Modeling Function-Valued Stochastic Processes, With Applications to Fertility Dynamics

We introduce a simple and interpretable model for functional data analysis for situations where the observations at each location are functional rather than scalar. This new approach is based on a tensor product representation of the function-valued process and utilizes eigenfunctions of marginal kernels. The resulting marginal principal components and product principal components are shown to provide optimal representations in a welldefined sense. Given a sample of independent realizations of the underlying function-valued stochastic process, we propose straightforward fitting methods to obtain the components of this model and to establish asymptotic consistency and rates of convergence for the proposed estimates. The methods are illustrated by modeling the dynamics of annual fertility profile functions for 17 countries. This analysis demonstrates that the proposed approach leads to insightful interpretations of the model components and interesting conclusions.