SMOOTHING SPLINES ESTIMATORS FOR FUNCTIONAL LINEAR REGRESSION
SMOOTHING SPLINES ESTIMATORS FOR FUNCTIONAL LINEAR REGRESSION
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DOI：
10.1214/07-aos563
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发表时间：
2009-02-01
影响因子：
4.5
通讯作者：
Sarda, Pascal
中科院分区：
文献类型：
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作者：
Crambes, Christophe;Kneip, Alois;Sarda, Pascal
本文考虑函数线性回归，其中标量响应 Y-1, ... , Y-n 根据随机函数 X-1, ... , X-n 进行建模。我们基于对通常惩罚的轻微修改，提出了函数斜率参数的平滑样条估计器。理论分析集中于新随机函数 Xn+1 响应的所有样本外预测中的误差。结果表明，预测误差的收敛速度取决于斜率函数的平滑度和预测变量的结构。然后，我们证明这些速率是最优的，因为它们在大类可能的斜率函数和预测曲线的分布上是极小极大的。对于具有变量误差的模型的情况，通过使用离散曲线的协方差矩阵的去噪校正来修改平滑样条估计器。然后将该方法应用于实际案例研究，其目的是通过使用前一天测量的臭氧浓度曲线来预测臭氧浓度的最大值。
The paper considers functional linear regression, where scalar responses Y-1, ... , Y-n are modeled in dependence of random functions X-1, ... , X-n. We propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty. Theoretical analysis concentrates on the error in all out-of-sample prediction of the response for a new random function Xn+1. It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the Structure Of the predictors. We then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. For the case of models with errors-in-variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves. The methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day.