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More efficient approximation of smoothing splines via space-filling basis selection. (English) Zbl 07263823
Summary: 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 $$p \in [1,2]$$ 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 $$d \leq pr + 1$$. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods.
##### MSC:
 62G08 Nonparametric regression and quantile regression 62K15 Factorial statistical designs 65D07 Numerical computation using splines 62P12 Applications of statistics to environmental and related topics
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