论文标题
RKHS的非线性功能在功能上回归
Nonlinear function-on-function regression by RKHS
论文作者
论文摘要
我们提出了一个非线性功能在功能回归模型中，其中协变量和响应都是随机函数。非线性回归分为两个步骤：我们首先构建希尔伯特空间以适应功能协变量和功能响应，然后构建一个第二层希尔伯特空间，以捕获非线性。假定第二层空间是一个繁殖的内核希尔伯特空间，该空间是由由$ x $的第一层希尔伯特空间的内部产物确定的积极确定的内核生成的 - 这种结构称为嵌套的希尔伯特空间。我们开发了实施提出的方法的估算程序，该方法允许在不同主题的不同时间点观察功能数据。此外，我们建立了估计器的收敛速率以及希尔伯特空间中预测响应的弱收敛性。进行包括模拟和数据应用在内的数值研究是为了研究有限样本中估计器的性能。
We propose a nonlinear function-on-function regression model where both the covariate and the response are random functions. The nonlinear regression is carried out in two steps: we first construct Hilbert spaces to accommodate the functional covariate and the functional response, and then build a second-layer Hilbert space for the covariate to capture nonlinearity. The second-layer space is assumed to be a reproducing kernel Hilbert space, which is generated by a positive definite kernel determined by the inner product of the first-layer Hilbert space for $X$--this structure is known as the nested Hilbert spaces. We develop estimation procedures to implement the proposed method, which allows the functional data to be observed at different time points for different subjects. Furthermore, we establish the convergence rate of our estimator as well as the weak convergence of the predicted response in the Hilbert space. Numerical studies including both simulations and a data application are conducted to investigate the performance of our estimator in finite sample.
