标题： 关于函数响应的变化平滑模型 显示英文标题

标题： Varying-smoother models for functional responses

摘要： 本文研究了当给定形式为 $f(t,\cdot)$ + 误差的功能响应时，光滑函数 $f(t,s)$ 的估计问题，但科学兴趣集中在不同 $s$ 对应的函数集合 \($f(\cdot,s)$\}。 研究动机来源于人类大脑发育的研究，其中 $t$ 表示年龄，而 $s$ 指的是大脑位置。 类似于变系数模型（其中响应均值在$t$上是线性的），我们所考虑的“变平滑器”模型表现出对$t$的非线性依赖，并且这种依赖关系随$s$平滑变化。 我们讨论了估计变平滑器模型的三种方法：(a) 使用张量积惩罚的方法；(b) 基于平滑函数主成分得分的方法；以及(c) 两步法，首先对每个$s$中的$t$进行初步平滑，然后进行后处理步骤。 对于第一种方法，我们推导了Wood提出的惩罚项的一个精确表达式，并提出了一种自适应惩罚项，允许平滑度更灵活地随$s$变化。 我们还开发了“逐点自由度”这一新工具，用于研究在每个$s$处对$f(\cdot,s)$估计的复杂性。 三种变光滑模型的方法在模拟和扩散张量成像数据集中进行了比较。 显示英文摘要

摘要： This paper studies estimation of a smooth function $f(t,s)$ when we are given functional responses of the form $f(t,\cdot)$ + error, but scientific interest centers on the collection of functions $f(\cdot,s)$ for different $s$. The motivation comes from studies of human brain development, in which $t$ denotes age whereas $s$ refers to brain locations. Analogously to varying-coefficient models, in which the mean response is linear in $t$, the "varying-smoother" models that we consider exhibit nonlinear dependence on $t$ that varies smoothly with $s$. We discuss three approaches to estimating varying-smoother models: (a) methods that employ a tensor product penalty; (b) an approach based on smoothed functional principal component scores; and (c) two-step methods consisting of an initial smooth with respect to $t$ at each $s$, followed by a postprocessing step. For the first approach, we derive an exact expression for a penalty proposed by Wood, and an adaptive penalty that allows smoothness to vary more flexibly with $s$. We also develop "pointwise degrees of freedom," a new tool for studying the complexity of estimates of $f(\cdot,s)$ at each $s$. The three approaches to varying-smoother models are compared in simulations and with a diffusion tensor imaging data set.