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More efficient approximation of smoothing splines via space-filling basis selection

Author

Listed:
  • Cheng Meng
  • Xinlian Zhang
  • Jingyi Zhang
  • Wenxuan Zhong
  • Ping Ma

Abstract

SummaryWe 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.

Suggested Citation

  • Cheng Meng & Xinlian Zhang & Jingyi Zhang & Wenxuan Zhong & Ping Ma, 2020. "More efficient approximation of smoothing splines via space-filling basis selection," Biometrika, Biometrika Trust, vol. 107(3), pages 723-735.
  • Handle: RePEc:oup:biomet:v:107:y:2020:i:3:p:723-735.
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    File URL: http://hdl.handle.net/10.1093/biomet/asaa019
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    Cited by:

    1. Jun Yu & HaiYing Wang, 2022. "Subdata selection algorithm for linear model discrimination," Statistical Papers, Springer, vol. 63(6), pages 1883-1906, December.

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