Dynamic Regression Models for Time-Ordered Functional Data

Abstract

For time-ordered functional data, an important yet challenging task is to forecast functional observations with uncertainty quantification. Scalar predictors are often observed concurrently with functional data and provide valuable information about the dynamics of the functional time series. We develop a fully Bayesian framework for dynamic functional regression, which employs scalar predictors to model the time-evolution of functional data. Functional within-curve dependence is modeled using unknown basis functions, which are learned from the data. The unknown basis provides substantial dimension reduction, which is essential for scalable computing, and may incorporate prior knowledge such as smoothness or periodicity. The dynamics of the time-ordered functional data are specified using a time-varying parameter regression model in which the effects of the scalar predictors evolve over time. To guard against overfitting, we design shrinkage priors that regularize irrelevant predictors and shrink toward time-invariance. Simulation studies decisively confirm the utility of these modeling and prior choices. Posterior inference is available via a customized Gibbs sampler, which offers unrivaled scalability for Bayesian dynamic functional regression. The methodology is applied to model and forecast yield curves using macroeconomic predictors, and demonstrates exceptional forecasting accuracy and uncertainty quantification over the span of four decades.