Functional Autoregression for Sparsely Sampled Data

Abstract

We develop a hierarchical Gaussian process model for forecasting and inference of functional time series data. Unlike existing methods, our approach is especially suited for sparsely or irregularly sampled curves and for curves sampled with nonnegligible measurement error. The latent process is dynamically modeled as a functional autoregression (FAR) with Gaussian process innovations. We propose a fully nonparametric dynamic functional factor model for the dynamic innovation process, with broader applicability and improved computational efficiency over standard Gaussian process models. We prove finite-sample forecasting and interpolation optimality properties of the proposed model, which remain valid with the Gaussian assumption relaxed. An efficient Gibbs sampling algorithm is developed for estimation, inference, and forecasting, with extensions for FAR(p) models with model averaging over the lag p. Extensive simulations demonstrate substantial improvements in forecasting performance and recovery of the autoregressive surface over competing methods, especially under sparse designs. We apply the proposed methods to forecast nominal and real yield curves using daily U.S. data. Real yields are observed more sparsely than nominal yields, yet the proposed methods are highly competitive in both settings. Supplementary materials, including R code and the yield curve data, are available online.

KEY WORDS:

• Functional factor analysis
• Gaussian process
• Hierarchical Bayes
• Model averaging
• Time series

SUPPLEMENTARY MATERIALS

Supplementary materials, including R code and the yield curve data, are available online.

ACKNOWLEDGMENTS

We thank Professor Robert A. Jarrow for suggesting the yield curve application and for his helpful discussions. The authors thank the editors and two referees for very helpful comments. Financial support from NSF grant AST-1312903 (Kowal and Ruppert) and a Xerox PARC Faculty Research Award, and NSF grant DMS-1455172 (Matteson) is gratefully acknowledged.