Optimal smoothing spline surfaces with constraints on derivatives

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

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On the equivalence of two stochastic approaches to spline smoothing

Journal of Applied Probability ◽

10.2307/3214367 ◽

1986 ◽

Vol 23 (A) ◽

pp. 391-405 ◽

Cited By ~ 7

Author(s):

Craig F. Ansley ◽

Robert Kohn

Keyword(s):

Stochastic Process ◽

Hilbert Space ◽

Minimization Problem ◽

Stochastic Models ◽

Conditional Expectation ◽

Smoothing Spline ◽

Computational Algorithms ◽

Spline Smoothing ◽

Optimal Smoothing ◽

Space Formulation

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

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An asymptotic and empirical smoothing parameters selection method for smoothing spline ANOVA models in large samples

Biometrika ◽

10.1093/biomet/asaa047 ◽

2020 ◽

Author(s):

Xiaoxiao Sun ◽

Wenxuan Zhong ◽

Ping Ma

Keyword(s):

Risk Function ◽

Computational Cost ◽

Selection Method ◽

Smoothing Spline ◽

Unknown Constant ◽

Large Samples ◽

Parameters Selection ◽

Optimal Smoothing ◽

Smoothing Parameters ◽

Smoothing Spline Anova

Summary Large samples are generated routinely from various sources. Classic statistical models, such as smoothing spline ANOVA models, are not well equipped to analyse such large samples because of high computational costs. In particular, the daunting computational cost of selecting smoothing parameters renders smoothing spline ANOVA models impractical. In this article, we develop an asympirical, i.e., asymptotic and empirical, smoothing parameters selection method for smoothing spline ANOVA models in large samples. The idea of our approach is to use asymptotic analysis to show that the optimal smoothing parameter is a polynomial function of the sample size and an unknown constant. The unknown constant is then estimated through empirical subsample extrapolation. The proposed method significantly reduces the computational burden of selecting smoothing parameters in high-dimensional and large samples. We show that smoothing parameters chosen by the proposed method tend to the optimal smoothing parameters that minimize a specific risk function. In addition, the estimator based on the proposed smoothing parameters achieves the optimal convergence rate. Extensive simulation studies demonstrate the numerical advantage of the proposed method over competing methods in terms of relative efficacy and running time. In an application to molecular dynamics data containing nearly one million observations, the proposed method has the best prediction performance.

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