Smoothing and Mean-Covariance Estimation of Functional Data with a Bayesian Hierarchical Model.
Smoothing and Mean-Covariance Estimation of Functional Data with a Bayesian Hierarchical Model.
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DOI：
10.1214/15-ba967
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发表时间：
2016-09
影响因子：
4.4
通讯作者：
Cox DD
中科院分区：
文献类型：
--
作者：
Yang J;Zhu H;Choi T;Cox DD
在各种应用中经常遇到功能数据，基本观测单位是连续体上变化的函数（如曲线、曲面）。虽然已经开发了许多用于功能数据分析的统计工具，但同时平滑所有功能观测值的问题研究较少。现有的方法通常侧重于分别平滑每个单独的功能，这样做的风险是删除了功能之间常见的重要系统模式。我们提出了一种非参数贝叶斯方法来同时非参数地平滑所有的函数观测值。在提出的方法中，我们假设函数观测值是独立的高斯过程，受到共同测量误差水平的影响，从而能够借用所有观测值的强度。与大多数依赖于协方差核预先指定结构的高斯过程回归模型不同，我们采用分层框架，假设均值函数为高斯过程先验，协方差函数为逆wishart过程先验。这些先验假设在所有观测值的同时平滑之外，还在后验推理中诱导自动平均协方差估计。这样的层次框架足够灵活，可以合并具有不同特征的功能数据，包括在普通或不常见网格上测量的数据，以及具有平稳或非平稳协方差结构的数据。仿真和实际数据分析表明，与其他方法相比，贝叶斯方法具有更好的平滑精度和可比较的均值-协方差估计结果。此外，它可以成功地保留功能观测中的系统模式，这些模式通常被现有的基于单个曲线平滑的功能数据分析所忽略。
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.
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影响因子：
4.5
作者：
Yao, F;Müller, HG;Wang, JL
通讯作者：
Wang, JL
影响因子：
5.8
作者：
Leng, XY;Müller, HG
通讯作者：
Müller, HG
DOI：
10.1111/j.1467-9868.2008.00663.x
发表时间：
2008-09-01
期刊：
Journal of the Royal Statistical Society. Series B, Statistical methodology
影响因子：
--
作者：
Banerjee S;Gelfand AE;Finley AO;Sang H
通讯作者：
Sang H
影响因子：
--
作者：
Buys, Timon P. H.;Cantor, Scott B.;Follen, Michele
通讯作者：
Follen, Michele
影响因子：
2
作者：
Shi, J. Q.;Wang, B.;West, R. M.
通讯作者：
West, R. M.