Optimal Penalized Function-on-Function Regression Under a Reproducing Kernel Hilbert Space Framework
Optimal Penalized Function-on-Function Regression Under a Reproducing Kernel Hilbert Space Framework
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DOI：
10.1080/01621459.2017.1356320
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发表时间：
2018-01-01
影响因子：
3.7
通讯作者：
Ma, Ping
中科院分区：
文献类型：
--
作者：
Sun, Xiaoxiao;Du, Pang;Ma, Ping
许多科学研究收集的数据中，响应变量和预测变量都是时间、位置或其他协变量的函数。了解这些函数变量之间的关系是这些研究的共同目标。从两个实际例子出发，本文提出了一个函数对函数回归模型，可以用来分析这类函数数据。我们的2D系数函数估计器是一种惩罚最小二乘形式的优化器，其中惩罚对估计器施加一定程度的平滑度。我们的第一个结果是表示定理，该定理指出，精确的优化的惩罚最小二乘实际上驻留在一个数据自适应有限维子空间，虽然优化问题是定义在一个函数空间的无限维。这个定理，然后允许我们很容易地将高斯积分到优化的惩罚最小二乘，这可以通过标准的数值程序进行。我们还表明，我们的估计实现了极小极大收敛速度的均值预测的框架下的功能回归。广泛的模拟研究表明，我们的方法比现有的，其中还引入了稀疏的功能数据扩展的数值优势。所提出的方法，然后应用到我们的激励的基准加拿大天气数据和组蛋白调节研究的例子。本文的补充材料可在网上查阅。
Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this article a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the representer theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite-dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study. Supplementary materials for this article are available online.