More efficient approximation of smoothing splines via space-filling basis selection
More efficient approximation of smoothing splines via space-filling basis selection
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DOI：
10.1093/biomet/asaa019
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发表时间：
2020-09-01
期刊：
影响因子：
2.7
通讯作者：
Ma, Ping
中科院分区：
文献类型：
--
作者：
Meng, Cheng;Zhang, Xinlian;Ma, Ping
研究非参数回归模型中光滑样条估计的逼近问题。当应用于大小为n的样本时，平滑样条估计量可以表示为n个基函数的线性组合，当预测变量的数量d为2或更多时，需要O（n（3））计算时间。这样一个相当大的计算成本阻碍了平滑样条的广泛适用性。在实践中，全样本平滑样条估计器可以通过基于q个随机选择的基函数的估计器来近似，从而导致O（nq（2））的计算成本。已知当q为O{n（2/（pr+1））}阶时，这两个估计以相同的速度收敛，其中p是[1，2]中的元素，依赖于真函数，r > 1依赖于样条类型.这样的q被称为基函数的本质数。在本文中，我们开发了一种更有效的基选择方法。该方法通过选取与近似等间距观测值相对应的基函数，选取一组具有极大多样性的基函数。渐近分析表明，当d为0时，所提出的光滑样条估计可使q降到O{n（1/（pr+1））}左右
We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size n, the smoothing spline estimator can be expressed as a linear combination of n basis functions, requiring O(n(3)) computational time when the number d of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on q randomly selected basis functions, resulting in a computational cost of O(nq(2)). It is known that these two estimators converge at the same rate when q is of order O{n(2/(pr+1))}, where p is an element of [1, 2] depends on the true function and r > 1 depends on the type of spline. Such a q is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease q to around O{n(1/(pr+1))} when d