hsieh, W. W. (2003). Nonlinear modes of decadal and interannual variability of the subsurface thermal structure in the Pacific Ocean. Journal of the Geophysical Research 108. DOI: 10.1029/2001JC001236Google Scholar
  55. Villmann, T., Merenyi, E., & Hammer, B. (2003). Neural maps in remote sensing image analysis. Neural Networks 16389– 403CrossRefGoogle Scholar
  56. von Storch, H., & Zwiers, F. W. (1999). Statistical analysis in climate research (484 pp.). Cambridge: Cambridge University PressGoogle Scholar
  57. Webb, A. R. (1999). A loss function approach to model selection in nonlinear principal components. Neural Networks 12339–345CrossRefGoogle Scholar
  58. Yacoub, M., Badran, F., & Thiria, S. (2001). A topological hierarchical clustering: Application to ocean color classification. Artificial Neural Networks-ICANN 2001, Proceedings. Lecture Notes in Computer Science492–499Google Scholar

Copyright information

© Springer Science+Business Media B.V 2009

Authors and Affiliations

  1. 1.Department of Earth and Ocean SciencesUniversity of British ColumbiaVancouverCanada

Personalised recommendations