Publication Date: April 2009
We investigate a method for extracting nonlinear principal components. These principal components maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate densities, including densities without compact support and even densities with algebraic tails. We provide primitive suﬀicient conditions for the existence of these principal components. We characterize the limiting behavior of the associated eigenvalues, the objects used to quantify the incremental importance of the principal components. By exploiting the theory of continuous-time, reversible Markov processes, we give a diﬀerent interpretation of the principal components and the smoothness constraints. When the diﬀusion matrix is used to enforce smoothness, the principal components maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the principal components behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identiﬁcation of a multivariate reversible diﬀusion process and tests of the overidentifying restrictions implied by such a process from low frequency data. We also explore implications for stationary, possibly non-reversible diﬀusion processes.
Nonlinear principal components, Discrete spectrum, Eigenvalue decay rates, Multivariate diﬀusion, Quadratic form, Conditional expectations operator
JEL Classification Codes: C12, C22