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Efficient Bayesian hierarchical functional data analysis with basis function approximations using Gaussian–Wishart processes
Author(s) -
Yang Jingjing,
Cox Dennis D.,
Lee Jong Soo,
Ren Peng,
Choi Taeryon
Publication year - 2017
Publication title -
biometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.298
H-Index - 130
eISSN - 1541-0420
pISSN - 0006-341X
DOI - 10.1111/biom.12705
Subject(s) - wishart distribution , functional data analysis , basis function , computer science , smoothing , gaussian process , bayesian probability , markov chain monte carlo , curse of dimensionality , bayesian inference , algorithm , covariance , mathematics , artificial intelligence , machine learning , gaussian , statistics , multivariate statistics , mathematical analysis , physics , quantum mechanics
Summary 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.