The Annals of Statistics

Nonlinear principal components I. Absolutely continuous random variables with positive bounded densities

Ernesto Salinelli

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Abstract

Nonlinear principal components for an absolutely continuous random vector X with positive bounded density are defined as the solution of a variational problem in a suitable function space. In this way transformations depending on all the components of X are obtained. Some properties of nonlinear principal components are proved: in particular, it is shown that the set of nonlinear principal transformations of X is an orthonormal basis for the function space associated with the optimal problem. The spectral decomposition of X and its covariance matrix with respect to this basis are given. A notion of marginal nonlinear principal components is sketched and the relations with nonlinear principal components are shown. Finally, treating the case of random vectors distributed on unbounded domains, the existence problem is shown to be related to the global existence of the moment generating function of X. Since it is not restrictive, definitions and results are stated in terms of a uniformly distributed random vector.

Article information

Source
Ann. Statist., Volume 26, Number 2 (1998), 596-616.

Dates
First available in Project Euclid: 31 July 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1028144850

Digital Object Identifier
doi:10.1214/aos/1028144850

Mathematical Reviews number (MathSciNet)
MR1626079

Zentralblatt MATH identifier
0929.62067

Subjects
Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 35J20: Variational methods for second-order elliptic equations

Keywords
Nonlinear principal components moment generating function Sobolev spaces Laplacian

Citation

Salinelli, Ernesto. Nonlinear principal components I. Absolutely continuous random variables with positive bounded densities. Ann. Statist. 26 (1998), no. 2, 596--616. doi:10.1214/aos/1028144850. https://projecteuclid.org/euclid.aos/1028144850


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