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Precise Tensor Product Smoothing via Spectral Splines

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  • Nathaniel E. Helwig

    (Department of Psychology, University of Minnesota, 75 E River Road, Minneapolis, MN 55455, USA
    School of Statistics, University of Minnesota, 224 Church Street SE, Minneapolis, MN 55455, USA)

Abstract

Tensor product smoothers are frequently used to include interaction effects in multiple nonparametric regression models. Current implementations of tensor product smoothers either require using approximate penalties, such as those typically used in generalized additive models, or costly parameterizations, such as those used in smoothing spline analysis of variance models. In this paper, I propose a computationally efficient and theoretically precise approach for tensor product smoothing. Specifically, I propose a spectral representation of a univariate smoothing spline basis, and I develop an efficient approach for building tensor product smooths from marginal spectral spline representations. The developed theory suggests that current tensor product smoothing methods could be improved by incorporating the proposed tensor product spectral smoothers. Simulation results demonstrate that the proposed approach can outperform popular tensor product smoothing implementations, which supports the theoretical results developed in the paper.

Suggested Citation

  • Nathaniel E. Helwig, 2024. "Precise Tensor Product Smoothing via Spectral Splines," Stats, MDPI, vol. 7(1), pages 1-20, January.
  • Handle: RePEc:gam:jstats:v:7:y:2024:i:1:p:3-53:d:1316790
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    References listed on IDEAS

    as
    1. Yuedong Wang, 1998. "Mixed effects smoothing spline analysis of variance," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 159-174.
    2. Bates, Douglas & Mächler, Martin & Bolker, Ben & Walker, Steve, 2015. "Fitting Linear Mixed-Effects Models Using lme4," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 67(i01).
    3. Young‐Ju Kim & Chong Gu, 2004. "Smoothing spline Gaussian regression: more scalable computation via efficient approximation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(2), pages 337-356, May.
    4. Zack W. Almquist & Nathaniel E. Helwig & Yun You, 2020. "Connecting Continuum of Care point-in-time homeless counts to United States Census areal units," Mathematical Population Studies, Taylor & Francis Journals, vol. 27(1), pages 46-58, January.
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