title: Efficient Bayesian hierarchical functional data analysis with basis function approximations using Gaussian-Wishart processes.
abstract: Functional data are defined as realizations of random functions (mostly smooth functions) varying over a continuum, which are usually collected on discretized grids with measurement errors. In order to accurately smooth noisy functional observations and deal with the issue of high-dimensional observation grids, we propose a novel Bayesian method based on the Bayesian hierarchical model with a Gaussian-Wishart process prior and basis function representations. We first derive an induced model for the basis-function coefficients of the functional data, and then use this model to conduct posterior inference through Markov chain Monte Carlo methods. Compared to the standard Bayesian inference that suffers serious computational burden and instability in analyzing high-dimensional functional data, our method greatly improves the computational scalability and stability, while inheriting the advantage of simultaneously smoothing raw observations and estimating the mean-covariance functions in a nonparametric way. In addition, our method can naturally handle functional data observed on random or uncommon grids. Simulation and real studies demonstrate that our method produces similar results to those obtainable by the standard Bayesian inference with low-dimensional common grids, while efficiently smoothing and estimating functional data with random and high-dimensional observation grids when the standard Bayesian inference fails. In conclusion, our method can efficiently smooth and estimate high-dimensional functional data, providing one way to resolve the curse of dimensionality for Bayesian functional data analysis with Gaussian-Wishart processes.
authors.affiliation: Department of Biostatistics, University of Michigan, Ann Arbor, Michigan 48109, U.S.A.
authors.forename: Jingjing
authors.identifier: 0000-0002-4191-4138
authors.initials: J
authors.lastname: Yang
authors.affiliation: Department of Statistics, Rice University, Houston, Texas 77005, U.S.A.
authors.forename: Dennis D
authors.initials: DD
authors.lastname: Cox
authors.affiliation: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, Massachusetts 01854, U.S.A.
authors.forename: Jong Soo
authors.initials: JS
authors.lastname: Lee
authors.affiliation: Suntrust Banks Inc, Atlanta, Georgia 30308, U.S.A.
authors.forename: Peng
authors.initials: P
authors.lastname: Ren
authors.affiliation: Department of Statistics, Korea University, Seoul 136-701, Republic of Korea.
authors.forename: Taeryon
authors.initials: T
authors.lastname: Choi
mesh_terms: D001499:Bayes Theorem; D003198:Computer Simulation; D008390:Markov Chains; D015233:Models, Statistical; D009010:Monte Carlo Method
keywords: Basis function; Bayesian hierarchical model; Functional data analysis; Gaussian-Wishart process; Smoothing