发表机构
Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对函数型协变量存在测量误差时的标量对函数回归问题，提出模拟-外推方法校正系数函数估计的衰减，并通过模拟和扩散张量成像数据验证其有效性。
AI 中文摘要
我们考虑标量对函数回归的问题。大多数现有方法隐含地假设函数型协变量被精确观测，但在实践中，它们常常受到测量误差的污染。因此，我们的目标是处理当回归函数被观测到带有误差时的标量对函数回归问题。在本文中，我们提出了一种模拟-外推方法来校正由误差引起的估计系数函数的衰减。该方法首先估计误差方差，建立一系列增加的误差方差与相应的系数函数估计之间的关系，然后外推到零误差。我们描述了三种方法来外推估计系数函数的序列。在一项模拟研究中，我们将模拟-外推方法与基于平滑样条和函数主成分分析的两种预平滑方法进行了比较。接下来，我们讨论了该方法在几个方向上的扩展，允许更复杂的噪声协方差结构、函数型预测因子的多次重复、广义响应以及二维和三维函数型预测因子。最后，我们通过一个扩散张量成像数据的应用来展示这些方法。
英文摘要
We consider the problem of scalar-on-function regression. Most existing methods implicitly assume that the functional covariates are observed exactly, but in practice, they are often contaminated by measurement error. Our goal, therefore, is to deal with the problem of scalar-on-function regression when the regressor functions are observed with error. In this paper, we propose a simulation-extrapolation method to correct for the attenuation of estimated coefficient functions caused by the error. The method first estimates the error variance, establishes the relationship between a sequence of added error variance and the corresponding estimates of coefficient functions, and then extrapolates to the zero-error. We describe three methods to extrapolate the sequence of estimated coefficient functions. In a simulation study, we compare the performance of the simulation-extrapolation method with two pre-smoothing methods based on smoothing splines and functional principal component analysis. Next, we discuss the extension of the method in several directions, allowing for more complex noise covariance structures, multiple replications of functional predictors, generalized responses, and 2D and 3D functional predictors. Finally, we illustrate the methods by an application to diffusion tensor imaging data.
