Journal Article

Efficient computation of smoothing splines via adaptive basis sampling

Ping Ma, Jianhua Z. Huang and Nan Zhang

in Biometrika

Published on behalf of Biometrika Trust

Volume 102, issue 3, pages 631-645
Published in print September 2015 | ISSN: 0006-3444
Published online June 2015 | e-ISSN: 1464-3510 | DOI: http://dx.doi.org/10.1093/biomet/asv009
Efficient computation of smoothing splines via adaptive basis sampling

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Smoothing splines provide flexible nonparametric regression estimators. However, the high computational cost of smoothing splines for large datasets has hindered their wide application. In this article, we develop a new method, named adaptive basis sampling, for efficient computation of smoothing splines in super-large samples. Except for the univariate case where the Reinsch algorithm is applicable, a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is generally O(n3). We achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions, obtained by an adaptive sampling scheme that uses values of the response variable. Our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines. Using simulation studies and a large-scale deep earth core-mantle boundary imaging study, we show that the proposed method outperforms a sampling method that does not use the values of response variables.

Keywords: Bayesian confidence interval; Core-mantle boundary; Nonparametric regression; Penalized least squares; Reproducing kernel Hilbert space; Sampling

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Subjects: Probability and Statistics

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