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Principal curves and principal oriented points

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  • Pedro Delicado

Abstract

Principal curves have been defined Hastie and Stuetzle (JASA, 1989) 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 for the principal curves definition. In this paper we propose an alternative approach based on a different property of principal components. Consider a point in the space where a multivariate normal is defined and, for each hyperplane containing that point, compute the total variance of the normal distribution conditioned to belong to that hyperplane. Choose now the hyperplane minimizing this conditional total variance and look for the corresponding conditional mean. The first principal component of the original distribution passes by this conditional mean and it is orthogonal to that hyperplane. This property is easily generalized to data sets with nonlinear structure. Repeating the search from different starting points, many points analogous to conditional means are found. We call them principal oriented points. When a one-dimensional curve runs the set of these special points it is called principal curve of oriented points. Successive principal curves are recursively defined from a generalization of the total variance.

Suggested Citation

  • Pedro Delicado, 1998. "Principal curves and principal oriented points," Economics Working Papers 309, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:309
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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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    More about this item

    Keywords

    Fixed points; generalized total variance; nonlinear multivariate analysis; principal components; smoothing techniques;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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