Functional Horseshoe Smoothing for Functional Trend Estimation

Abstract

Due to developments in instruments and computers, functional observations are increasingly popular. However, effective methodologies for flexibly estimating the underlying trends with valid uncertainty quantification for a sequence of functional data (e.g. functional time series) are still scarce. In this work, we develop a locally adaptive smoothing method, called functional horseshoe smoothing, by introducing a shrinkage prior to the general order of differences of functional variables. This allows us to capture abrupt changes by taking advantage of the shrinkage capability and also to assess uncertainty by Bayesian inference. The fully Bayesian framework also allows the selection of the number of basis functions via the posterior predictive loss. Also, by taking advantage of the nature of functional data, this method is able to handle heterogeneously observed data without data augmentation. We show the theoretical properties of the proposed prior distribution and the posterior mean, and finally demonstrate them through simulation studies and applications to a real-world dataset.

Spatiotemporal factor models for functional data with application to population map forecast

“…where ε ts is an error term, which is independent of t and s, e 2 s is an unknown variance, and z ts is the focus. Such models are widely adopted in the context of Bayesian modeling of functional data (Yang et al, 2016(Yang et al, , 2017Wakayama and Sugasawa, 2022). Assume function z ts follows the Gaussian process.…”

“…Such prior is used in the horseshoe prior (Carvalho et al, 2009(Carvalho et al, , 2010 for a univariate parameter, and the resulting distribution of b •s is a multivariate version of the horseshoe prior. A similar multivariate prior is adopted in Shin et al (2020) and Wakayama and Sugasawa (2022) in non-spatial settings. The horseshoe distribution is known for strong shrinking ability, which allows the coefficients of singular factors to be zero.…”