Modelling function-valued stochastic processes, with applications to fertility dynamics

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2017-01, Vol.79 (1), p.177-196
• Main Authors: Chen, Kehui, Delicado, Pedro, Müller, Hans-Georg
• Format: Article
• Language: English
• Subjects: 90 Operations research, mathematical programming; Asymptotic methods; Asymptotics; Classificació AMS; Data analysis; Demography; Dynamics; Fertility; Fittings; Functional data analysis; Investigació operativa; Marginal kernel; Matemàtiques i estadística; Modelling; Product principal component analysis; Samples; Statistical methods; Statistics; Stochastic models; Stochastic processes; Studies; Tensor product representation; Àrees temàtiques de la UPC
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/rssb.12160

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 have nice properties. 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 estimates proposed. The methods are illustrated by modelling the dynamics of annual fertility profile functions for 17 countries. This analysis demonstrates that the approach proposed leads to insightful interpretations of the model components and interesting conclusions.