Bayesian Analysis

Smoothing and Mean–Covariance Estimation of Functional Data with a Bayesian Hierarchical Model

Jingjing Yang, Hongxiao Zhu, Taeryon Choi, and Dennis D. Cox

Full-text: Open access

Abstract

Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean–covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean–covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.

Article information

Source
Bayesian Anal., Volume 11, Number 3 (2016), 649-670.

Dates
First available in Project Euclid: 26 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.ba/1440594947

Digital Object Identifier
doi:10.1214/15-BA967

Mathematical Reviews number (MathSciNet)
MR3498041

Zentralblatt MATH identifier
1357.62182

Keywords
functional data smoothing Bayesian hierarchical model Gaussian process Matérn covariance function empirical Bayes

Citation

Yang, Jingjing; Zhu, Hongxiao; Choi, Taeryon; Cox, Dennis D. Smoothing and Mean–Covariance Estimation of Functional Data with a Bayesian Hierarchical Model. Bayesian Anal. 11 (2016), no. 3, 649--670. doi:10.1214/15-BA967. https://projecteuclid.org/euclid.ba/1440594947


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