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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.

Suggested Citation

  • Delicado, Pedro, 2001. "Another Look at Principal Curves and Surfaces," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 84-116, April.
    • Handle: RePEc:eee:jmvana:v:77:y:2001:i:1:p:84-116
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    References listed on IDEAS

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

    1. Pulkkinen, Seppo, 2015. "Ridge-based method for finding curvilinear structures from noisy data," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 89-109.
    2. 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.
    3. Serge Iovleff, 2015. "Probabilistic auto-associative models and semi-linear PCA," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 9(3), pages 267-286, September.
    4. Pedro Delicado, 1998. "Statistics in archaeology: New directions," Economics Working Papers 310, Department of Economics and Business, Universitat Pompeu Fabra.
    5. Pedro Delicado & Mario Huerta, 2003. "Principal Curves of Oriented Points: theoretical and computational improvements," Computational Statistics, Springer, vol. 18(2), pages 293-315, July.
    6. Salinelli, Ernesto, 2009. "Nonlinear principal components, II: Characterization of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 652-660, April.
    7. Youness Aliyari Ghassabeh & Frank Rudzicz, 2021. "Modified Subspace Constrained Mean Shift Algorithm," Journal of Classification, Springer;The Classification Society, vol. 38(1), pages 27-43, April.
    8. Cholaquidis, Alejandro & Fraiman, Ricardo & Moreno, Leonardo, 2022. "Level set and density estimation on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    9. Girard, Stéphane & Iovleff, Serge, 2005. "Auto-associative models and generalized principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 21-39, March.

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