2014/02/28 by Jingjing Yang, Hongxiao Zhu, Taeryon Choi +1 · 48 citations
Computer Science · Environmental Science · Mathematics · #Algorithm #Artificial intelligence #Bayesian probability #Computer science #Covariance #Covariance function #Functional data analysis #Gaussian #Gaussian process #Geochemistry and Geologic Mapping #Kernel (algebra) #Kernel method #Kernel smoother #Kriging #Mathematics #Smoothing #Soil Geostatistics and Mapping #Statistical Methods and Inference #Statistics #stat.ME
paper · pdf · doi:10.1214/15-ba967
published in Bayesian Analysis 11(3), 649-670 (International Society for Bayesian Analysis) · Submitted to Bayesian Analysis
arxiv created 2015/07/07 · openalex publication_date 2015/08/26 · arxiv updated 2016/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.