非参数局部多项式回归估计的最优子抽样算法

Optimal Subsampling Algorithm for Nonparametric Local Polynomial Regression Estimation

随着科学技术的发展,虽然人们提高了收集和处理数据的能力,但仍存在一些大数据集超出了现有计算机的计算能力.目前,抽取一部分样本来替代全样本进行建模计算是减轻计算负担的一种方法.大数据背景下线性模型的子抽样方法已经得到了相对成熟的研究,在减轻计算量方面获得了很大的优势.文章将线性模型下的子抽样方法推广到非参数回归模型,并推导出了基于子样本的加权最小二乘参数估计对全样本加权最小二乘参数估计的收敛速度,以及子样本参数估计的条件渐近正态性.通过最小化渐近方差的准则,提出了非参数局部多项式回归模型下的OPT和PL两种抽样方案,最后在均方误差、计算成本和拟合效果等方面进行数值模拟,比较了OPT子抽样和PL子抽样相对于均匀子抽样和杠杆子抽样的差别,其结果表明于OPT准则和PL准则的子抽样方法在提高估计精确性和减少计算负担方面具有很大优势.

In this paper,we extend the subsampling method under the linear model to the nonparametric regression model and propose two subsampling methods for the nonparametric local polynomial regression model.First,we derive the convergence rate of subsampling based weighted least squares parameter estimation to full sample weighted least squares parameter estimation,and the asymptotic normality of the subsample parameter estimation are derived.Then,we use the criterion of minimizing the asymptotic variance,and two subsampling methods of OPT and PL under nonparametric local polynomial regression model are proposed.Finally,numerical simulation of OPT subsampling and PL subsampling,uniform subsampling and Basic Leveraging subsampling are carried out respectively,in terms of mean square error,fitting effect and computational cost.The results show that the subsampling method based on OPT criterion and PL criterion has great advantages in improving estimation accuracy and reducing calculation burden.