Robust, Adaptive Functional Regression in Functional Mixed Model Framework.
Robust, Adaptive Functional Regression in Functional Mixed Model Framework.
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DOI：
10.1198/jasa.2011.tm10370
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发表时间：
2011-09-01
影响因子：
3.7
通讯作者：
Morris JS
中科院分区：
文献类型：
--
作者：
Zhu H;Brown PJ;Morris JS
在科学研究中，函数数据越来越多地遇到，其高维和复杂性导致了许多分析挑战。已经开发了用于功能数据分析的各种方法，包括涉及将功能响应回归到具有非参数表示的功能系数的单变量/多变量预测值上的功能响应回归方法。然而，在现有的方法中，函数回归会对边缘曲线和边缘区域敏感，因此不具有稳健性。本文介绍了一种新的贝叶斯方法--稳健函数混合模型(R-FMM)，用于在一般函数混合模型框架内进行稳健函数回归，该模型包括多个连续或分类预测因子和适应实验设计引起的潜在函数间相关性的随机效应函数。基本模型包括固定效应、随机效应和残差函数的分级尺度混合模型。这些跨曲线的建模假设导致对固定和随机效应函数的稳健的非参数估计器，该估计器降低边缘曲线和曲线区域的权重，并产生可用于标记全局和局部异常值的统计数据。这些假设还导致了具有突出的稀疏性和自适应收缩特性的子波系数的分布，使得数据具有很大的灵活性来确定尾部的稀疏性和重度性。再加上异常值的降低，这些曲线内的属性导致了固定和随机的效果函数估计，在我们的模拟中，这些估计在去除虚假特征同时保留函数的真实特征方面具有显著的自适应能力。我们已经开发了通用代码来实现这种完全自动的贝叶斯方法，要求用户只提供功能数据和设计矩阵。它的效率足以处理大数据集，并产生所有模型参数的后验样本，可用于执行所需的贝叶斯估计和推断。虽然我们使用分层模型、一维函数和小波变换中的特定分布选择来给出R-FMM的具体实现的细节，但是该方法可以更普遍地使用其他重尾分布、更高维函数(例如图像)以及使用其他可逆变换作为小波的替代。
Functional data are increasingly encountered in scientific studies, and their high dimensionality and complexity lead to many analytical challenges. Various methods for functional data analysis have been developed, including functional response regression methods that involve regression of a functional response on univariate/multivariate predictors with nonparametrically represented functional coefficients. In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions of curves, so is not robust. In this paper, we introduce a new Bayesian method, robust functional mixed models (R-FMM), for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation induced by the experimental design. The underlying model involves a hierarchical scale mixture model for the fixed effects, random effect and residual error functions. These modeling assumptions across curves result in robust nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with great flexibility for the data to determine the sparsity and the heaviness of the tails. Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large data sets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be applied more generally using other heavy-tailed distributions, higher dimensional functions (e.g. images), and using other invertible transformations as alternatives to wavelets.
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通讯作者：
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