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Smoothing spline Gaussian regression: more scalable computation via efficient approximation

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  • Young‐Ju Kim
  • Chong Gu

Abstract

Summary. Smoothing splines via the penalized least squares method provide versatile and effective nonparametric models for regression with Gaussian responses. The computation of smoothing splines is generally of the order O(n3), n being the sample size, which severely limits its practical applicability. We study more scalable computation of smoothing spline regression via certain low dimensional approximations that are asymptotically as efficient. A simple algorithm is presented and the Bayes model that is associated with the approximations is derived, with the latter guiding the porting of Bayesian confidence intervals. The practical choice of the dimension of the approximating space is determined through simulation studies, and empirical comparisons of the approximations with the exact solution are presented. Also evaluated is a simple modification of the generalized cross‐validation method for smoothing parameter selection, which to a large extent fixes the occasional undersmoothing problem that is suffered by generalized cross‐validation.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:66:y:2004:i:2:p:337-356
    DOI: 10.1046/j.1369-7412.2003.05316.x
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    Cited by:

    1. Elena Geminiani & Giampiero Marra & Irini Moustaki, 2021. "Single- and Multiple-Group Penalized Factor Analysis: A Trust-Region Algorithm Approach with Integrated Automatic Multiple Tuning Parameter Selection," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 65-95, March.
    2. Massimiliano Mazzanti & Antonio Musolesi, 2011. "Income and time related effects in EKC," Working Papers 201105, University of Ferrara, Department of Economics.
    3. Kim, Young-Ju, 2011. "A comparative study of nonparametric estimation in Weibull regression: A penalized likelihood approach," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1884-1896, April.
    4. Zlatana Nenova & Jennifer Shang, 2022. "Chronic Disease Progression Prediction: Leveraging Case‐Based Reasoning and Big Data Analytics," Production and Operations Management, Production and Operations Management Society, vol. 31(1), pages 259-280, January.
    5. Xin Fang & Bo Fang & Chunfang Wang & Tian Xia & Matteo Bottai & Fang Fang & Yang Cao, 2019. "Comparison of Frequentist and Bayesian Generalized Additive Models for Assessing the Association between Daily Exposure to Fine Particles and Respiratory Mortality: A Simulation Study," IJERPH, MDPI, vol. 16(5), pages 1-20, March.
    6. Nathaniel E. Helwig, 2024. "Precise Tensor Product Smoothing via Spectral Splines," Stats, MDPI, vol. 7(1), pages 1-20, January.
    7. Nathaniel E. Helwig, 2022. "Robust Permutation Tests for Penalized Splines," Stats, MDPI, vol. 5(3), pages 1-18, September.
    8. Eduardo L. Montoya, 2020. "On the Number of Independent Pieces of Information in a Functional Linear Model with a Scalar Response," Stats, MDPI, vol. 3(4), pages 1-16, November.
    9. Gu, Chong, 2014. "Smoothing Spline ANOVA Models: R Package gss," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 58(i05).
    10. Massimiliano Mazzanti & Antonio Musolesi, 2010. "Carbon Abatement Leaders and Laggards Non Parametric Analyses of Policy Oriented Kuznets Curves," Working Papers 2010.149, Fondazione Eni Enrico Mattei.
    11. Nikolay Markov & Dr. Thomas Nitschka, 2013. "Estimating Taylor Rules for Switzerland: Evidence from 2000 to 2012," Working Papers 2013-08, Swiss National Bank.
    12. Simon N. Wood & Mark V. Bravington & Sharon L. Hedley, 2008. "Soap film smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 931-955, November.
    13. Marcillo-Delgado, J.C. & Ortego, M.I. & Pérez-Foguet, A., 2019. "A compositional approach for modelling SDG7 indicators: Case study applied to electricity access," Renewable and Sustainable Energy Reviews, Elsevier, vol. 107(C), pages 388-398.
    14. Pang Du & Yihua Jiang & Yuedong Wang, 2011. "Smoothing Spline ANOVA Frailty Model for Recurrent Event Data," Biometrics, The International Biometric Society, vol. 67(4), pages 1330-1339, December.
    15. Lauren N. Berry & Nathaniel E. Helwig, 2021. "Cross-Validation, Information Theory, or Maximum Likelihood? A Comparison of Tuning Methods for Penalized Splines," Stats, MDPI, vol. 4(3), pages 1-24, September.
    16. Nitschka Thomas & Markov Nikolay, 2016. "Semi-Parametric Estimates of Taylor Rules for a Small, Open Economy – Evidence from Switzerland," German Economic Review, De Gruyter, vol. 17(4), pages 478-490, December.
    17. Longhi, Christian & Musolesi, Antonio & Baumont, Catherine, 2014. "Modeling structural change in the European metropolitan areas during the process of economic integration," Economic Modelling, Elsevier, vol. 37(C), pages 395-407.
    18. Kim, Young-Ju, 2013. "A partial spline approach for semiparametric estimation of varying-coefficient partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 62(C), pages 181-187.
    19. Stoklosa, Jakub & Huggins, Richard M., 2012. "A robust P-spline approach to closed population capture–recapture models with time dependence and heterogeneity," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 408-417.
    20. Simon N. Wood, 2006. "Low-Rank Scale-Invariant Tensor Product Smooths for Generalized Additive Mixed Models," Biometrics, The International Biometric Society, vol. 62(4), pages 1025-1036, December.
    21. Geminiani, Elena & Marra, Giampiero & Moustaki, Irini, 2021. "Single and multiple-group penalized factor analysis: a trust-region algorithm approach with integrated automatic multiple tuning parameter selection," LSE Research Online Documents on Economics 108873, London School of Economics and Political Science, LSE Library.
    22. Longhi, C. & Musolesi, A. & Baumont, C., 2013. "Modeling the industrial dynamics of the European metropolitan areas during the process of economic integration: a semiparametric approach," Working Papers 2013-10, Grenoble Applied Economics Laboratory (GAEL).
    23. Wojtyś, Magorzata & Marra, Giampiero & Radice, Rosalba, 2016. "Copula Regression Spline Sample Selection Models: The R Package SemiParSampleSel," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 71(i06).
    24. Young-Ju Kim, 2010. "Semiparametric analysis for case-control studies: a partial smoothing spline approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(6), pages 1015-1025.
    25. Du, Pang & Gu, Chong, 2006. "Penalized likelihood hazard estimation: Efficient approximation and Bayesian confidence intervals," Statistics & Probability Letters, Elsevier, vol. 76(3), pages 244-254, February.

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