Abstract
We introduce two new methods of deriving the classical PCA in the framework of minimizing the mean square error upon performing a lower-dimensional approximation of the data. These methods are based on two forms of the mean square error function. One of the novelties of the presented methods is that the commonly employed process of subtraction of the mean of the data becomes part of the solution of the optimization problem and not a pre-analysis heuristic. We also derive the optimal basis and the minimum error of approximation in this framework and demonstrate the elegance of our solution in comparison with a recent solution in the framework.
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References
Bishop CM (2006) Pattern recognition and machine learning. Information science and statistics. Springer, New York
Diamantaras KI, Kung SY (1996) Principal component neural networks: theory and applications. John wiley, NewYork
Duda RO, Hart PE, Stork DG (2001) Pattern classification. 2nd edn. Wiley Interscience, New York
Fukunaga K (1990) Introduction to statistical pattern recognition. Computer science and scientific computing, 2nd edn. Academic Press, San Diego
Fukunaga K, Koontz WLG (1970) Application of the Karhunen–Loeve expansion to feature selection and ordering. IEEE Transac Comput C- 19(4): 311–318
Harsanyi JC, Chang C-I (1994) Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach. IEEE Transac Geosci Remote Sens 32(4): 779–785
Hotelling H (1933) Analysis of a complex of statistical variables into principal components. J Educ Psychol 24: 417–441
Huo X, Elad M, Flesia AG, Muise B, Stanfill R, Mahalanobis A et al (2003) Optimal reduced-rank quadratic classifiers using the Fukunaga–Koontz transform with applications to automated target recognition. Proc SPIE 5094: 59–72
Hyvarinen A, Karhunen J, Oja E (2001) Independent component analysis, vol 27 of adaptive and learning systems for signal processing, communications and control. Wiley-Interscience, New York
Johnson RA, Wichern DW (1992) Applied multivariate statistical analysis, 3rd edn. Prentice-Hall, Inc., Upper Saddle River
Jolliffe IT (2002) Principal component analysis, 2nd edn. Springer, New York
Mahanalobis A, Muise RR, Stanfill SR, Van Nevel A (2004) Design and application of quadratic correlation filters for target detection. IEEE Transac Aerosp Electron Syst 40(3): 837–850
Mann ME, Bradley RS, Hughes MK (1998) Global-scale temperature patterns and climate forcing over the past six centuries. Nature 392: 779–788
Mardia K, Kent J, Bibby J (1979) Multivariate analysis. Academic Press, London
McIntyre S, McKitrick R (2005) Reply to comment by Huybers on “hockey sticks, principal components, and spurious significance”. Geophys Res Lett 32: L20713
Miranda AA, Whelan PF (2005) Fukunaga–Koontz transform for small sample size problems. In: Proceedings of the IEE Irish signals and systems conference, pp 156–161, Dublin
Noy-Meir I (1973) Data transformations in ecological ordination: I. some advantages of non-centering. J Ecol 61(2): 329–341
Pearson K (1901) On lines and planes of closest fit to systems of points in space. Philos Mag 2: 559–572
Plett GL, Doi T, Torrieri D (1997) Mine detection using scattering parameters and an artificial neural network. IEEE Transac Neural Netw 8(6): 1456–1467
Ripley BD (1996) Pattern recognition and neural networks. Cambridge University Press, Cambridge
Van Huffel S (ed) (1997) Recent advances in total least squares techniques and errors-in-variables modeling. SIAM, Philadelphia
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Miranda, A.A., Le Borgne, YA. & Bontempi, G. New Routes from Minimal Approximation Error to Principal Components. Neural Process Lett 27, 197–207 (2008). https://doi.org/10.1007/s11063-007-9069-2
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DOI: https://doi.org/10.1007/s11063-007-9069-2