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Model Selection for High-Dimensional Quadratic Regression via Regularization

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  • Ning Hao
  • Yang Feng
  • Hao Helen Zhang

Abstract

Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high-dimensional data. This article focuses on scalable regularization methods for model selection in high-dimensional QR. We first consider two-stage regularization methods and establish theoretical properties of the two-stage LASSO. Then, a new regularization method, called regularization algorithm under marginality principle (RAMP), is proposed to compute a hierarchy-preserving regularization solution path efficiently. Both methods are further extended to solve generalized QR models. Numerical results are also shown to demonstrate performance of the methods. Supplementary materials for this article are available online.

Suggested Citation

  • Ning Hao & Yang Feng & Hao Helen Zhang, 2018. "Model Selection for High-Dimensional Quadratic Regression via Regularization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 615-625, April.
    • Handle: RePEc:taf:jnlasa:v:113:y:2018:i:522:p:615-625
    DOI: 10.1080/01621459.2016.1264956
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    Citations

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    Cited by:

    1. Bonnie R. Joubert & Marianthi-Anna Kioumourtzoglou & Toccara Chamberlain & Hua Yun Chen & Chris Gennings & Mary E. Turyk & Marie Lynn Miranda & Thomas F. Webster & Katherine B. Ensor & David B. Dunson, 2022. "Powering Research through Innovative Methods for Mixtures in Epidemiology (PRIME) Program: Novel and Expanded Statistical Methods," IJERPH, MDPI, vol. 19(3), pages 1-24, January.
    2. Ning Hao & Hao Helen Zhang, 2017. "A Note on High-Dimensional Linear Regression With Interactions," The American Statistician, Taylor & Francis Journals, vol. 71(4), pages 291-297, October.
    3. Ryan A. Peterson & Joseph E. Cavanaugh, 2022. "Ranked sparsity: a cogent regularization framework for selecting and estimating feature interactions and polynomials," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(3), pages 427-454, September.
    4. Feng Li & Yajie Li & Sanying Feng, 2021. "Estimation for Varying Coefficient Models with Hierarchical Structure," Mathematics, MDPI, vol. 9(2), pages 1-18, January.
    5. Bhatnagar, Sahir R. & Lu, Tianyuan & Lovato, Amanda & Olds, David L. & Kobor, Michael S. & Meaney, Michael J. & O'Donnell, Kieran & Yang, Archer Y. & Greenwood, Celia M.T., 2023. "A sparse additive model for high-dimensional interactions with an exposure variable," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
    6. He Jiang, 2022. "A novel robust structural quadratic forecasting model and applications," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 41(6), pages 1156-1180, September.
    7. Wang, Cheng & Chen, Haozhe & Jiang, Binyan, 2024. "HiQR: An efficient algorithm for high-dimensional quadratic regression with penalties," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    8. Randall Reese & Guifang Fu & Geran Zhao & Xiaotian Dai & Xiaotian Li & Kenneth Chiu, 2022. "Epistasis Detection via the Joint Cumulant," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 14(3), pages 514-532, December.
    9. Sanying Feng & Menghan Zhang & Tiejun Tong, 2022. "Variable selection for functional linear models with strong heredity constraint," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 321-339, April.

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