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Another Look at Principal Curves and Surfaces

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  • Delicado, Pedro

Abstract

Principal curves have been defined as smooth curves passing through the "middle" of a multidimensional data set. They are nonlinear generalizations of the first principal component, a characterization of which is the basis of the definition of principal curves. We establish a new characterization of the first principal component and base our new definition of a principal curve on this property. We introduce the notion of principal oriented points and we prove the existence of principal curves passing through these points. We extend the definition of principal curves to multivariate data sets and propose an algorithm to find them. The new notions lead us to generalize the definition of total variance. Successive principal curves are recursively defined from this generalization. The new methods are illustrated on simulated and real data sets.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 77 (2001)
Issue (Month): 1 (April)
Pages: 84-116

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Handle: RePEc:eee:jmvana:v:77:y:2001:i:1:p:84-116

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Related research

Keywords: fixed points; generalized total variance; nonlinear multivariate analysis; principal components; smoothing techniques;

References

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  1. Jamshid Etezadi-Amoli & Roderick McDonald, 1983. "A second generation nonlinear factor analysis," Psychometrika, Springer, vol. 48(3), pages 315-342, September.
  2. Victor Yohai & Werner Ackermann & Cristina Haigh, 1985. "Nonlinear principal components," Quality & Quantity: International Journal of Methodology, Springer, vol. 19(1), pages 53-69, January.
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Citations

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Cited by:
  1. Salinelli, Ernesto, 2009. "Nonlinear principal components, II: Characterization of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 652-660, April.
  2. Pedro Delicado, 1998. "Statistics in archaeology: New directions," Economics Working Papers 310, Department of Economics and Business, Universitat Pompeu Fabra.
  3. Berrendero, J.R. & Justel, A. & Svarc, M., 2011. "Principal components for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2619-2634, September.

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