Nonlinear Principal Components I: Absolutely Continuous Random Variables with Positive Bounded Densities

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.